On a conjecture of S. Chowla

Author:
D. Suryanarayana

Journal:
Proc. Amer. Math. Soc. **56** (1976), 27-33

MSC:
Primary 10H25

DOI:
https://doi.org/10.1090/S0002-9939-1976-0401682-7

MathSciNet review:
0401682

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Abstract: Let $\psi (x) = x - [x] - \tfrac {1}{2}$. It has been conjectured by S. Chowla that ${\Sigma _{n \leqslant \surd x}}\{ {\psi ^2}(x/n) - 1/12\} = o({x^{1/4 + \epsilon }})$, for every $\epsilon > 0$. In this paper we show that this conjecture is equivalent to ${\Sigma _{n \leqslant \surd x}}{n^2}\{ {\psi ^2}(x/n) - 1/12\}$ by proving that \[ \sum \limits _{n \leqslant \surd x} {\left \{ {{\psi ^2}\left ( {\tfrac {x}{n}} \right ) - \tfrac {1}{{12}}} \right \} + \tfrac {1}{x}\sum \limits _{n \leqslant \surd x} {{n^2}\left \{ {{\psi ^2}\left ( {\tfrac {x}{n}} \right ) - \tfrac {1}{{12}}} \right \} = o({x^{1/4}}).} } \]

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Keywords:
Dirichlet’s divisor problem,
average order of number-theoretic error terms

Article copyright:
© Copyright 1976
American Mathematical Society