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(cite.Coh1960) cvn H.B /ANN pdfmark end 41081 15725 a Black Fo(,)p 0 .5 0 TeXcolorrgb 41830 15725 a SDict begin H.S end 41830 15725 a 0 .5 0 TeXcolorrgb Fo(6)p 0 .5 0 TeXcolorrgb 42480 14869 a SDict begin H.R end 42480 14869 a 42480 15725 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Lal1974) cvn H.B /ANN pdfmark end 42480 15725 a Black Fo(,)p 0 .5 0 TeXcolorrgb 43229 15725 a SDict begin H.S end 43229 15725 a 0 .5 0 TeXcolorrgb Fo(9)p 0 .5 0 TeXcolorrgb 43879 14869 a SDict begin H.R end 43879 14869 a 43879 15725 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.McC1986) cvn H.B /ANN pdfmark end 43879 15725 a Black Fo(,)p 0 .5 0 TeXcolorrgb 44628 15725 a SDict begin H.S end 44628 15725 a 0 .5 0 TeXcolorrgb Fo(11)p 0 .5 0 TeXcolorrgb 45929 14869 a SDict begin H.R end 45929 14869 a 45929 15725 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.SitSur1973) cvn H.B /ANN pdfmark end 45929 15725 a Black Fo(,)p 0 .5 0 TeXcolorrgb 46678 15725 a SDict begin H.S end 46678 15725 a 0 .5 0 TeXcolorrgb Fo(12)p 0 .5 0 TeXcolorrgb 47978 14869 a SDict begin H.R end 47978 14869 a 47978 15725 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.SitSub2007) cvn H.B /ANN pdfmark end 47978 15725 a Black Fo(,)p 0 .5 0 TeXcolorrgb 48727 15725 a SDict begin H.S end 48727 15725 a 0 .5 0 TeXcolorrgb Fo(13)p 0 .5 0 TeXcolorrgb 50028 14869 a SDict begin H.R end 50028 14869 a 50028 15725 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.SivDix2006) cvn H.B /ANN pdfmark end 50028 15725 a Black Fo(,)p 0 .5 0 TeXcolorrgb 50777 15725 a SDict begin H.S end 50777 15725 a 0 .5 0 TeXcolorrgb Fo(14)p 0 .5 0 TeXcolorrgb 52077 14869 a SDict begin H.R end 52077 14869 a 52077 15725 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Sko2008) cvn H.B /ANN pdfmark end 52077 15725 a Black Fo(].)2751 17330 y(F)-108 b(urthermore,)432 b(consider)h(the)h(gcd-sum)e(t)-36 b(yp)36 b(e)434 b(function)f(giv)-36 b(en)434 b(b)-36 b(y)36312 15725 y SDict begin H.S end 36312 15725 a 36312 15725 a SDict begin 14.5 H.A end 36312 15725 a 36312 15725 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6) cvn H.B /DEST pdfmark end 36312 15725 a 21129 21201 a Fn(P)22148 20652 y Fh(\003)22674 21201 y Fo(\()p Fn(n)p Fo(\))369 b(:=)27247 19540 y Fk(n)26573 19939 y Fj(X)26675 22767 y Fk(k)24 b Fi(=1)28492 21201 y Fo(\()p Fn(k)45 b(;)221 b(n)p Fo(\))31583 21400 y Fh(\003)32109 21201 y Fn(;)18668 b Fo(\(6\))800 25334 y(whic)-36 b(h)411 b(w)-36 b(as)411 b(in)-36 b(tro)36 b(duced)409 b(in)i(our)f(pap)36 b(er)411 b([)p 0 .5 0 TeXcolorrgb 21228 25334 a SDict begin H.S end 21228 25334 a 0 .5 0 TeXcolorrgb Fo(17)p 0 .5 0 TeXcolorrgb 22529 24478 a SDict begin H.R end 22529 24478 a 22529 25334 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Tot1989) cvn H.B /ANN pdfmark end 22529 25334 a Black Fo(];)419 b(also)411 b(see)g([)p 0 .5 0 TeXcolorrgb 28696 25334 a SDict begin H.S end 28696 25334 a 0 .5 0 TeXcolorrgb Fo(18)p 0 .5 0 TeXcolorrgb 29997 24478 a SDict begin H.R end 29997 24478 a 29997 25334 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Tot1996) cvn H.B /ANN pdfmark end 29997 25334 a Black Fo(,)p 0 .5 0 TeXcolorrgb 30769 25334 a SDict begin H.S end 30769 25334 a 0 .5 0 TeXcolorrgb Fo(19)p 0 .5 0 TeXcolorrgb 32070 24478 a SDict begin H.R end 32070 24478 a 32070 25334 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Tot1998) cvn H.B /ANN pdfmark end 32070 25334 a Black Fo(].)571 b(The)411 b(function)f Fn(P)42104 24852 y Fh(\003)43041 25334 y Fo(\()p 0 1 0 0 TeXcolorcmyk 43547 25600 a SDict begin H.S end 43547 25600 a 0 1 0 0 TeXcolorcmyk -266 x Fo(A145388)p 43547 25546 4878 54 v 0 1 0 0 TeXcolorcmyk 48423 24426 a SDict begin H.R end 48423 24426 a 48423 25334 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A145388) >> /Subtype /Link H.B /ANN pdfmark end 48423 25334 a Black Fo(\))h(is)g(also)800 26939 y(m)-36 b(ultiplicativ)g(e;)435 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)433 b Fn(P)15295 26457 y Fh(\003)15821 26939 y Fo(\()p Fn(p)16980 26457 y Fk(\027)17556 26939 y Fo(\))368 b(=)h(2)p Fn(p)21114 26457 y Fk(\027)21985 26939 y Fm(\000)295 b Fo(1)434 b(for)g(ev)-36 b(ery)435 b(prime)e(p)36 b(o)-36 b(w)g(er)434 b Fn(p)37949 26457 y Fk(\027)38958 26939 y Fo(\()p Fn(\027)452 b Fm(\025)369 b Fo(1\))434 b(and)46080 25334 y SDict begin H.S end 46080 25334 a 46080 25334 a SDict begin 14.5 H.A end 46080 25334 a 46080 25334 a SDict begin [ /View [/XYZ H.V] /Dest (equation.7) cvn H.B /DEST pdfmark end 46080 25334 a 20465 30103 a Fn(P)21484 29555 y Fh(\003)22010 30103 y Fo(\()p Fn(n)p Fo(\))369 b(=)25548 28841 y Fj(X)25718 31719 y Fk(d)p Fh(jj)p Fk(n)27688 30103 y Fn(d\036)29134 29555 y Fh(\003)29659 30103 y Fo(\()p 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a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 11220 44887 a SDict begin H.R end 11220 44887 a 11220 45743 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 11220 45743 a Black Fo(\))g(and)f(of)i(the)f (gcd-function)f(\()p 0 .5 0 TeXcolorrgb 25917 45743 a SDict begin H.S end 25917 45743 a 0 .5 0 TeXcolorrgb Fo(2)p 0 .5 0 TeXcolorrgb 26567 44887 a SDict begin H.R end 26567 44887 a 26567 45743 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2) cvn H.B /ANN pdfmark end 26567 45743 a Black Fo(\),)357 b(as)339 b(it)f(turns)f(out)g(from)i(the)e(results)h (of)h(the)e(presen)-36 b(t)800 47348 y(pap)36 b(er.)2751 48953 y(Let)460 b(\()p Fn(k)45 b(;)221 b(n)p Fo(\))8199 49152 y Fh(\003\003)9610 48953 y Fo(=)415 b(max)p Fm(f)p Fn(d)g Fm(2)f Fl(N)h Fo(:)f Fn(d)h Fm(jj)f Fn(k)45 b(;)221 b(d)415 b Fm(jj)g Fn(n)p Fm(g)461 b Fo(stand)e(for)i(the)f(greatest)h (common)f(unitary)g(divisor)800 50558 y(of)434 b Fn(k)479 b Fo(and)433 b Fn(n)p Fo(.)579 b(The)433 b(function)15449 48953 y SDict begin H.S end 15449 48953 a 15449 48953 a SDict begin 14.5 H.A end 15449 48953 a 15449 48953 a SDict begin [ /View [/XYZ H.V] /Dest (equation.8) cvn H.B /DEST pdfmark end 15449 48953 a 13999 53490 a Fn(\036)14769 52941 y Fh(\003\003)15765 53490 y Fo(\()p Fn(n)p Fo(\))369 b(:=)g(#)p Fm(f)p Fn(k)414 b Fm(2)368 b Fl(N)i Fo(:)f(1)g Fm(\024)h Fn(k)414 b Fm(\024)369 b Fn(n;)221 b Fo(\()p Fn(k)45 b(;)221 b(n)p Fo(\))35178 53689 y Fh(\003\003)36545 53490 y Fo(=)368 b(1)p Fm(g)p Fn(;)11538 b Fo(\(8\))800 56422 y(called)610 b(the)e(bi-unitary)g(Euler)g(function,)653 b(w)-36 b(as)610 b(in)-36 b(tro)36 b(duced)607 b(b)-36 b(y)609 b(Subbarao)f(and)g(Sury)-36 b(anara)g(y)g(ana)609 b(in)800 58027 y(1971,)435 b(cf.)g([)p 0 .5 0 TeXcolorrgb 6328 58027 a SDict begin H.S end 6328 58027 a 0 .5 0 TeXcolorrgb Fo(16)p 0 .5 0 TeXcolorrgb 7629 57171 a SDict begin H.R end 7629 57171 a 7629 58027 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.SurSub1980) cvn H.B /ANN pdfmark end 7629 58027 a Black Fo(])e(and)g([)p 0 .5 0 TeXcolorrgb 11313 58027 a SDict begin H.S end 11313 58027 a 0 .5 0 TeXcolorrgb Fo(2)p 0 .5 0 TeXcolorrgb 11964 57171 a SDict begin H.R end 11964 57171 a 11964 58027 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Hau1998) cvn H.B /ANN pdfmark end 11964 58027 a Black Fo(].)579 b(W)-108 b(e)433 b(ha)-36 b(v)g(e)18503 56422 y SDict begin H.S end 18503 56422 a 18503 56422 a SDict begin 14.5 H.A end 18503 56422 a 18503 56422 a SDict begin [ /View [/XYZ H.V] /Dest (equation.9) cvn H.B /DEST pdfmark end 18503 56422 a 18828 61191 a Fn(\036)19598 60642 y Fh(\003\003)20594 61191 y Fo(\()p Fn(n)p Fo(\))369 b(=)24132 59929 y Fj(X)24302 62807 y Fk(d)p Fh(jj)p Fk(n)26272 61191 y Fn(\026)27055 60642 y Fh(\003)27580 61191 y Fo(\()p Fn(d)p Fo(\))p Fn(\036)p Fo(\()p Fn(n=d;)221 b(d)p Fo(\))p Fn(;)16367 b Fo(\(9\))800 65595 y(where)470 b Fn(\036)p 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Fj(\022)32000 13069 y Fo(1)296 b Fm(\000)34671 12170 y Fo(2)p 34407 12763 1179 54 v 34407 13980 a Fn(p)35060 13596 y Fi(2)36014 13069 y Fo(+)37717 12170 y(2)p 37453 12763 V 37453 13980 a Fn(p)38106 13596 y Fi(3)39060 13069 y Fm(\000)40785 12170 y Fo(1)p 40521 12763 V 40521 13980 a Fn(p)41174 13596 y Fi(4)41832 11196 y Fj(\023)43032 13069 y Fn(;)7095 b Fo(\(12\))800 16728 y(the)612 b(pro)36 b(ducts)611 b(b)36 b(eing)612 b(o)-36 b(v)g(er)612 b(the)g(primes)g(and)g Fn(\020)712 b Fo(denoting)612 b(the)f(Riemann)i(zeta)f(function;)701 b(see)613 b([)p 0 .5 0 TeXcolorrgb 51138 16728 a SDict begin H.S end 51138 16728 a 0 .5 0 TeXcolorrgb Fo(16)p 0 .5 0 TeXcolorrgb 52439 15872 a SDict begin H.R end 52439 15872 a 52439 16728 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.SurSub1980) cvn H.B /ANN pdfmark end 52439 16728 a Black Fo(,)800 18333 y(Corollary)577 b(3.6.2].)1006 b(Here)575 b Fn(A)611 b Fm(\031)f Fo(0)p Fn(:)p Fo(8073,)j(this)575 b(v)-72 b(alue)576 b(is)g(the)e(asymptotic)i(densit)-36 b(y)575 b(of)h(ordered)f(pairs)800 19938 y(\()p Fn(m;)221 b(n)p