\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand*\HyPL@Entry[1]{}
\citation{Hardy}
\citation{Hardy}
\citation{Tenenbaum}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\newlabel{E1}{{1}{2}{Introduction}{equation.1.1}{}}
\newlabel{E1@cref}{{[section][1][]1}{2}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Main Results}{2}{section.2}}
\newlabel{T1}{{1}{2}{}{theorem.1}{}}
\newlabel{T1@cref}{{[theorem][1][]1}{2}}
\newlabel{E2}{{2}{2}{}{equation.2.2}{}}
\newlabel{E2@cref}{{[theorem][1][]1}{2}}
\newlabel{E3}{{3}{2}{}{equation.2.3}{}}
\newlabel{E3@cref}{{[theorem][1][]1}{2}}
\newlabel{E4}{{4}{2}{Main Results}{equation.2.4}{}}
\newlabel{E4@cref}{{[section][2][]2}{2}}
\newlabel{E5}{{5}{3}{Main Results}{equation.2.5}{}}
\newlabel{E5@cref}{{[section][2][]2}{3}}
\newlabel{E6}{{6}{3}{Main Results}{equation.2.6}{}}
\newlabel{E6@cref}{{[section][2][]2}{3}}
\newlabel{E7}{{7}{3}{Main Results}{equation.2.7}{}}
\newlabel{E7@cref}{{[section][2][]2}{3}}
\newlabel{E8}{{8}{3}{Main Results}{equation.2.8}{}}
\newlabel{E8@cref}{{[section][2][]2}{3}}
\newlabel{E9}{{9}{3}{Main Results}{equation.2.9}{}}
\newlabel{E9@cref}{{[section][2][]2}{3}}
\newlabel{E10}{{10}{4}{Main Results}{equation.2.10}{}}
\newlabel{E10@cref}{{[section][2][]2}{4}}
\newlabel{E11}{{11}{4}{Main Results}{equation.2.11}{}}
\newlabel{E11@cref}{{[section][2][]2}{4}}
\newlabel{E12}{{12}{4}{Main Results}{equation.2.12}{}}
\newlabel{E12@cref}{{[section][2][]2}{4}}
\newlabel{E13}{{13}{4}{Main Results}{equation.2.13}{}}
\newlabel{E13@cref}{{[section][2][]2}{4}}
\newlabel{E14}{{14}{4}{Main Results}{equation.2.14}{}}
\newlabel{E14@cref}{{[section][2][]2}{4}}
\newlabel{E15}{{15}{4}{Main Results}{equation.2.15}{}}
\newlabel{E15@cref}{{[section][2][]2}{4}}
\newlabel{E16}{{16}{4}{Main Results}{equation.2.16}{}}
\newlabel{E16@cref}{{[section][2][]2}{4}}
\citation{Srivastava}
\newlabel{E17}{{17}{5}{Main Results}{equation.2.17}{}}
\newlabel{E17@cref}{{[section][2][]2}{5}}
\newlabel{E18}{{18}{5}{Main Results}{equation.2.18}{}}
\newlabel{E18@cref}{{[section][2][]2}{5}}
\newlabel{E19}{{19}{5}{Main Results}{equation.2.19}{}}
\newlabel{E19@cref}{{[section][2][]2}{5}}
\newlabel{T3}{{3}{5}{}{theorem.3}{}}
\newlabel{T3@cref}{{[theorem][3][]3}{5}}
\citation{Bordelles}
\newlabel{T4}{{4}{6}{}{theorem.4}{}}
\newlabel{T4@cref}{{[theorem][4][]4}{6}}
\citation{Glaisher}
\bibcite{Bordelles}{1}
\bibcite{Glaisher}{2}
\bibcite{Hardy}{3}
\bibcite{Srivastava}{4}
\bibcite{Tenenbaum}{5}
\newlabel{R5}{{5}{7}{}{theorem.5}{}}
\newlabel{R5@cref}{{[theorem][5][]5}{7}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Acknowledgments}{7}{section.3}}
