On Residues of Rounded Shifted Fractions with a Common Numerator

The proportion of integers

$\displaystyle \lfloor n/1 \rfloor , \lfloor n/2 \rfloor , \dots, \lfloor n/n \rfloor$

that are odd is asymptotically $\log2$ . If instead of the floor function, one uses the nearest-integer function, the proportion drops to $\pi/2-1$ . In this work, we prove these facts and, more generally, give an integral formula for the proportion of

$\displaystyle \lfloor (n-\nu)/1+\alpha \rfloor ,\, \lfloor (n-\nu)/2+\alpha \rfloor ,\, \dots,\, \lfloor (n-\nu)/n+\alpha \rfloor$

that are congruent to $r$

modulo

. We conclude by linking this problem to the Dirichlet divisor and Gauss circle problems, and counting lattice points in other quadratic regions.

On Residues of Rounded Shifted Fractions with a Common Numerator

Nicholas Dent Department of Mathematics Boston University Academy Boston, MA 02215 USA Caleb M. Shor Department of Mathematics Western New England University Springfield, MA 01119 USA

Nicholas Dent
Department of Mathematics
Boston University Academy
Boston, MA 02215
USA
Caleb M. Shor
Department of Mathematics
Western New England University
Springfield, MA 01119
USA