Fo(\))403 b Fm(2)f Fl(N)6958 19456 y Fi(2)7937 19938 y Fo(suc)-36 b(h)453 b(that)g(\()p Fn(m;)221 b(n)p Fo(\))17237 20137 y Fh(\003\003)18636 19938 y Fo(=)402 b(1,)459 b(i.e.,)g(there)453 b(is)g(no)g(prime)g(p)36 b(o)-36 b(w)g(er)453 b Fn(p)38669 19456 y Fk(\027)39698 19938 y Fo(app)36 b(earing)454 b(in)f(the)f(prime)800 21543 y(factorizations)435 b(of)f(b)36 b(oth)433 b Fn(m)h Fo(and)f Fn(n)p Fo(;)h(see)g([)p 0 .5 0 TeXcolorrgb 21552 21543 a SDict begin H.S end 21552 21543 a 0 .5 0 TeXcolorrgb Fo(16)p 0 .5 0 TeXcolorrgb 22853 20687 a SDict begin H.R end 22853 20687 a 22853 21543 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.SurSub1980) cvn H.B /ANN pdfmark end 22853 21543 a Black Fo(,)g(Corollary)h(3.6.3],)h([)p 0 .5 0 TeXcolorrgb 33583 21543 a SDict begin H.S end 33583 21543 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 34884 20687 a SDict begin H.R end 34884 20687 a 34884 21543 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Tot2001) cvn H.B /ANN pdfmark end 34884 21543 a Black Fo(,)e(Theorems)f(2.2,) i(2.3].)2751 23148 y(No)-36 b(w)434 b(let)7592 21543 y SDict begin H.S end 7592 21543 a 7592 21543 a SDict begin 14.5 H.A end 7592 21543 a 7592 21543 a SDict begin [ /View [/XYZ H.V] /Dest (equation.13) cvn H.B /DEST pdfmark end 7592 21543 a 20840 25433 a Fn(P)21859 24885 y Fh(\003\003)22855 25433 y Fo(\()p Fn(n)p Fo(\))369 b(:=)27428 23773 y Fk(n)26754 24171 y Fj(X)26855 27000 y Fk(k)24 b Fi(=1)28672 25433 y Fo(\()p Fn(k)45 b(;)221 b(n)p Fo(\))31763 25632 y Fh(\003\003)50488 25433 y Fo(\(13\))800 28960 y(b)36 b(e)432 b(the)h(bi-unitary)f (gcd-sum)f(function,)i(in)-36 b(tro)36 b(duced)431 b(recen)-36 b(tly)433 b(b)-36 b(y)433 b(Haukk)-72 b(anen)433 b([)p 0 .5 0 TeXcolorrgb 41842 28960 a SDict begin H.S end 41842 28960 a 0 .5 0 TeXcolorrgb Fo(4)p 0 .5 0 TeXcolorrgb 42492 28104 a SDict begin H.R end 42492 28104 a 42492 28960 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Hau2008) cvn H.B /ANN pdfmark end 42492 28960 a Black Fo(].)579 b(It)432 b(is)h(kno)-36 b(wn)433 b(\(see)800 30565 y([)p 0 .5 0 TeXcolorrgb 1161 30565 a SDict begin H.S end 1161 30565 a 0 .5 0 TeXcolorrgb Fo(4)p 0 .5 0 TeXcolorrgb 1812 29709 a SDict begin H.R end 1812 29709 a 1812 30565 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Hau2008) cvn H.B /ANN pdfmark end 1812 30565 a Black Fo(,)h(Corollary)h(3.1]\))g(that)14132 28960 y SDict begin H.S end 14132 28960 a 14132 28960 a SDict begin 14.5 H.A end 14132 28960 a 14132 28960 a SDict begin [ /View [/XYZ H.V] /Dest (equation.14) cvn H.B /DEST pdfmark end 14132 28960 a 18710 32402 a Fn(P)19729 31854 y Fh(\003\003)20725 32402 y Fo(\()p Fn(n)p Fo(\))369 b(=)24263 31140 y Fj(X)24433 34018 y Fk(d)p Fh(jj)p Fk(n)26403 32402 y Fn(\036)27173 31854 y Fh(\003)27699 32402 y Fo(\()p Fn(d)p Fo(\))p Fn(\036)p Fo(\()p Fn(n=d;)221 b(d)p Fo(\))p Fn(:)15598 b Fo(\(14\))2751 36125 y(Note)434 b(that)f(for)h(ev)-36 b(ery)434 b Fn(n)370 b Fm(\025)f Fo(1,)18124 34520 y SDict begin H.S end 18124 34520 a 18124 34520 a SDict begin 14.5 H.A end 18124 34520 a 18124 34520 a SDict begin [ /View [/XYZ H.V] /Dest (equation.15) cvn H.B /DEST pdfmark end 18124 34520 a 19877 39059 a Fn(P)20896 38510 y Fh(\003\003)21892 39059 y Fo(\()p Fn(n)p Fo(\))g Fm(\024)g Fn(P)26470 38510 y Fh(\003)26996 39059 y Fo(\()p Fn(n)p Fo(\))g Fm(\024)g Fn(P)181 b Fo(\()p Fn(n)p Fo(\))p Fn(:)16765 b Fo(\(15\))2751 41992 y(Haukk)-72 b(anen)434 b([)p 0 .5 0 TeXcolorrgb 9868 41992 a SDict begin H.S end 9868 41992 a 0 .5 0 TeXcolorrgb Fo(4)p 0 .5 0 TeXcolorrgb 10518 41136 a SDict begin H.R end 10518 41136 a 10518 41992 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Hau2008) cvn H.B /ANN pdfmark end 10518 41992 a Black Fo(])g(considered)f(also)i(the)e(function)27622 40387 y SDict begin H.S end 27622 40387 a 27622 40387 a SDict begin 14.5 H.A end 27622 40387 a 27622 40387 a SDict begin [ /View [/XYZ H.V] /Dest (equation.16) cvn H.B /DEST pdfmark end 27622 40387 a 19651 45938 a Fn(P)20670 45389 y Fh(\003\003)20489 46266 y Fk(f)21666 45938 y Fo(\()p Fn(n)p Fo(\))369 b(:=)26239 44277 y Fk(n)25565 44676 y Fj(X)25667 47505 y Fk(k)24 b Fi(=1)27705 45938 y Fn(f)142 b Fo(\(\()p Fn(k)45 b(;)221 b(n)p Fo(\))32085 46137 y Fh(\003\003)33082 45938 y Fo(\))p Fn(;)16539 b Fo(\(16\))800 49999 y(and)433 b(sho)-36 b(w)g(ed)433 b(that)h(for)g(an)-36 b(y)433 b(arithmetical)i(function)e Fn(f)142 b Fo(,)29049 48394 y SDict begin H.S end 29049 48394 a 29049 48394 a SDict begin 14.5 H.A end 29049 48394 a 29049 48394 a SDict begin [ /View [/XYZ H.V] /Dest (equation.17) cvn H.B /DEST pdfmark end 29049 48394 a 17105 53091 a Fn(P)18124 52543 y Fh(\003\003)17943 53420 y Fk(f)19121 53091 y Fo(\()p Fn(n)p Fo(\))369 b(=)22658 51829 y Fj(X)22829 54708 y Fk(d)p Fh(jj)p Fk(n)24577 53091 y Fo(\()p Fn(f)437 b Fm(\002)295 b Fn(\026)28272 52543 y Fh(\003)28798 53091 y Fo(\)\()p Fn(d)p Fo(\))p Fn(\036)p Fo(\()p Fn(n=d;)221 b(d)p Fo(\))p Fn(;)13993 b Fo(\(17\))800 57423 y(where)399 b Fm(\002)h Fo(stands)f(for)h(the)f(unitary)g(con)-36 b(v)g(olution.)567 b(The)400 b(unitary)f(con)-36 b(v)g(olution)400 b(of)g(the)f(functions)g Fn(f)542 b Fo(and)800 59028 y Fn(g)481 b Fo(is)434 b(giv)-36 b(en)434 b(b)-36 b(y)8415 57423 y SDict begin H.S end 8415 57423 a 8415 57423 a SDict begin 14.5 H.A end 8415 57423 a 8415 57423 a SDict begin [ /View [/XYZ H.V] /Dest (equation.18) cvn H.B /DEST pdfmark end 8415 57423 a 18609 60792 a Fo(\()p Fn(f)437 b Fm(\002)295 b Fn(g)48 b Fo(\)\()p Fn(n)p Fo(\))368 b(=)26235 59530 y Fj(X)26405 62408 y Fk(d)p Fh(jj)p Fk(n)28375 60792 y Fn(f)142 b Fo(\()p Fn(d)p Fo(\))p Fn(g)48 b Fo(\()p Fn(n=d)p Fo(\))p Fn(:)15496 b Fo(\(18\))2751 64589 y(F)-108 b(or)396 b(the)f(prop)36 b(erties)396 b(of)h(con)-36 b(v)g(olution)397 b(\()p 0 .5 0 TeXcolorrgb 22208 64589 a SDict begin H.S end 22208 64589 a 0 .5 0 TeXcolorrgb Fo(18)p 0 .5 0 TeXcolorrgb 23508 63733 a SDict begin H.R end 23508 63733 a 23508 64589 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.18) cvn H.B /ANN pdfmark end 23508 64589 a Black Fo(\))f(and)g(for)g(bac)-36 b(kground)396 b(material)h(on)f (arithmetical)h(func-)800 66194 y(tions)434 b(in)-36 b(v)g(olving)435 b(unitary)e(divisors)i(w)-36 b(e)433 b(refer)h(to)g(the)f(b)36 b(o)g(ok)434 b([)p 0 .5 0 TeXcolorrgb 31206 66194 a SDict begin H.S end 31206 66194 a 0 .5 0 TeXcolorrgb Fo(9)p 0 .5 0 TeXcolorrgb 31857 65338 a SDict begin H.R end 31857 65338 a 31857 66194 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.McC1986) cvn H.B /ANN pdfmark end 31857 66194 a Black Fo(].)p Black 26475 70521 a(3)p Black eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto 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b(v)g(estigate)434 b(the)f(a)-36 b(v)g(erage)435 b(orders)e(of)i(the)e (functions)g Fn(P)36646 5908 y Fh(\003\003)38076 6390 y Fo(and)g Fn(S)41482 5908 y Fh(\003\003)42478 6390 y Fo(.)800 8649 y SDict begin H.S end 800 8649 a 800 8649 a SDict begin 14.5 H.A end 800 8649 a 800 8649 a SDict begin [ /View [/XYZ H.V] /Dest (section.2) cvn H.B /DEST pdfmark end 800 8649 a 2103 x Fp(2)2152 b(Com)-60 b(binatorial-t)g(yp)60 b(e)715 b(form)-60 b(ulae)800 13673 y Fo(Let)615 b Fn(\036)4082 13191 y Fh(\003)4608 13673 y Fo(\()p Fn(x;)221 b(n)p Fo(\))679 b(=)g(#)p Fm(f)p Fn(k)724 b Fm(2)678 b Fl(N)h Fo(:)h(1)f Fm(\024)g Fn(k)724 b Fm(\024)679 b Fn(x;)221 b Fo(\()p Fn(k)45 b(;)221 b(n)p Fo(\))28043 13872 y Fh(\003)29249 13673 y Fo(=)678 b(1)p Fm(g)616 b Fo(b)36 b(e)616 b(the)f(unitary)g (Legendre)g(function,)800 15278 y(considered)461 b(b)-36 b(y)462 b(Haukk)-72 b(anen)462 b([)p 0 .5 0 TeXcolorrgb 16174 15278 a SDict begin H.S end 16174 15278 a 0 .5 0 TeXcolorrgb Fo(3)p 0 .5 0 TeXcolorrgb 16825 14422 a SDict begin H.R end 16825 14422 a 16825 15278 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Hau2002) cvn H.B /ANN pdfmark end 16825 15278 a Black Fo(],)469 b(and)462 b(let)g Fn(\036)23251 14796 y Fh(\003\003)24247 15278 y Fo(\()p Fn(x;)221 b(n)p Fo(\))418 b(=)f(#)p Fm(f)p Fn(k)462 b Fm(2)417 b Fl(N)g Fo(:)h(1)f Fm(\024)h Fn(k)462 b Fm(\024)417 b Fn(x;)221 b Fo(\()p Fn(k)45 b(;)221 b(n)p Fo(\))45065 15477 y Fh(\003\003)46480 15278 y Fo(=)417 b(1)p Fm(g)463 b Fo(stand)800 16883 y(for)369 b(the)e(bi-unitary)h(Legendre)f (function,)381 b(in)-36 b(v)g(estigated)369 b(in)f([)p 0 .5 0 TeXcolorrgb 30801 16883 a SDict begin H.S end 30801 16883 a 0 .5 0 TeXcolorrgb Fo(16)p 0 .5 0 TeXcolorrgb 32101 16027 a SDict begin H.R end 32101 16027 a 32101 16883 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.SurSub1980) cvn H.B /ANN pdfmark end 32101 16883 a Black Fo(].)557 b(Observ)-36 b(e)368 b(that)g Fn(\036)41831 16401 y Fh(\003)42356 16883 y Fo(\()p Fn(n;)221 b(n)p Fo(\))370 b(=)f Fn(\036)48023 16401 y Fh(\003)48548 16883 y Fo(\()p Fn(n)p Fo(\))g(and)800 18488 y Fn(\036)1570 18006 y Fh(\003\003)2566 18488 y Fo(\()p Fn(n;)221 b(n)p Fo(\))370 b(=)f Fn(\036)8233 18006 y Fh(\003\003)9229 18488 y Fo(\()p Fn(n)p Fo(\).)2751 20093 y(Let)433 b Fn(n)370 b Fo(=)e Fn(p)8260 19513 y Fk(\027)8723 19636 y Ff(1)8260 20419 y Fi(1)9460 20093 y Fm(\001)221 b(\001)g(\001)i Fn(p)11885 19611 y Fk(\027)12348 19722 y Fe(r)11885 20421 y Fk(r)13233 20093 y Fn(>)369 b Fo(1)434 b(b)36 b(e)433 b(a)h(\014xed)f(in)-36 b(teger)434 b(and)f(let)g Fn(A)369 b Fo(=)g Fn(A)34102 20292 y Fk(n)35097 20093 y Fo(=)g Fm(f)p Fo(1)p Fn(;)221 b(:)g(:)g(:)j(;)d(r)36 b Fm(g)p Fo(,)435 b Fn(r)405 b Fo(=)369 b Fn(!)48 b Fo(\()p Fn(n)p Fo(\).)2751 21698 y(W)-108 b(e)646 b(giv)-36 b(e)647 b(explicit)f(form)-36 b(ulae)646 b(for)h(the)e(v)-72 b(alues)646 b Fn(\036)p Fo(\()p Fn(x;)221 b(n)p Fo(\),)700 b Fn(\036)32895 21216 y Fh(\003)33420 21698 y Fo(\()p Fn(x;)221 b(n)p Fo(\))647 b(and)e Fn(\036)40687 21216 y Fh(\003\003)41683 21698 y Fo(\()p Fn(x;)221 b(n)p Fo(\))647 b(whic)-36 b(h)645 b(are)h(of)800 23303 y(com)-36 b(binatorial)549 b(t)-36 b(yp)36 b(e)547 b(and)g(do)h(not)f(in)-36 b(v)g(olv)g(e)549 b(the)e(M\177)-650 b(obius)548 b(function)f Fn(\026)p Fo(,)576 b(its)548 b(unitary)f(analogue)i Fn(\026)50571 22821 y Fh(\003)51644 23303 y Fo(or)800 24908 y(other)433 b(similar)i(arithmetical)f(functions.)2751 26513 y(In)h(the)g(follo)-36 b(wing)438 b(form)-36 b(ulae)436 b Fm(b)p Fn(y)48 b Fm(c)436 b Fo(stands)f(for)h(the)f(largest)h(in)-36 b(teger)435 b(less)h(than)f(or)h(equal)g(to)g Fn(y)483 b Fo(\(\015o)36 b(or)800 28119 y(function\))433 b(and)g(w)-36 b(e)434 b(use)f(the)g(con)-36 b(v)g(en)g(tion)434 b(that)f(the)g(empt)-36 b(y)433 b(pro)36 b(ducts)433 b(are)g(equal)i(to)e(1.)800 28451 y SDict begin H.S end 800 28451 a 800 28451 a SDict begin 14.5 H.A end 800 28451 a 800 28451 a SDict begin [ /View [/XYZ H.V] /Dest (thm.1) cvn H.B /DEST pdfmark end 800 28451 a 800 29121 a SDict begin H.S end 800 29121 a 800 29121 a SDict begin 14.5 H.A end 800 29121 a 800 29121 a SDict begin [ /View [/XYZ H.V] /Dest (equation.20) cvn H.B /DEST pdfmark end 800 29121 a Black 1273 x Fd(Theorem)499 b(1.)p Black 17044 32830 a Fn(\036)p Fo(\()p Fn(x;)221 b(n)p Fo(\))369 b(=)22808 31568 y Fj(X)22673 34387 y Fk(B)44 b Fh(\022)p Fk(A)24863 32830 y Fo(\()p Fm(\000)p Fo(1\))27558 32281 y Fi(#)p Fk(B)29372 30558 y Fj($)30221 32830 y Fn(x)31239 31568 y Fj(Y)31181 34387 y Fk(j)51 b Fh(2)p Fk(B)33215 32830 y Fn(p)33868 32280 y Fh(\000)p Fi(1)33868 33170 y Fk(j)35125 30558 y Fj(\045)36195 32830 y Fn(;)13932 b Fo(\(20\))800 33699 y SDict begin H.S end 800 33699 a 800 33699 a SDict begin 14.5 H.A end 800 33699 a 800 33699 a SDict begin [ /View [/XYZ H.V] /Dest (equation.21) cvn H.B /DEST pdfmark end 800 33699 a 16568 37739 a Fn(\036)17338 37191 y Fh(\003)17863 37739 y Fo(\()p Fn(x;)221 b(n)p Fo(\))370 b(=)22858 36477 y Fj(X)22723 39296 y Fk(B)44 b Fh(\022)p Fk(A)24913 37739 y Fo(\()p Fm(\000)p Fo(1\))27608 37191 y Fi(#)p Fk(B)29422 35468 y Fj($)30271 37739 y Fn(x)31289 36477 y Fj(Y)31231 39296 y Fk(j)51 b Fh(2)p Fk(B)33265 37739 y Fn(p)33918 37017 y Fh(\000)p Fk(\027)35113 37152 y Fe(j)33918 38080 y Fk(j)35601 35468 y Fj(\045)36671 37739 y Fn(;)13456 b Fo(\(21\))800 38608 y SDict begin H.S end 800 38608 a 800 38608 a SDict begin 14.5 H.A end 800 38608 a 800 38608 a SDict begin [ /View [/XYZ H.V] /Dest (equation.22) cvn H.B /DEST pdfmark end 800 38608 a 10906 42649 a Fn(\036)11676 42100 y Fh(\003\003)12672 42649 y Fo(\()p Fn(x;)221 b(n)p Fo(\))370 b(=)17667 41387 y Fj(X)17532 44206 y Fk(B)44 b Fh(\022)p Fk(A)19722 42649 y Fo(\()p Fm(\000)p Fo(1\))22417 42100 y Fi(#)p Fk(B)24381 41387 y Fj(X)24231 44206 y Fk(C)63 b Fh(\022)p Fk(B)26449 42649 y Fo(\()p Fm(\000)p Fo(1\))29144 42100 y Fi(#)p Fk(C)30938 40377 y Fj($)31787 42649 y Fn(x)32805 41387 y Fj(Y)32747 44206 y Fk(j)51 b Fh(2)p Fk(B)34781 42649 y Fn(p)35434 41927 y Fh(\000)p Fk(\027)36629 42062 y Fe(j)35434 42989 y Fk(j)37386 41387 y Fj(Y)37339 44206 y Fk(j)g Fh(2)p Fk(C)39352 42649 y Fn(p)40005 42099 y Fh(\000)p Fi(1)40005 42989 y Fk(j)41263 40377 y Fj(\045)42333 42649 y Fn(:)7794 b Fo(\(22\))p Black 800 46288 a Fg(Pr)-66 b(o)g(of.)p Black 649 w Fo(It)550 b(is)g(w)-36 b(ell)550 b(kno)-36 b(wn)550 b(that)f Fn(\036)p Fo(\()p Fn(x;)221 b(n)p Fo(\))567 b(=)23767 45292 y Fj(P)25169 46676 y Fk(d)p Fh(j)p Fk(n)26762 46288 y Fn(\026)p Fo(\()p Fn(d)p Fo(\))p Fm(b)p Fn(x=d)p Fm(c)p Fo(.)926 b(W)-108 b(e)550 b(giv)-36 b(e)550 b(its)g(pro)36 b(of)550 b(for)g(the)f(sak)-36 b(e)550 b(of)800 48004 y(completeness)446 b(and)g(in)h(order)f(to)g (compare)h(it)f(with)h(the)f(follo)-36 b(wing)448 b(pro)36 b(ofs.)618 b(Using)447 b(that)f Fn(d)390 b Fm(j)h Fo(gcd\()p Fn(k)45 b(;)221 b(n)p Fo(\))800 49609 y(i\013)434 b Fn(d)368 b Fm(j)h Fn(k)45 b Fo(,)434 b Fn(d)369 b Fm(j)g Fn(n)p Fo(,)5533 52266 y Fn(\036)p Fo(\()p Fn(x;)221 b(n)p Fo(\))370 b(=)11162 51004 y Fj(X)11234 53833 y Fk(k)24 b Fh(\024)p Fk(x)14460 51004 y Fj(X)13302 53882 y Fk(d)p Fh(j)p Fi(gcd)q(\()p Fk(k)g(;n)p Fi(\))17758 52266 y Fn(\026)p Fo(\()p Fn(d)p Fo(\))368 b(=)21978 51004 y Fj(X)22050 53833 y Fk(k)24 b Fh(\024)p Fk(x)24118 51004 y Fj(X)24448 53882 y Fk(d)p Fh(j)p Fk(k)24419 54888 y(d)p Fh(j)p Fk(n)26258 52266 y Fn(\026)p Fo(\()p Fn(d)p Fo(\))368 b(=)30478 51004 y Fj(X)30779 53882 y Fk(d)p Fh(j)p Fk(n)32618 52266 y Fn(\026)p Fo(\()p Fn(d)p Fo(\))35310 51004 y Fj(X)35381 53833 y Fk(k)24 b Fh(\024)p Fk(x)35639 54766 y(d)p Fh(j)p Fk(k)37450 52266 y Fo(1)369 b(=)39849 51004 y Fj(X)40151 53882 y Fk(d)p Fh(j)p Fk(n)41990 52266 y Fn(\026)p Fo(\()p Fn(d)p Fo(\))p Fm(b)p Fn(x=d)p Fm(c)p Fn(:)2751 57416 y Fo(No)-36 b(w)473 b(let)f Fn(d)434 b Fo(=)g Fn(p)10878 56787 y Fk(\014)11406 56910 y Ff(1)10878 57742 y Fi(1)12143 57416 y Fm(\001)221 b(\001)g(\001)h Fn(p)14567 56934 y Fk(\014)15095 57045 y Fe(r)14567 57745 y Fk(r)16084 57416 y Fo(stand)472 b(for)g(a)g(square-free)h(divisor)f(of)h Fn(n)p Fo(,)482 b(i.e.,)h Fn(\014)40090 57615 y Fk(j)41011 57416 y Fo(=)434 b(1)473 b(or)f(0)g(for)g(ev)-36 b(ery)473 b Fn(j)75 b Fo(,)800 59022 y(and)434 b(let)g Fn(B)502 b Fo(b)36 b(e)434 b(the)g(set)g(of)h(those)f Fn(j)509 b Fo(for)435 b(whic)-36 b(h)434 b Fn(\014)25119 59221 y Fk(j)25976 59022 y Fo(=)370 b(1)435 b(and)f(w)-36 b(e)434 b(obtain)h(\()p 0 .5 0 TeXcolorrgb 37442 59022 a SDict begin H.S end 37442 59022 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 38742 58166 a SDict begin H.R end 38742 58166 a 38742 59022 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.20) cvn H.B /ANN pdfmark end 38742 59022 a Black Fo(\))f(b)-36 b(y)434 b(the)g(de\014nition)g(of)h(the)800 60627 y(M\177)-650 b(obius)433 b(function.)2751 62232 y(F)-108 b(or)433 b(the)g(pro)36 b(of)434 b(of)527 b(\()p 0 .5 0 TeXcolorrgb 12797 62232 a SDict begin H.S end 12797 62232 a 0 .5 0 TeXcolorrgb Fo(21)p 0 .5 0 TeXcolorrgb 14098 61376 a SDict begin H.R end 14098 61376 a 14098 62232 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.21) cvn H.B /ANN pdfmark end 14098 62232 a Black Fo(\))433 b(w)-36 b(e)434 b(use)f(that)g Fn(d)369 b Fm(jj)g Fo(\()p Fn(k)45 b(;)221 b(n)p Fo(\))27260 62431 y Fh(\003)28220 62232 y Fo(i\013)434 b Fn(d)369 b Fm(j)g Fn(k)45 b Fo(,)434 b Fn(d)368 b Fm(jj)h Fn(n)434 b Fo(and)f(ha)-36 b(v)g(e)4554 64889 y Fn(\036)5324 64340 y Fh(\003)5849 64889 y Fo(\()p Fn(x;)221 b(n)p Fo(\))370 b(=)10709 63627 y Fj(X)10781 66456 y Fk(k)24 b Fh(\024)p Fk(x)13672 63627 y Fj(X)12849 66505 y Fk(d)p Fh(jj)p Fi(\()p Fk(k)g(;n)p Fi(\))15933 66616 y Fc(\003)16634 64889 y Fn(\026)17417 64340 y Fh(\003)17943 64889 y Fo(\()p Fn(d)p Fo(\))368 b(=)21380 63627 y Fj(X)21452 66456 y Fk(k)24 b Fh(\024)p Fk(x)23520 63627 y Fj(X)23850 66505 y Fk(d)p Fh(j)p Fk(k)23691 67510 y(d)p Fh(jj)p Fk(n)25660 64889 y Fn(\026)26443 64340 y Fh(\003)26969 64889 y Fo(\()p Fn(d)p Fo(\))368 b(=)30406 63627 y Fj(X)30576 66505 y Fk(d)p Fh(jj)p Fk(n)32546 64889 y Fn(\026)33329 64340 y Fh(\003)33854 64889 y Fo(\()p Fn(d)p Fo(\))35763 63627 y Fj(X)35835 66456 y Fk(k)24 b Fh(\024)p Fk(x)36093 67389 y(d)p Fh(j)p Fk(k)37903 64889 y Fo(1)369 b(=)40303 63627 y Fj(X)40473 66505 y Fk(d)p Fh(jj)p Fk(n)42443 64889 y Fn(\026)43226 64340 y Fh(\003)43751 64889 y Fo(\()p Fn(d)p Fo(\))p Fm(b)p Fn(x=d)p Fm(c)p Fn(:)p Black 26475 70521 a Fo(4)p Black eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.5) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 2751 -2672 a Fo(No)-36 b(w)451 b(let)f Fn(d)397 b Fo(=)g Fn(p)10760 -3301 y Fk(\015)11245 -3178 y Ff(1)10760 -2346 y Fi(1)11982 -2672 y Fm(\001)221 b(\001)g(\001)h Fn(p)14406 -3154 y Fk(\015)14891 -3043 y Fe(r)14406 -2343 y Fk(r)15858 -2672 y Fo(b)36 b(e)450 b(a)h(unitary)f(divisor)h(of)g Fn(n)p Fo(,)k(i.e.,)g Fn(\015)33844 -2473 y Fk(j)34728 -2672 y Fo(=)397 b Fn(\027)36780 -2473 y Fk(j)37716 -2672 y Fo(or)451 b(0)f(for)h(ev)-36 b(ery)451 b Fn(j)75 b Fo(,)454 b(and)c(let)g Fn(B)800 -1067 y Fo(b)36 b(e)433 b(the)g(set)h(of)g(those)g Fn(j)508 b Fo(for)434 b(whic)-36 b(h)433 b Fn(\015)19155 -868 y Fk(j)20011 -1067 y Fo(=)369 b Fn(\027)22035 -868 y Fk(j)22955 -1067 y Fo(and)433 b(w)-36 b(e)434 b(obtain)f(the)g(form)-36 b(ula)434 b(\()p 0 .5 0 TeXcolorrgb 38960 -1067 a SDict begin H.S end 38960 -1067 a 0 .5 0 TeXcolorrgb Fo(21)p 0 .5 0 TeXcolorrgb 40261 -1923 a SDict begin H.R end 40261 -1923 a 40261 -1067 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.21) cvn H.B /ANN pdfmark end 40261 -1067 a Black Fo(\).)2751 539 y(F)-108 b(or)433 b(the)g(pro)36 b(of)434 b(of)527 b(\()p 0 .5 0 TeXcolorrgb 12797 539 a SDict begin H.S end 12797 539 a 0 .5 0 TeXcolorrgb Fo(22)p 0 .5 0 TeXcolorrgb 14098 -318 a SDict begin H.R end 14098 -318 a 14098 539 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.22) cvn H.B /ANN pdfmark end 14098 539 a Black Fo(\))433 b(w)-36 b(e)434 b(apply)g(that)f Fn(d)369 b Fm(jj)g Fo(\()p Fn(k)45 b(;)221 b(n)p Fo(\))28590 738 y Fh(\003\003)30020 539 y Fo(i\013)434 b Fn(d)369 b Fm(jj)g Fn(k)45 b Fo(,)434 b Fn(d)368 b Fm(jj)h Fn(n)434 b Fo(and)f(obtain)7533 3509 y Fn(\036)8303 2961 y Fh(\003\003)9299 3509 y Fo(\()p Fn(x;)221 b(n)p Fo(\))369 b(=)14158 2247 y Fj(X)14230 5076 y Fk(k)24 b Fh(\024)p Fk(x)17333 2247 y Fj(X)16298 5125 y Fk(d)p Fh(jj)p Fi(\()p Fk(k)g(;n)p Fi(\))19382 5236 y Fc(\003\003)20508 3509 y Fn(\026)21291 2961 y Fh(\003)21816 3509 y Fo(\()p Fn(d)p Fo(\))369 b(=)25254 2247 y Fj(X)25325 5076 y Fk(k)24 b Fh(\024)p Fk(x)27394 2247 y Fj(X)27593 5125 y Fk(d)p Fh(jj)p Fk(k)27564 6131 y(d)p Fh(jj)p Fk(n)29534 3509 y Fn(\026)30317 2961 y Fh(\003)30842 3509 y Fo(\()p Fn(d)p Fo(\))369 b(=)34279 2247 y Fj(X)34450 5125 y Fk(d)p Fh(jj)p Fk(n)36419 3509 y Fn(\026)37202 2961 y Fh(\003)37728 3509 y Fo(\()p Fn(d)p Fo(\))41457 2247 y Fj(X)41529 5076 y Fk(k)24 b Fh(\024)p Fk(x)41787 6009 y(d)p Fh(j)p Fk(k)39637 7014 y Fi(gcd\()p Fk(d;k)g(=d)p Fi(\)=1)45417 3509 y Fo(1)12556 10007 y(=)13937 8745 y Fj(X)14107 11623 y Fk(d)p Fh(jj)p Fk(n)16077 10007 y Fn(\026)16860 9459 y Fh(\003)17385 10007 y Fo(\()p Fn(d)p Fo(\))p Fn(\036)p Fo(\()p Fn(x=d;)221 b(d)p Fo(\))369 b(=)25928 8745 y Fj(X)26098 11623 y Fk(d)p Fh(jj)p Fk(n)28068 10007 y Fn(\026)28851 9459 y Fh(\003)29376 10007 y Fo(\()p Fn(d)p Fo(\))31285 8745 y Fj(X)31653 11623 y Fk(e)p Fh(j)p Fk(d)33425 10007 y Fn(\026)p Fo(\()p Fn(e)p Fo(\))p Fm(b)p Fn(x=)p Fo(\()p Fn(de)p Fo(\))p Fm(c)p Fn(:)2751 13735 y Fo(Let)493 b Fn(d)472 b Fo(=)f Fn(p)8425 13105 y Fk(\015)8910 13228 y Ff(1)8425 14061 y Fi(1)9647 13735 y Fm(\001)221 b(\001)g(\001)i Fn(p)12072 13253 y Fk(\015)12557 13364 y Fe(r)12072 14064 y Fk(r)13567 13735 y Fo(b)36 b(e)494 b(a)g(unitary)g(divisor)g(of)h Fn(n)f Fo(i.e.,)510 b Fn(\015)31504 13934 y Fk(j)32462 13735 y Fo(=)472 b Fn(\027)34589 13934 y Fk(j)35569 13735 y Fo(or)494 b(0)g(for)g(ev)-36 b(ery)495 b Fn(j)75 b Fo(,)509 b(and)493 b(let)h Fn(B)561 b Fo(b)36 b(e)800 15340 y(the)435 b(set)g(of)h(those)f Fn(j)510 b Fo(for)435 b(whic)-36 b(h)435 b Fn(\015)17396 15539 y Fk(j)18255 15340 y Fo(=)371 b Fn(\027)20281 15539 y Fk(j)20767 15340 y Fo(.)584 b(F)-108 b(urthermore,)434 b(let)h Fn(e)372 b Fo(=)f Fn(p)34499 14711 y Fk(\021)34971 14834 y Ff(1)34499 15666 y Fi(1)35709 15340 y Fm(\001)221 b(\001)g(\001)h Fn(p)38133 14858 y Fk(\021)38605 14969 y Fe(r)38133 15669 y Fk(r)39557 15340 y Fo(b)36 b(e)435 b(a)h(square-free)f(divisor)800 16945 y(of)456 b Fn(d)p Fo(,)461 b(where)455 b Fn(\021)8222 17144 y Fk(k)9196 16945 y Fo(=)406 b(1)455 b(or)h(0)f(for)h(ev)-36 b(ery)456 b Fn(k)501 b Fo(and)454 b(let)i Fn(C)550 b Fo(b)36 b(e)455 b(the)g(set)g(of)h(those)f Fn(k)500 b Fo(for)456 b(whic)-36 b(h)455 b Fn(\021)45612 17144 y Fk(k)46586 16945 y Fo(=)406 b(1,)461 b(where)800 18550 y Fn(C)j Fm(\022)369 b Fn(B)67 b Fo(.)p 51860 18550 45 878 v 51905 17717 781 45 v 51905 18550 V 52684 18550 45 878 v 800 19672 a SDict begin H.S end 800 19672 a 800 19672 a SDict begin 14.5 H.A end 800 19672 a 800 19672 a SDict begin [ /View [/XYZ H.V] /Dest (cor.1) cvn H.B /DEST pdfmark end 800 19672 a 800 19671 a SDict begin H.S end 800 19671 a 800 19671 a SDict begin 14.5 H.A end 800 19671 a 800 19671 a SDict begin [ /View [/XYZ H.V] /Dest (equation.23) cvn H.B /DEST pdfmark end 800 19671 a Black 1424 x Fd(Corollary)521 b(1.)p Black 11789 24187 a Fn(\036)12559 23639 y Fh(\003\003)13555 24187 y Fo(\()p Fn(n)p Fo(\))369 b(=)17228 22925 y Fj(X)17092 25744 y Fk(B)44 b Fh(\022)p Fk(A)19282 24187 y Fo(\()p Fm(\000)p Fo(1\))21977 23639 y Fi(#)p Fk(B)23941 22925 y Fj(X)23792 25744 y Fk(C)63 b Fh(\022)p Fk(B)26010 24187 y Fo(\()p Fm(\000)p Fo(1\))28705 23639 y Fi(#)p Fk(C)30499 21464 y Fj(6)30499 22261 y(6)30499 23058 y(6)30499 23908 y(4)32030 22925 y(Y)31385 25803 y Fk(j)51 b Fh(2)p Fk(A)p Fh(n)p Fk(B)34594 24187 y Fn(p)35247 23466 y Fk(\027)35710 23601 y Fe(j)35247 24528 y Fk(j)36467 22925 y Fj(Y)36419 25744 y Fk(j)g Fh(2)p Fk(C)38433 24187 y Fn(p)39086 23637 y Fh(\000)p Fi(1)39086 24528 y Fk(j)40343 21464 y Fj(7)40343 22261 y(7)40343 23058 y(7)40343 23908 y(5)41450 24187 y Fn(:)8677 b Fo(\(23\))2751 28293 y(This)423 b(is)g(obtained)f(from)h (\()p 0 .5 0 TeXcolorrgb 15909 28293 a SDict begin H.S end 15909 28293 a 0 .5 0 TeXcolorrgb Fo(22)p 0 .5 0 TeXcolorrgb 17209 27437 a SDict begin H.R end 17209 27437 a 17209 28293 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.22) cvn H.B /ANN pdfmark end 17209 28293 a Black Fo(\))g(for)g Fn(x)369 b Fo(=)g Fn(n)p Fo(.)575 b(If)423 b Fn(r)405 b Fo(=)369 b(1,)425 b(then)d(w)-36 b(e)423 b(deduce)e(that)h Fn(\036)42169 27811 y Fh(\003\003)43165 28293 y Fo(\()p Fn(p)44324 27712 y Fk(\027)44787 27835 y Ff(1)44324 28618 y Fi(1)45303 28293 y Fo(\))369 b(=)g Fn(p)48212 27712 y Fk(\027)48675 27835 y Ff(1)48212 28618 y Fi(1)49463 28293 y Fm(\000)273 b Fo(1)369 b(=)800 29898 y Fn(\036)1570 29416 y Fh(\003)2096 29898 y Fo(\()p Fn(p)3255 29317 y Fk(\027)3718 29440 y Ff(1)3255 30224 y Fi(1)4233 29898 y Fo(\),)434 b(whic)-36 b(h)433 b(can)h(b)36 b(e)433 b(seen)h(directly)g(from)g(the)f(de\014nition)f(\()p 0 .5 0 TeXcolorrgb 32566 29898 a SDict begin H.S end 32566 29898 a 0 .5 0 TeXcolorrgb Fo(8)p 0 .5 0 TeXcolorrgb 33216 29042 a SDict begin H.R end 33216 29042 a 33216 29898 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.8) cvn H.B /ANN pdfmark end 33216 29898 a Black Fo(\).)579 b(F)-108 b(or)433 b Fn(r)405 b Fo(=)369 b(2)434 b(w)-36 b(e)433 b(obtain)46409 28293 y SDict begin H.S end 46409 28293 a 46409 28293 a SDict begin 14.5 H.A end 46409 28293 a 46409 28293 a SDict begin [ /View [/XYZ H.V] /Dest (equation.24) cvn H.B /DEST pdfmark end 46409 28293 a 1464 32636 a Fn(\036)2234 32088 y Fh(\003\003)3230 32636 y Fo(\()p Fn(p)4389 32056 y Fk(\027)4852 32179 y Ff(1)4389 32962 y Fi(1)5368 32636 y Fn(p)6021 32056 y Fk(\027)6484 32179 y Ff(2)6021 32962 y Fi(2)7000 32636 y Fo(\))369 b(=)f(\()p Fn(p)10414 32056 y Fk(\027)10877 32179 y Ff(1)10414 32962 y Fi(1)11458 32636 y Fm(\000)65 b Fo(1\)\()p Fn(p)14871 32056 y Fk(\027)15334 32179 y Ff(2)14871 32962 y Fi(2)15914 32636 y Fm(\000)g Fo(1\))g(+)19310 31560 y Fj(\004)19937 32636 y Fn(p)20590 32056 y Fk(\027)21053 32179 y Ff(1)20590 32962 y Fi(1)21569 32636 y Fn(p)22222 32088 y Fh(\000)p Fi(1)22222 32965 y(2)23479 31560 y Fj(\005)24171 32636 y Fo(+)25248 31560 y Fj(\004)25875 32636 y Fn(p)26528 32056 y Fk(\027)26991 32179 y Ff(2)26528 32962 y Fi(2)27507 32636 y Fn(p)28160 32088 y Fh(\000)p Fi(1)28160 32965 y(1)29417 31560 y Fj(\005)30413 32636 y Fo(=)369 b Fn(\036)32564 32088 y Fh(\003)33089 32636 y Fo(\()p Fn(p)34248 32056 y Fk(\027)34711 32179 y Ff(1)34248 32962 y Fi(1)35227 32636 y Fn(p)35880 32056 y Fk(\027)36343 32179 y Ff(2)35880 32962 y Fi(2)36859 32636 y Fo(\))65 b(+)38507 31560 y Fj(\004)39133 32636 y Fn(p)39786 32056 y Fk(\027)40249 32179 y Ff(1)39786 32962 y Fi(1)40765 32636 y Fn(p)41418 32088 y Fh(\000)p Fi(1)41418 32965 y(2)42676 31560 y Fj(\005)43368 32636 y Fo(+)44445 31560 y Fj(\004)45071 32636 y Fn(p)45724 32056 y Fk(\027)46187 32179 y Ff(2)45724 32962 y Fi(2)46703 32636 y Fn(p)47356 32088 y Fh(\000)p Fi(1)47356 32965 y(1)48614 31560 y Fj(\005)49462 32636 y Fn(;)665 b Fo(\(24\))800 35374 y(therefore)434 b(the)f(function)g Fn(\036)14360 34892 y Fh(\003\003)15790 35374 y Fo(is)g(not)h(m)-36 b(ultiplicativ)g(e.)2751 36979 y(F)-108 b(or)381 b Fn(P)6048 36497 y Fh(\003\003)7045 36979 y Fo(\()p Fn(n)p Fo(\))h(and)f Fn(S)12569 36497 y Fh(\003\003)13566 36979 y Fo(\()p Fn(n)p Fo(\))g(w)-36 b(e)383 b(ha)-36 b(v)g(e)382 b(the)f(follo)-36 b(wing)384 b(form)-36 b(ulae.)562 b(First)381 b(note)h(that)f Fn(\034)338 b Fm(\002)190 b Fn(\026)45640 36497 y Fh(\003)46534 36979 y Fo(=)369 b Fn(\014)74 b Fo(,)392 b(where)800 38585 y Fn(\014)74 b Fo(\()p Fn(n)p Fo(\))427 b(is)h(the)f(n)-36 b(um)g(b)36 b(er)426 b(of)i(square-full)g(divisors)h(of)f Fn(n)g Fo(\()p 0 1 0 0 TeXcolorcmyk 27848 38850 a SDict begin H.S end 27848 38850 a 0 1 0 0 TeXcolorcmyk -265 x Fo(A005361)p 27848 38797 4878 54 v 0 1 0 0 TeXcolorcmyk 32725 37677 a SDict begin H.R end 32725 37677 a 32725 38585 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A005361) >> /Subtype /Link H.B /ANN pdfmark end 32725 38585 a Black Fo(\),)g Fn(\014)502 b Fo(is)428 b(m)-36 b(ultiplicativ)g(e)428 b(and)f Fn(\014)74 b Fo(\()p Fn(p)49243 38102 y Fk(\027)49818 38585 y Fo(\))369 b(=)g Fn(\027)800 40190 y Fo(for)434 b(an)-36 b(y)434 b(prime)f(p)36 b(o)-36 b(w)g(er)434 b Fn(p)13376 39708 y Fk(\027)14385 40190 y Fo(\()p Fn(\027)452 b Fm(\025)369 b Fo(1\).)579 b(Therefore)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(from)g(\()p 0 .5 0 TeXcolorrgb 33864 40190 a SDict begin H.S end 33864 40190 a 0 .5 0 TeXcolorrgb Fo(17)p 0 .5 0 TeXcolorrgb 35164 39334 a SDict begin H.R end 35164 39334 a 35164 40190 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.17) cvn H.B /ANN pdfmark end 35164 40190 a Black Fo(\),)36465 38585 y SDict begin H.S end 36465 38585 a 36465 38585 a SDict begin 14.5 H.A end 36465 38585 a 36465 38585 a SDict begin [ /View [/XYZ H.V] /Dest (equation.25) cvn H.B /DEST pdfmark end 36465 38585 a 19025 43160 a Fn(S)19902 42612 y Fh(\003\003)20898 43160 y Fo(\()p Fn(n)p Fo(\))369 b(=)24436 41898 y Fj(X)24606 44776 y Fk(d)p Fh(jj)p Fk(n)26576 43160 y Fn(\014)74 b Fo(\()p Fn(d)p Fo(\))p Fn(\036)p Fo(\()p Fn(n=d;)221 b(d)p Fo(\))p Fn(:)15913 b Fo(\(25\))800 46264 y SDict begin H.S end 800 46264 a 800 46264 a SDict begin 14.5 H.A end 800 46264 a 800 46264 a SDict begin [ /View [/XYZ H.V] /Dest (thm.2) cvn H.B /DEST pdfmark end 800 46264 a 800 46264 a SDict begin H.S end 800 46264 a 800 46264 a SDict begin 14.5 H.A end 800 46264 a 800 46264 a SDict begin [ /View [/XYZ H.V] /Dest (equation.26) cvn H.B /DEST pdfmark end 800 46264 a Black 1033 x Fd(Theorem)499 b(2.)p Black 5501 51264 a Fn(P)6520 50715 y Fh(\003\003)7516 51264 y Fo(\()p Fn(n)p Fo(\))369 b(=)11189 50002 y Fj(X)11054 52821 y Fk(B)44 b Fh(\022)p Fk(A)13465 48594 y Fj(0)13465 50985 y(@)14810 50002 y(X)14627 52821 y Fk(D)26 b Fh(\022)p Fk(B)16911 51264 y Fo(\()p Fm(\000)p Fo(1\))19606 50715 y Fi(#)p Fk(D)22156 50002 y Fj(Y)21465 52880 y Fk(j)51 b Fh(2)p Fk(B)44 b Fh(n)p Fk(D)24767 51264 y Fn(p)25420 50542 y Fk(\027)25883 50677 y Fe(j)25420 51604 y Fk(j)26371 48594 y Fj(1)26371 50985 y(A)27755 48594 y(0)27755 50985 y(@)29067 50002 y(X)28917 52821 y Fk(C)63 b Fh(\022)p Fk(B)31136 51264 y Fo(\()p Fm(\000)p Fo(1\))33831 50715 y Fi(#)p Fk(C)35625 48541 y Fj(6)35625 49338 y(6)35625 50135 y(6)35625 50985 y(4)37155 50002 y(Y)36510 52880 y Fk(j)51 b Fh(2)p Fk(A)p Fh(n)p Fk(B)39719 51264 y Fn(p)40372 50542 y Fk(\027)40835 50677 y Fe(j)40372 51604 y Fk(j)41592 50002 y Fj(Y)41545 52821 y Fk(j)g Fh(2)p Fk(C)43558 51264 y Fn(p)44211 50714 y Fh(\000)p Fi(1)44211 51604 y Fk(j)45469 48541 y Fj(7)45469 49338 y(7)45469 50135 y(7)45469 50985 y(5)46354 48594 y(1)46354 50985 y(A)47738 51264 y Fn(;)2389 b Fo(\(26\))800 52962 y SDict begin H.S end 800 52962 a 800 52962 a SDict begin 14.5 H.A end 800 52962 a 800 52962 a SDict begin [ /View [/XYZ H.V] /Dest (equation.27) cvn H.B /DEST pdfmark end 800 52962 a 11135 57401 a Fn(S)12012 56852 y Fh(\003\003)13009 57401 y Fo(\()p Fn(n)p Fo(\))369 b(=)16682 56139 y Fj(X)16546 58958 y Fk(B)44 b Fh(\022)p Fk(A)18958 55129 y Fj( )20067 56139 y(Y)20009 58958 y Fk(j)51 b Fh(2)p Fk(B)22043 57401 y Fn(\027)22686 57600 y Fk(j)23172 55129 y Fj(!)24595 56139 y(X)24445 58958 y Fk(C)63 b Fh(\022)p Fk(B)26664 57401 y Fo(\()p Fm(\000)p Fo(1\))29359 56852 y Fi(#)p Fk(C)31153 54677 y Fj(6)31153 55474 y(6)31153 56271 y(6)31153 57122 y(4)32683 56139 y(Y)32038 59017 y Fk(j)51 b Fh(2)p Fk(A)p Fh(n)p Fk(B)35247 57401 y Fn(p)35900 56679 y Fk(\027)36363 56814 y Fe(j)35900 57741 y Fk(j)37120 56139 y Fj(Y)37073 58958 y Fk(j)g Fh(2)p Fk(C)39086 57401 y Fn(p)39739 56851 y Fh(\000)p Fi(1)39739 57741 y Fk(j)40997 54677 y Fj(7)40997 55474 y(7)40997 56271 y(7)40997 57122 y(5)42104 57401 y Fn(:)8023 b Fo(\(27\))p Black 800 61506 a Fg(Pr)-66 b(o)g(of.)p Black 649 w Fo(F)-108 b(rom)433 b(\()p 0 .5 0 TeXcolorrgb 8792 61506 a SDict begin H.S end 8792 61506 a 0 .5 0 TeXcolorrgb Fo(14)p 0 .5 0 TeXcolorrgb 10093 60650 a SDict begin H.R end 10093 60650 a 10093 61506 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.14) cvn H.B /ANN pdfmark end 10093 61506 a Black Fo(\))g(and)g(\()p 0 .5 0 TeXcolorrgb 14067 61506 a SDict begin H.S end 14067 61506 a 0 .5 0 TeXcolorrgb Fo(5)p 0 .5 0 TeXcolorrgb 14717 60650 a SDict begin H.R end 14717 60650 a 14717 61506 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5) cvn H.B /ANN pdfmark end 14717 61506 a Black Fo(\))h(w)-36 b(e)433 b(obtain)6476 65805 y Fn(P)7495 65257 y Fh(\003\003)8492 65805 y Fo(\()p Fn(n)p Fo(\))369 b(=)12029 64543 y Fj(X)12200 67421 y Fk(d)p Fh(jj)p Fk(n)14169 65805 y Fn(\036)14939 65257 y Fh(\003)15465 65805 y Fo(\()p Fn(d)p Fo(\))p Fn(\036)p Fo(\()p Fn(n=d;)221 b(d)p Fo(\))369 b(=)24045 64543 y Fj(X)24215 67421 y Fk(d)p Fh(jj)p Fk(n)26185 63135 y Fj(0)26185 65526 y(@)27347 64543 y(X)27585 67421 y Fk(e)p Fh(jj)p Fk(d)29487 65805 y Fn(\026)30270 65257 y Fh(\003)30796 65805 y Fo(\()p Fn(e)p Fo(\))32544 64907 y Fn(d)p 32544 65500 676 54 v 32581 66716 a(e)33352 63135 y Fj(1)33352 65526 y(A)34735 63135 y(0)34735 65526 y(@)35898 64543 y(X)36259 67421 y Fk(\016)33 b Fh(j)p Fk(d)38038 65805 y Fn(\026)p Fo(\()p Fn(\016)50 b Fo(\))p Fm(b)p Fn(n=)p Fo(\()p Fn(d\016)g Fo(\))p Fm(c)45379 63135 y Fj(1)45379 65526 y(A)46762 65805 y Fn(;)p Black 26475 70521 a Fo(5)p Black eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Fo(and)433 b(use)g(similar)i(argumen)-36 b(ts)433 b(as)h(in)f(Theorem)h(1.)579 b(T)-108 b(o)433 b(obtain)h(\()p 0 .5 0 TeXcolorrgb 32840 -2672 a SDict begin H.S end 32840 -2672 a 0 .5 0 TeXcolorrgb Fo(27)p 0 .5 0 TeXcolorrgb 34140 -3528 a SDict begin H.R end 34140 -3528 a 34140 -2672 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.27) cvn H.B /ANN pdfmark end 34140 -2672 a Black Fo(\))g(use)f(\()p 0 .5 0 TeXcolorrgb 37833 -2672 a SDict begin H.S end 37833 -2672 a 0 .5 0 TeXcolorrgb Fo(25)p 0 .5 0 TeXcolorrgb 39134 -3528 a SDict begin H.R end 39134 -3528 a 39134 -2672 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.25) cvn H.B /ANN pdfmark end 39134 -2672 a Black Fo(\))o(.)2751 -1067 y(F)-108 b(or)495 b Fn(r)511 b Fo(=)474 b(1)495 b(w)-36 b(e)496 b(deduce)f(from)g(\()p 0 .5 0 TeXcolorrgb 18884 -1067 a SDict begin H.S end 18884 -1067 a 0 .5 0 TeXcolorrgb Fo(26)p 0 .5 0 TeXcolorrgb 20185 -1923 a SDict begin H.R end 20185 -1923 a 20185 -1067 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.26) cvn H.B /ANN pdfmark end 20185 -1067 a Black Fo(\))g(that)g Fn(P)25085 -1549 y Fh(\003\003)26081 -1067 y Fo(\()p Fn(p)27240 -1647 y Fk(\027)27703 -1524 y Ff(1)27240 -741 y Fi(1)28219 -1067 y Fo(\))474 b(=)g(2)p Fn(p)31988 -1647 y Fk(\027)32451 -1524 y Ff(1)31988 -741 y Fi(1)33305 -1067 y Fm(\000)337 b Fo(1)475 b(=)f Fn(P)38305 -1549 y Fh(\003)38831 -1067 y Fo(\()p Fn(p)39990 -1647 y Fk(\027)40453 -1524 y Ff(1)39990 -741 y Fi(1)40968 -1067 y Fo(\),)511 b(whic)-36 b(h)496 b(can)f(b)36 b(e)495 b(seen)800 539 y(also)435 b(directly)f(from)g(the)f(de\014nitions.)577 b(F)-108 b(or)433 b Fn(r)406 b Fo(=)368 b(2)434 b(w)-36 b(e)434 b(obtain)5542 3472 y Fn(P)6561 2924 y Fh(\003\003)7558 3472 y Fo(\()p Fn(p)8717 2891 y Fk(\027)9180 3014 y Ff(1)8717 3798 y Fi(1)9695 3472 y Fn(p)10348 2891 y Fk(\027)10811 3014 y Ff(2)10348 3798 y Fi(2)11327 3472 y Fo(\))369 b(=)g(\(2)p Fn(p)15392 2891 y Fk(\027)15855 3014 y Ff(1)15392 3798 y Fi(1)16666 3472 y Fm(\000)295 b Fo(1\))221 b(\()q(2)p Fn(p)21181 2891 y Fk(\027)21644 3014 y Ff(2)21181 3798 y Fi(2)22455 3472 y Fm(\000)295 b Fo(1\))h Fm(\000)f Fo(\()p Fn(p)27722 2891 y Fk(\027)28185 3014 y Ff(1)27722 3798 y Fi(1)28996 3472 y Fm(\000)g Fo(1\))31701 2396 y Fj(\004)32329 3472 y Fn(p)32982 2891 y Fk(\027)33445 3014 y Ff(2)32982 3798 y Fi(2)33961 3472 y Fn(p)34614 2924 y Fh(\000)p Fi(1)34614 3800 y(1)35871 2396 y Fj(\005)36794 3472 y Fm(\000)g Fo(\()p Fn(p)39281 2891 y Fk(\027)39744 3014 y Ff(2)39281 3798 y Fi(2)40555 3472 y Fm(\000)g Fo(1\))43260 2396 y Fj(\004)43888 3472 y Fn(p)44541 2891 y Fk(\027)45004 3014 y Ff(1)44541 3798 y Fi(1)45520 3472 y Fn(p)46173 2924 y Fh(\000)p Fi(1)46173 3800 y(2)47431 2396 y Fj(\005)11349 6405 y Fo(=)369 b Fn(P)13749 5857 y Fh(\003)14275 6405 y Fo(\()p Fn(p)15434 5825 y Fk(\027)15897 5948 y Ff(1)15434 6731 y Fi(1)16412 6405 y Fn(p)17065 5825 y Fk(\027)17528 5948 y Ff(2)17065 6731 y Fi(2)18044 6405 y Fo(\))295 b Fm(\000)g Fo(\()p Fn(p)21332 5825 y Fk(\027)21795 5948 y Ff(1)21332 6731 y Fi(1)22606 6405 y Fm(\000)h Fo(1\))25312 5329 y Fj(\004)25940 6405 y Fn(p)26593 5825 y Fk(\027)27056 5948 y Ff(2)26593 6731 y Fi(2)27571 6405 y Fn(p)28224 5857 y Fh(\000)p Fi(1)28224 6734 y(1)29482 5329 y Fj(\005)30404 6405 y Fm(\000)g Fo(\()p Fn(p)32892 5825 y Fk(\027)33355 5948 y Ff(2)32892 6731 y Fi(2)34166 6405 y Fm(\000)f Fo(1\))36871 5329 y Fj(\004)37499 6405 y Fn(p)38152 5825 y Fk(\027)38615 5948 y Ff(1)38152 6731 y Fi(1)39131 6405 y Fn(p)39784 5857 y Fh(\000)p Fi(1)39784 6734 y(2)41041 5329 y Fj(\005)41890 6405 y Fn(:)2751 8730 y Fo(Also,)435 b(for)f Fn(r)405 b Fo(=)369 b(1)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(from)g(\()p 0 .5 0 TeXcolorrgb 19979 8730 a SDict begin H.S end 19979 8730 a 0 .5 0 TeXcolorrgb Fo(27)p 0 .5 0 TeXcolorrgb 21280 7874 a SDict begin H.R end 21280 7874 a 21280 8730 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.27) cvn H.B /ANN pdfmark end 21280 8730 a Black Fo(\))o(,)g Fn(S)23457 8248 y Fh(\003\003)24454 8730 y Fo(\()p Fn(p)25613 8150 y Fk(\027)26076 8273 y Ff(1)25613 9056 y Fi(1)26591 8730 y Fo(\))369 b(=)g Fn(p)29500 8150 y Fk(\027)29963 8273 y Ff(1)29500 9056 y Fi(1)30774 8730 y Fo(+)295 b Fn(\027)32724 8929 y Fi(1)33249 8730 y Fo(.)579 b(F)-108 b(or)433 b Fn(r)405 b Fo(=)369 b(2)433 b(w)-36 b(e)434 b(obtain)9627 11663 y Fn(S)10504 11115 y Fh(\003\003)11500 11663 y Fo(\()p Fn(p)12659 11083 y Fk(\027)13122 11206 y Ff(1)12659 11989 y Fi(1)13638 11663 y Fn(p)14291 11083 y Fk(\027)14754 11206 y Ff(2)14291 11989 y Fi(2)15270 11663 y Fo(\))368 b(=)h(\()p Fn(p)18684 11083 y Fk(\027)19147 11206 y Ff(1)18684 11989 y Fi(1)19958 11663 y Fo(+)295 b Fn(\027)21908 11862 y Fi(1)22433 11663 y Fo(\)\()p Fn(p)24098 11083 y Fk(\027)24561 11206 y Ff(2)24098 11989 y Fi(2)25372 11663 y Fo(+)g Fn(\027)27322 11862 y Fi(2)27847 11663 y Fo(\))g Fm(\000)h Fn(\027)30620 11862 y Fi(1)31366 10588 y Fj(\004)31994 11663 y Fn(p)32647 11083 y Fk(\027)33110 11206 y Ff(2)32647 11989 y Fi(2)33626 11663 y Fn(p)34279 11115 y Fh(\000)p Fi(1)34279 11992 y(1)35536 10588 y Fj(\005)36458 11663 y Fm(\000)g Fn(\027)38430 11862 y Fi(2)39177 10588 y Fj(\004)39804 11663 y Fn(p)40457 11083 y Fk(\027)40920 11206 y Ff(1)40457 11989 y Fi(1)41436 11663 y Fn(p)42089 11115 y Fh(\000)p Fi(1)42089 11992 y(2)43346 10588 y Fj(\005)14934 14597 y Fo(=)368 b Fn(S)17191 14048 y Fh(\003\003)18188 14597 y Fo(\()p Fn(p)19347 14016 y Fk(\027)19810 14139 y Ff(1)19347 14923 y Fi(1)20326 14597 y Fn(p)20979 14016 y Fk(\027)21442 14139 y Ff(2)20979 14923 y Fi(2)21957 14597 y Fo(\))295 b Fm(\000)h Fn(\027)24730 14796 y Fi(1)25477 13521 y Fj(\004)26104 14597 y Fn(p)26757 14016 y Fk(\027)27220 14139 y Ff(2)26757 14923 y Fi(2)27736 14597 y Fn(p)28389 14048 y Fh(\000)p Fi(1)28389 14925 y(1)29646 13521 y Fj(\005)30569 14597 y Fm(\000)f Fn(\027)32540 14796 y Fi(2)33287 13521 y Fj(\004)33914 14597 y Fn(p)34567 14016 y Fk(\027)35030 14139 y Ff(1)34567 14923 y Fi(1)35546 14597 y Fn(p)36199 14048 y Fh(\000)p Fi(1)36199 14925 y(2)37456 13521 y Fj(\005)38305 14597 y Fn(:)p 51860 16922 45 878 v 51905 16088 781 45 v 51905 16922 V 52684 16922 45 878 v 2751 19523 a Fo(Therefore)434 b(the)f(functions)h Fn(P)17507 19041 y Fh(\003\003)18936 19523 y Fo(and)f Fn(S)22342 19041 y Fh(\003\003)23773 19523 y Fo(are)g(not)h(m)-36 b(ultiplicativ)g(e.)800 21783 y SDict begin H.S end 800 21783 a 800 21783 a SDict begin 14.5 H.A end 800 21783 a 800 21783 a SDict begin [ /View [/XYZ H.V] /Dest (section.3) cvn H.B /DEST pdfmark end 800 21783 a 2177 x Fp(3)2152 b(The)716 b(a)-60 b(v)g(erage)718 b(order)e(of)h Fb(P)25810 23266 y Fm(\003\003)800 26880 y Fo(W)-108 b(e)434 b(sho)-36 b(w)433 b(that)800 26880 y SDict begin H.S end 800 26880 a 800 26880 a SDict begin 14.5 H.A end 800 26880 a 800 26880 a SDict begin [ /View [/XYZ H.V] /Dest (thm.3) cvn H.B /DEST pdfmark end 800 26880 a 800 27987 a SDict begin H.S end 800 27987 a 800 27987 a SDict begin 14.5 H.A end 800 27987 a 800 27987 a SDict begin [ /View [/XYZ H.V] /Dest (equation.28) cvn H.B /DEST pdfmark end 800 27987 a Black 1606 x Fd(Theorem)499 b(3.)p Black 17237 30196 a Fj(X)17281 33002 y Fk(n)p Fh(\024)p Fk(x)19378 31458 y Fn(P)20397 30909 y Fh(\003\003)21393 31458 y Fo(\()p Fn(n)p Fo(\))369 b(=)25063 30559 y(1)p 25063 31152 651 54 v 25063 32369 a(2)25846 31458 y Fn(B)67 b(x)27640 30909 y Fi(2)28388 31458 y Fo(log)240 b Fn(x)296 b Fo(+)e Fm(O)37 b Fo(\()p Fn(x)34970 30909 y Fi(2)35496 31458 y Fo(\))p Fn(;)14125 b Fo(\(28\))800 35037 y Fg(wher)-66 b(e)4470 33432 y SDict begin H.S end 4470 33432 a 4470 33432 a SDict begin 14.5 H.A end 4470 33432 a 4470 33432 a SDict begin [ /View [/XYZ H.V] /Dest (equation.29) cvn H.B /DEST pdfmark end 4470 33432 a 8350 37099 a Fn(B)436 b Fo(:=)11516 35837 y Fj(Y)12128 38627 y Fk(p)13434 35226 y Fj(\022)14412 37099 y Fo(1)296 b Fm(\000)17578 36200 y Fo(3)p Fn(p)f Fm(\000)h Fo(1)p 16819 36793 5096 54 v 16819 38010 a Fn(p)17472 37626 y Fi(2)17998 38010 y Fo(\()p Fn(p)e Fo(+)h(1\))22047 35226 y Fj(\023)23394 37099 y Fo(=)369 b Fn(\020)100 b Fo(\(2\))27326 35837 y Fj(Y)27938 38627 y Fk(p)29245 35226 y Fj(\022)30223 37099 y Fo(1)295 b Fm(\000)32630 36200 y Fo(\(2)p Fn(p)g Fm(\000)g Fo(1\))37218 35718 y Fi(2)p 32630 36793 5115 54 v 34598 38010 a Fn(p)35251 37626 y Fi(4)37877 35226 y Fj(\023)39224 37099 y Fm(\031)369 b Fo(0)p Fn(:)p Fo(35823)p Fn(:)5240 b Fo(\(29\))p Black 800 41072 a Fg(Pr)-66 b(o)g(of.)p Black 649 w Fo(W)-108 b(e)434 b(will)h(use)e(the)g(fact)h(that)f(for)h(an)-36 b(y)434 b(real)g(n)-36 b(um)g(b)36 b(ers)432 b Fn(s)369 b Fm(\025)g Fo(0)434 b(and)f(0)369 b Fm(\024)g Fn(")g(<)g Fo(1,)43880 39467 y SDict begin H.S end 43880 39467 a 43880 39467 a SDict begin 14.5 H.A end 43880 39467 a 43880 39467 a SDict begin [ /View [/XYZ H.V] /Dest (equation.30) cvn H.B /DEST pdfmark end 43880 39467 a 13235 43458 a Fj(X)13279 46265 y Fk(n)p Fh(\024)p Fk(x)11849 47198 y Fi(gcd\()p Fk(n;k)24 b Fi(\)=1)16762 44720 y Fn(n)17538 44172 y Fk(s)18398 44720 y Fo(=)19911 43822 y Fn(x)20650 43340 y Fk(s)p Fi(+1)22343 43822 y Fn(\036)p Fo(\()p Fn(k)45 b Fo(\))p 19911 44415 4935 54 v 20079 45632 a(\()p Fn(s)295 b Fo(+)g(1\))p Fn(k)25273 44720 y Fo(+)g Fm(O)37 b Fo(\()p Fn(x)28920 44172 y Fk(s)29631 44720 y Fo(min\()p Fn(x)33044 44172 y Fk(")33534 44720 y Fn(\033)34273 44919 y Fh(\000)p Fk(")35495 44720 y Fo(\()p Fn(k)45 b Fo(\))p Fn(;)221 b Fo(2)38460 44172 y Fk(!)32 b Fi(\()p Fk(k)24 b Fi(\))40378 44720 y Fo(\)\))p Fn(;)8737 b Fo(\(30\))800 49987 y(this)590 b(estimate)g(b)36 b(eing)591 b(uniform)f(for)g Fn(x)h Fo(and)e Fn(k)45 b Fo(,)630 b(where)590 b Fn(\033)30078 50186 y Fh(\000)p Fk(")31300 49987 y Fo(\()p Fn(k)45 b Fo(\))590 b(is)g(the)g(sum)g(of)h Fn(")p Fo(-th)d(p)36 b(o)-36 b(w)g(ers)591 b(of)g(the)800 51592 y(recipro)36 b(cals)463 b(of)f(divisors)h(of)f Fn(k)45 b Fo(.)664 b(A)461 b(similar)i(treatmen)-36 b(t)461 b(w)-36 b(as)462 b(applied)g(also)h(in)f([)p 0 .5 0 TeXcolorrgb 40293 51592 a SDict begin H.S end 40293 51592 a 0 .5 0 TeXcolorrgb Fo(17)p 0 .5 0 TeXcolorrgb 41593 50736 a SDict begin H.R end 41593 50736 a 41593 51592 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Tot1989) cvn H.B /ANN pdfmark end 41593 51592 a Black Fo(].)664 b(W)-108 b(e)462 b(ha)-36 b(v)g(e)462 b(b)-36 b(y)479 b(\()p 0 .5 0 TeXcolorrgb 50632 51592 a SDict begin H.S end 50632 51592 a 0 .5 0 TeXcolorrgb Fo(14)p 0 .5 0 TeXcolorrgb 51933 50736 a SDict begin H.R end 51933 50736 a 51933 51592 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.14) cvn H.B /ANN pdfmark end 51933 51592 a Black Fo(\),)800 53197 y(and)433 b(b)-36 b(y)452 b(\()p 0 .5 0 TeXcolorrgb 5660 53197 a SDict begin H.S end 5660 53197 a 0 .5 0 TeXcolorrgb Fo(30)p 0 .5 0 TeXcolorrgb 6960 52341 a SDict begin H.R end 6960 52341 a 6960 53197 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.30) cvn H.B /ANN pdfmark end 6960 53197 a Black Fo(\))434 b(applied)f(for)h Fn(s)369 b Fo(=)f(0)434 b(and)f Fn(s)369 b Fo(=)f(1,)435 b(resp)36 b(ectiv)-36 b(ely)-108 b(,)5439 55633 y Fj(X)5482 58439 y Fk(n)p Fh(\024)p Fk(x)7579 56895 y Fn(P)8598 56346 y Fh(\003\003)9594 56895 y Fo(\()p Fn(n)p Fo(\))369 b(=)14436 55633 y Fj(X)14304 58462 y Fk(de)p Fh(\024)p Fk(x)13132 59395 y Fi(gcd\()p Fk(d;e)p Fi(\)=1)17881 56895 y Fn(\036)18651 56346 y Fh(\003)19176 56895 y Fo(\()p Fn(d)p Fo(\))p Fn(\036)p Fo(\()p Fn(e;)221 b(d)p Fo(\))368 b(=)27560 55633 y Fj(X)27429 58462 y Fk(de)p Fh(\024)p Fk(x)26256 59395 y Fi(gcd\()p Fk(d;e)p Fi(\)=1)31005 56895 y Fn(\036)31775 56346 y Fh(\003)32301 56895 y Fo(\()p Fn(d)p Fo(\))34210 55022 y Fj(\022)35320 55996 y Fn(e\036)p Fo(\()p Fn(d)p Fo(\))p 35320 56589 3061 54 v 36512 57806 a Fn(d)38808 56895 y Fo(+)295 b Fm(O)258 b Fo(\()p Fn(e)42540 56346 y Fk(")43029 56895 y Fn(\033)43768 57094 y Fh(\000)p Fk(")44990 56895 y Fo(\()p Fn(d)p Fo(\)\))47184 55022 y Fj(\023)10299 63911 y Fo(=)11680 62649 y Fj(X)11766 65478 y Fk(d)p Fh(\024)p Fk(x)13953 63012 y Fn(\036)p Fo(\()p Fn(d)p Fo(\))p Fn(\036)17181 62530 y Fh(\003)17705 63012 y Fo(\()p Fn(d)p Fo(\))p 13953 63605 5441 54 v 16335 64822 a Fn(d)21052 62649 y Fj(X)20685 65527 y Fk(e)p Fh(\024)p Fk(x=d)19747 66532 y Fi(gcd)q(\()p Fk(e;d)p Fi(\)=1)24496 63911 y Fn(e)295 b Fo(+)g Fm(O)28017 61241 y Fj(0)28017 63632 y(@)29179 62649 y(X)29266 65478 y Fk(d)p Fh(\024)p Fk(x)31319 63911 y Fn(\036)32089 63363 y Fh(\003)32615 63911 y Fo(\()p Fn(d)p Fo(\))p Fn(\033)35042 64110 y Fh(\000)p Fk(")36263 63911 y Fo(\()p Fn(d)p Fo(\))38539 62649 y Fj(X)38172 65527 y Fk(e)p Fh(\024)p Fk(x=d)41046 63911 y Fn(e)41649 63363 y Fk(")42139 61241 y Fj(1)42139 63632 y(A)p Black 26475 70521 a Fo(6)p Black eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.7) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 4700 -1675 a Fo(=)6081 -2937 y Fj(X)6168 -108 y Fk(d)p Fh(\024)p Fk(x)8354 -2574 y Fn(\036)p Fo(\()p Fn(d)p Fo(\))p Fn(\036)11582 -3056 y Fh(\003)12107 -2574 y Fo(\()p Fn(d)p Fo(\))p 8354 -1981 5441 54 v 10736 -764 a Fn(d)14149 -3548 y Fj(\022)15259 -2574 y Fn(x)15998 -3056 y Fi(2)16524 -2574 y Fn(\036)p Fo(\()p Fn(d)p Fo(\))p 15259 -1981 3723 54 v 16194 -764 a(2)p Fn(d)17520 -1148 y Fi(3)19410 -1675 y Fo(+)294 b Fm(O)22032 -2751 y Fj(\000)22641 -1675 y Fo(\()p Fn(x=d)p Fo(\))25718 -2224 y Fk(")p Fi(+1)27410 -1675 y Fn(\033)28149 -1476 y Fh(\000)p Fk(")29371 -1675 y Fo(\()p Fn(d)p Fo(\))31059 -2751 y Fj(\001)31667 -3548 y(\023)32940 -1675 y Fo(+)h Fm(O)35563 -3947 y Fj( )36615 -2937 y(X)36701 -108 y Fk(d)p Fh(\024)p Fk(x)38755 -1675 y Fn(d\033)40170 -1476 y Fh(\000)p Fk(")41391 -1675 y Fo(\()p Fn(d)p Fo(\)\()p Fn(x=d)p Fo(\))46156 -2224 y Fk(")p Fi(+1)47848 -3947 y Fj(!)13892 3131 y Fo(=)15406 2232 y Fn(x)16145 1750 y Fi(2)p 15406 2825 1265 54 v 15713 4042 a Fo(2)17025 1869 y Fj(X)17112 4698 y Fk(d)p Fh(\024)p Fk(x)19298 2232 y Fn(\036)20068 1750 y Fi(2)20594 2232 y Fo(\()p Fn(d)p Fo(\))p Fn(\036)23052 1750 y Fh(\003)23577 2232 y Fo(\()p Fn(d)p Fo(\))p 19298 2825 5967 54 v 21680 4042 a Fn(d)22356 3658 y Fi(4)25692 3131 y Fo(+)g Fm(O)28315 859 y Fj( )29366 3131 y Fn(x)30105 2582 y Fk(")p Fi(+1)32019 1869 y Fj(X)32106 4698 y Fk(d)p Fh(\024)p Fk(x)34292 2232 y Fn(\033)35031 2431 y Fh(\000)p Fk(")36253 2232 y Fo(\()p Fn(d)p Fo(\))p 34292 2825 3649 54 v 35533 4042 a Fn(d)36209 3658 y Fk(")38073 859 y Fj(!)39346 3131 y Fn(;)800 6665 y Fo(where)434 b(the)f(error)g(term)g(is)h Fm(O)37 b Fo(\()p Fn(x)16732 6183 y Fk(")p Fi(+1)18719 6665 y Fm(\001)295 b Fn(x)20122 6183 y Fi(1)p Fh(\000)p Fk(")21814 6665 y Fo(\))369 b(=)f Fm(O)37 b Fo(\()p Fn(x)26409 6183 y Fi(2)26935 6665 y Fo(\),)433 b(cf.)i([)p 0 .5 0 TeXcolorrgb 30367 6665 a SDict begin H.S end 30367 6665 a 0 .5 0 TeXcolorrgb Fo(17)p 0 .5 0 TeXcolorrgb 31668 5809 a SDict begin H.R end 31668 5809 a 31668 6665 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Tot1989) cvn H.B /ANN pdfmark end 31668 6665 a Black Fo(,)e(Lemma)h(2.2].)2751 8270 y(No)-36 b(w)450 b(let)f Fn(E)8586 8469 y Fk(k)9155 8270 y Fo(\()p Fn(n)p Fo(\))395 b(=)g Fn(n)13521 7788 y Fk(k)14539 8270 y Fo(and)449 b Fn(\036)17854 7788 y Fi(2)18379 8270 y Fn(\036)19149 7788 y Fh(\003)20070 8270 y Fo(=)395 b Fn(f)448 b Fm(\003)306 b Fn(E)24499 8469 y Fi(3)25473 8270 y Fo(in)449 b(terms)g(of)h(the)e(Diric)-36 b(hlet)450 b(con)-36 b(v)g(olution.)625 b(Th)-36 b(us)448 b Fn(f)537 b Fo(=)800 9875 y Fn(\036)1570 9393 y Fi(2)2096 9875 y Fn(\036)2866 9393 y Fh(\003)3550 9875 y Fm(\003)158 b Fn(\026E)6118 10074 y Fi(3)6643 9875 y Fo(.)556 b(Here)367 b Fn(f)508 b Fo(is)367 b(m)-36 b(ultiplicativ)g(e)368 b(and)e(direct)g(computations)g(sho)-36 b(w)367 b(that)f Fn(f)142 b Fo(\()p Fn(p)p Fo(\))369 b(=)f Fm(\000)p Fo(3)p Fn(p)47642 9393 y Fi(2)48327 9875 y Fo(+)158 b(3)p Fn(p)g Fm(\000)g Fo(1)800 11480 y(and)377 b Fn(f)142 b Fo(\()p Fn(p)5215 10998 y Fk(\027)5791 11480 y Fo(\))368 b(=)h Fn(p)8699 10998 y Fi(2)p Fk(\027)58 b Fi(+1)10947 11480 y Fo(\(1)181 b Fm(\000)g Fo(1)p Fn(=p)p Fo(\))15957 10998 y Fi(3)16862 11480 y Fo(for)378 b(ev)-36 b(ery)378 b(prime)g Fn(p)f Fo(and)h(ev)-36 b(ery)378 b Fn(\027)452 b Fm(\025)370 b Fo(2.)560 b(W)-108 b(e)377 b(obtain)h(that)f(the)h(Diric)-36 b(hlet)800 13085 y(series)24801 13710 y Fh(1)24312 14109 y Fj(X)24384 16898 y Fk(n)p Fi(=1)26585 14472 y Fn(f)142 b Fo(\()p Fn(n)p Fo(\))p 26585 15065 2571 54 v 27237 16282 a Fn(n)28013 15898 y Fk(s)800 18716 y Fo(is)434 b(absolutely)g(con)-36 b(v)g(ergen)g(t)434 b(for)g(Re)221 b Fn(s)369 b(>)g Fo(3)434 b(and)f(b)-36 b(y)433 b Fn(\036)27092 18234 y Fi(2)27618 18716 y Fn(\036)28388 18234 y Fh(\003)28913 18716 y Fn(=E)30526 18915 y Fi(4)31421 18716 y Fo(=)369 b Fn(f)142 b(=E)35198 18915 y Fi(4)36019 18716 y Fm(\003)295 b Fn(E)37941 18915 y Fh(\000)p Fi(1)39199 18716 y Fo(,)8708 20715 y Fj(X)8751 23521 y Fk(n)p Fh(\024)p Fk(x)10981 21079 y Fn(\036)11751 20597 y Fi(2)12276 21079 y Fo(\()p Fn(n)p Fo(\))p Fn(\036)14834 20597 y Fh(\003)15360 21079 y Fo(\()p Fn(n)p Fo(\))p 10981 21672 6168 54 v 13413 22888 a Fn(n)14189 22505 y Fi(4)17650 21977 y Fo(=)19030 20715 y Fj(X)19117 23544 y Fk(d)p Fh(\024)p Fk(x)21303 21079 y Fn(f)142 b Fo(\()p Fn(d)p Fo(\))p 21303 21672 2471 54 v 21938 22888 a Fn(d)22614 22505 y Fi(4)24495 20715 y Fj(X)24128 23593 y Fk(e)p Fh(\024)p Fk(x=d)27135 21079 y Fo(1)p 27135 21672 651 54 v 27159 22888 a Fn(e)28287 21977 y Fo(=)29668 20715 y Fj(X)29754 23544 y Fk(d)p Fh(\024)p Fk(x)31941 21079 y Fn(f)g Fo(\()p Fn(d)p Fo(\))p 31941 21672 2471 54 v 32575 22888 a Fn(d)33251 22505 y Fi(4)34765 21977 y Fo(\(log)19 b(\()p Fn(x=d)p Fo(\))295 b(+)g Fm(O)37 b Fo(\(1\)\))3582 27250 y(=)369 b(log)240 b Fn(x)8313 25589 y Fh(1)7825 25988 y Fj(X)7941 28817 y Fk(d)p Fi(=1)10097 26351 y Fn(f)142 b Fo(\()p Fn(d)p Fo(\))p 10097 26944 V 10732 28161 a Fn(d)11408 27777 y Fi(4)12996 27250 y Fo(+)295 b Fm(O)15619 24978 y Fj( )16670 27250 y Fo(log)241 b Fn(x)19532 25988 y Fj(X)19619 28817 y Fk(d>x)21805 26351 y Fm(j)p Fn(f)142 b Fo(\()p Fn(d)p Fo(\))p Fm(j)p 21805 26944 3209 54 v 22808 28161 a Fn(d)23484 27777 y Fi(4)25146 24978 y Fj(!)26493 27250 y Fo(+)295 b Fm(O)29116 24978 y Fj( )30167 25988 y(X)30254 28817 y Fk(d)p Fh(\024)p Fk(x)32440 26351 y Fm(j)p Fn(f)142 b Fo(\()p Fn(d)p Fo(\))p Fm(j)221 b Fo(log)240 b Fn(d)p 32440 26944 6008 54 v 34843 28161 a(d)35519 27777 y Fi(4)38580 24978 y Fj(!)40001 27250 y Fo(=)368 b Fn(B)289 b Fo(log)240 b Fn(x)296 b Fo(+)e Fm(O)37 b Fo(\(1\))p Fn(;)800 30654 y Fo(therefore)6256 29049 y SDict begin H.S end 6256 29049 a 6256 29049 a SDict begin 14.5 H.A end 6256 29049 a 6256 29049 a SDict begin [ /View [/XYZ H.V] /Dest (equation.31) cvn H.B /DEST pdfmark end 6256 29049 a 17320 31454 a Fj(X)17364 34260 y Fk(n)p Fh(\024)p Fk(x)19593 31817 y Fn(\036)20363 31335 y Fi(2)20889 31817 y Fo(\()p Fn(n)p Fo(\))p Fn(\036)23447 31335 y Fh(\003)23973 31817 y Fo(\()p Fn(n)p Fo(\))p 19593 32410 6168 54 v 22026 33627 a Fn(n)22802 33243 y Fi(4)26262 32716 y Fo(=)369 b Fn(B)289 b Fo(log)240 b Fn(x)295 b Fo(+)g Fm(O)37 b Fo(\(1\))p Fn(;)14208 b Fo(\(31\))800 36097 y(b)-36 b(y)434 b(the)f(Euler)g(pro)36 b(duct)432 b(form)-36 b(ula.)579 b(This)434 b(completes)g(the)f(pro)36 b(of.)p 51860 36097 45 878 v 51905 35263 781 45 v 51905 36097 V 52684 36097 45 878 v 800 37220 a SDict begin H.S end 800 37220 a 800 37220 a SDict begin 14.5 H.A end 800 37220 a 800 37220 a SDict begin [ /View [/XYZ H.V] /Dest (rem.1) cvn H.B /DEST pdfmark end 800 37220 a Black 1193 x Fd(Remark)677 b(1.)p Black 633 w Fo(The)589 b(asymptotic)g(form) -36 b(ula)589 b(\()p 0 .5 0 TeXcolorrgb 23544 38413 a SDict begin H.S end 23544 38413 a 0 .5 0 TeXcolorrgb Fo(31)p 0 .5 0 TeXcolorrgb 24845 37557 a SDict begin H.R end 24845 37557 a 24845 38413 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.31) cvn H.B /ANN pdfmark end 24845 38413 a Black Fo(\))f(can)h(b)36 b(e)588 b(obtained)h(b)-36 b(y)588 b(using)h(kno)-36 b(wn)589 b(theorems)f(on)800 40018 y(mean)541 b(v)-72 b(alues)542 b(of)g(certain)f(m)-36 b(ultiplicativ)g(e)542 b(functions.)901 b(The)542 b(follo)-36 b(wing)543 b(useful)e(theorem,)568 b(whic)-36 b(h)541 b(is)h(a)800 41624 y(particular)434 b(case)g(of)g(a)g(more)f(general)i (result,)e(has)h(b)36 b(een)433 b(pro)-36 b(v)g(ed)433 b(in)g([)p 0 .5 0 TeXcolorrgb 35325 41624 a SDict begin H.S end 35325 41624 a 0 .5 0 TeXcolorrgb Fo(7)p 0 .5 0 TeXcolorrgb 35976 40768 a SDict begin H.R end 35976 40768 a 35976 41624 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Mar2002) cvn H.B /ANN pdfmark end 35976 41624 a Black Fo(])h(using)f(Iw)-36 b(aniec's)435 b(ideas.)2751 43229 y Fg(The)-66 b(or)g(em)455 b Fo([)p 0 .5 0 TeXcolorrgb 8500 43229 a SDict begin H.S end 8500 43229 a 0 .5 0 TeXcolorrgb Fo(7)p 0 .5 0 TeXcolorrgb 9151 42373 a SDict begin H.R end 9151 42373 a 9151 43229 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Mar2002) cvn H.B /ANN pdfmark end 9151 43229 a Black Fo(,)427 b(Prop)36 b(osition)425 b(A.3])p Fg(.)595 b(L)-66 b(et)456 b(f)g(b)-66 b(e)456 b(any)h(nonne)-66 b(gative)454 b(multiplic)-66 b(ative)454 b(function)h(such)i(that)800 44834 y Fn(f)142 b Fo(\()p Fn(n)p Fo(\))369 b Fm(\034)g Fn(n)6213 44352 y Fk(\013)7338 44834 y Fg(for)465 b(some)g Fn(\013)378 b(<)368 b Fo(1)p Fn(=)p Fo(2)466 b Fg(and)f(satisfying) 14911 46776 y Fj(X)15003 49582 y Fk(p)p Fh(\024)p Fk(x)17184 47140 y Fn(f)142 b Fo(\()p Fn(p)p Fo(\))221 b(log)241 b Fn(p)p 17184 47733 5223 54 v 19469 48949 a(p)22909 48038 y Fo(=)369 b Fn(\024)221 b Fo(log)240 b Fn(x)296 b Fo(+)e Fm(O)30560 48237 y Fk(f)31165 48038 y Fo(\(1\))1328 b(\()p Fn(x)369 b Fm(\025)g Fo(2\))p Fn(;)800 51787 y Fg(wher)-66 b(e)465 b Fn(\024)368 b Fo(=)h Fn(\024)7717 51986 y Fk(f)8691 51787 y Fn(>)f Fo(0)p Fg(.)598 b(Then)464 b(we)h(have)g(uniformly)f(for)h(al)66 b(l)465 b Fn(x)369 b Fm(\025)h Fo(2)p Fg(,)33907 50182 y SDict begin H.S end 33907 50182 a 33907 50182 a SDict begin 14.5 H.A end 33907 50182 a 33907 50182 a SDict begin [ /View [/XYZ H.V] /Dest (equation.32) cvn H.B /DEST pdfmark end 33907 50182 a 14792 53769 a Fj(X)14835 56575 y Fk(n)p Fh(\024)p Fk(x)17064 54132 y Fn(f)142 b Fo(\()p Fn(n)p Fo(\))p 17064 54726 2571 54 v 17962 55942 a Fn(n)20137 55031 y Fo(=)369 b Fm(M)23113 55230 y Fk(f)45 b(;\024)24469 55031 y Fo(\(log)241 b Fn(x)p Fo(\))28122 54483 y Fk(\024)29014 55031 y Fo(+)295 b Fm(O)31379 55230 y Fk(f)31984 55031 y Fo(\(\(log)240 b Fn(x)p Fo(\))36142 54483 y Fk(\024)p Fh(\000)p Fi(1)37941 55031 y Fo(\))p Fn(;)11680 b Fo(\(32\))800 58757 y Fg(wher)-66 b(e)12486 61193 y Fm(M)14081 61392 y Fk(f)45 b(;\024)15807 61193 y Fo(:=)19769 60294 y(1)p 17681 60887 4826 54 v 17681 62104 a(\000\()p Fn(\024)295 b Fo(+)g(1\))22861 59931 y Fj(Y)23473 62720 y Fk(p)24780 59320 y Fj(\022)25758 61193 y Fo(1)g Fm(\000)28166 60294 y Fo(1)p 28164 60887 653 54 v 28164 62104 a Fn(p)28950 59320 y Fj(\023)29928 59617 y Fk(\024)30747 58921 y Fj( )31798 61193 y Fo(1)h(+)34540 59532 y Fh(1)34051 59931 y Fj(X)34149 62720 y Fk(\027)58 b Fi(=1)36324 60294 y Fn(f)142 b Fo(\()p Fn(p)38266 59812 y Fk(\027)38841 60294 y Fo(\))p 36324 60887 3024 54 v 37221 62104 a Fn(p)37874 61720 y Fk(\027)39480 58921 y Fj(!)40753 61193 y Fn(:)2751 65263 y Fo(No)-36 b(w)489 b(\()p 0 .5 0 TeXcolorrgb 6274 65263 a SDict begin H.S end 6274 65263 a 0 .5 0 TeXcolorrgb Fo(31)p 0 .5 0 TeXcolorrgb 7574 64407 a SDict begin H.R end 7574 64407 a 7574 65263 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.31) cvn H.B /ANN pdfmark end 7574 65263 a Black Fo(\))470 b(follo)-36 b(ws)471 b(at)f(once)g(b)-36 b(y)470 b(applying)488 b(\()p 0 .5 0 TeXcolorrgb 25195 65263 a SDict begin H.S end 25195 65263 a 0 .5 0 TeXcolorrgb Fo(32)p 0 .5 0 TeXcolorrgb 26496 64407 a SDict begin H.R end 26496 64407 a 26496 65263 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.32) cvn H.B /ANN pdfmark end 26496 65263 a Black Fo(\))469 b(to)h Fn(f)142 b Fo(\()p Fn(n)p Fo(\))431 b(=)33674 64630 y Fk(\036)34233 64317 y Ff(2)34695 64630 y Fi(\()p Fk(n)p Fi(\))p Fk(\036)36557 64317 y Fc(\003)37036 64630 y Fi(\()p Fk(n)p Fi(\))p 33674 64957 4666 54 v 35490 65721 a Fk(n)36061 65469 y Ff(3)38942 65263 y Fo(with)470 b Fn(\024)430 b Fo(=)g(1)470 b(and)g(noticing)800 66868 y(that)433 b Fn(f)142 b Fo(\()p Fn(p)p Fo(\))368 b(=)h(\(1)295 b Fm(\000)h Fo(1)p Fn(=p)p Fo(\))13054 66386 y Fi(3)13949 66868 y Fo(=)368 b(1)296 b(+)f Fm(O)37 b Fo(\(1)p Fn(=p)p Fo(\).)p Black 26475 70521 a(7)p Black eop end end %%Page: 8 8 TeXDict begin HPSdict begin 8 7 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a 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31607 a Black 1864 x Fo([19])p Black 652 w(L.)315 b(T\023)-650 b(oth,)339 b(A)315 b(generalization)i(of)f(Pillai's)h(arithmetical)e (function)g(in)-36 b(v)g(olving)317 b(regular)e(con)-36 b(v)g(olutions,)3474 35076 y Fg(A)-66 b(cta)463 b(Math.)i(Inform.)e (Univ.)f(Ostr)-66 b(aviensis)433 b Fd(6)h Fo(\(1998\),)g(203{217,)j(a) -36 b(v)-72 b(ailable)435 b(at)p 0 1 0 0 TeXcolorcmyk 3474 36976 a SDict begin H.S end 3474 36976 a 0 1 0 0 TeXcolorcmyk -295 x Fa(http://www.ttk.pte.hu/matek/ltoth/TothGeneralizationPillai)o(\(19)o (98)o(\).)o(pd)o(f)p 0 1 0 0 TeXcolorcmyk 49955 35758 a SDict begin H.R end 49955 35758 a 49955 36681 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.ttk.pte.hu/matek/ltoth/TothGeneralizationPillai(1998).pdf) >> /Subtype /Link H.B /ANN pdfmark end 49955 36681 a Black 433 w Fo(.)800 37529 y SDict begin H.S end 800 37529 a 800 37529 a SDict begin 14.5 H.A end 800 37529 a 800 37529 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Tot2001) cvn H.B /DEST pdfmark end 800 37529 a Black 1864 x Fo([20])p Black 652 w(L.)456 b(T\023)-650 b(oth,)462 b(On)455 b(the)g(asymptotic)i (densities)f(of)h(certain)f(subsets)f(of)i Fl(N)38072 38911 y Fk(k)38641 39393 y Fo(,)462 b Fg(R)-33 b(iv.)484 b(Mat.)g(Univ.)g(Parma)3474 40998 y Fo(\(6\))433 b Fd(4)h Fo(\(2001\),)h(121{131,)h(a)-36 b(v)-72 b(ailable)435 b(at)p 0 1 0 0 TeXcolorcmyk 23442 41293 a SDict begin H.S end 23442 41293 a 0 1 0 0 TeXcolorcmyk -295 x Fa(http://arxiv.org/abs/math.NT/0610582)p 0 1 0 0 TeXcolorcmyk 48049 40075 a SDict begin H.R end 48049 40075 a 48049 40998 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://arxiv.org/abs/math.NT/0610582) >> /Subtype /Link H.B /ANN pdfmark end 48049 40998 a Black 434 w Fo(.)p 800 44142 52000 45 v 800 46393 a(2000)g Fg(Mathematics)464 b(Subje)-66 b(ct)463 b(Classi\014c)-66 b(ation)p Fo(:)577 b(Primary)434 b(11A25;)h(Secondary)e(11N37,)j(05A15.)800 47998 y Fg(Keywor)-66 b(ds:)777 b Fo(Euler's)496 b(arithmetical)g(function,)512 b(M\177)-650 b(obius)495 b(function,)512 b(divisor)496 b(function,)512 b(gcd-sum)495 b(func-)800 49603 y(tion,)434 b(unitary)f(divisor,)i(a)-36 b(v)g(erage)435 b(order.)p 800 51234 V 800 53558 a(\(Concerned)440 b(with)g(sequences)p 0 1 0 0 TeXcolorcmyk 16623 53824 a SDict begin H.S end 16623 53824 a 0 1 0 0 TeXcolorcmyk -266 x Fo(A000010)p 16623 53771 4878 54 v 0 1 0 0 TeXcolorcmyk 21500 52651 a SDict begin H.R end 21500 52651 a 21500 53558 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A000010) >> /Subtype /Link H.B /ANN pdfmark end 21500 53558 a Black Fo(,)p 0 1 0 0 TeXcolorcmyk 22303 53824 a SDict begin H.S end 22303 53824 a 0 1 0 0 TeXcolorcmyk -266 x Fo(A001221)p 22303 53771 4878 54 v 0 1 0 0 TeXcolorcmyk 27180 52651 a SDict begin H.R end 27180 52651 a 27180 53558 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A001221) >> /Subtype /Link H.B /ANN pdfmark end 27180 53558 a Black Fo(,)p 0 1 0 0 TeXcolorcmyk 27984 53824 a SDict begin H.S end 27984 53824 a 0 1 0 0 TeXcolorcmyk -266 x Fo(A005361)p 27984 53771 4878 54 v 0 1 0 0 TeXcolorcmyk 32861 52651 a SDict begin H.R end 32861 52651 a 32861 53558 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A005361) >> /Subtype /Link H.B /ANN pdfmark end 32861 53558 a Black Fo(,)p 0 1 0 0 TeXcolorcmyk 33664 53824 a SDict begin H.S end 33664 53824 a 0 1 0 0 TeXcolorcmyk -266 x Fo(A008683)p 33664 53771 4878 54 v 0 1 0 0 TeXcolorcmyk 38541 52651 a SDict begin H.R end 38541 52651 a 38541 53558 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A008683) >> /Subtype /Link H.B /ANN pdfmark end 38541 53558 a Black Fo(,)p 0 1 0 0 TeXcolorcmyk 39344 53824 a SDict begin H.S end 39344 53824 a 0 1 0 0 TeXcolorcmyk -266 x Fo(A018804)p 39344 53771 4878 54 v 0 1 0 0 TeXcolorcmyk 44221 52651 a SDict begin H.R end 44221 52651 a 44221 53558 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A018804) >> /Subtype /Link H.B /ANN pdfmark end 44221 53558 a Black Fo(,)p 0 1 0 0 TeXcolorcmyk 45024 53824 a SDict begin H.S end 45024 53824 a 0 1 0 0 TeXcolorcmyk -266 x Fo(A047994)p 45024 53771 4878 54 v 0 1 0 0 TeXcolorcmyk 49901 52651 a SDict begin H.R end 49901 52651 a 49901 53558 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A047994) >> /Subtype /Link H.B /ANN pdfmark end 49901 53558 a Black Fo(,)j(and)p 0 1 0 0 TeXcolorcmyk 800 55429 a SDict begin H.S end 800 55429 a 0 1 0 0 TeXcolorcmyk -266 x Fo(A145388)p 800 55376 4878 54 v 0 1 0 0 TeXcolorcmyk 5677 54256 a SDict begin H.R end 5677 54256 a 5677 55163 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A145388) >> /Subtype /Link H.B /ANN pdfmark end 5677 55163 a Black Fo(.\))p 800 56868 52000 45 v 800 59919 a(Receiv)-36 b(ed)597 b(April)f(24)h(2009;)679 b(revised)596 b(v)-36 b(ersion)597 b(receiv)-36 b(ed)597 b(June)e(22)i(2009.)1067 b(Published)596 b(in)g Fg(Journal)614 b(of)800 61524 y(Inte)-66 b(ger)463 b(Se)-66 b(quenc)g(es)p Fo(,)431 b(June)i(23)i(2009.)p 800 63155 V 800 65406 a(Return)e(to)p 0 1 0 0 TeXcolorcmyk 6815 65664 a SDict begin H.S end 6815 65664 a 0 1 0 0 TeXcolorcmyk -258 x Fo(Journal)h(of)g(In)-36 b(teger)434 b(Sequences)f(home)g(page)p 0 1 0 0 TeXcolorcmyk 29602 64483 a SDict begin H.R end 29602 64483 a 29602 65406 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.cs.uwaterloo.ca/journals/JIS/) >> /Subtype /Link H.B /ANN pdfmark end 29602 65406 a Black Fo(.)p Black 26150 70521 a(10)p Black eop end end %%Trailer end