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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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 \[ \sum \limits _{n \leqslant \surd x} {\left \{ {{\psi ^2}\left ( {\tfrac {x}{n}} \right ) - \tfrac {1}{{12}}} \right \} + \tfrac {1}{x}\sum \limits _{n \leqslant \surd x} {{n^2}\left \{ {{\psi ^2}\left ( {\tfrac {x}{n}} \right ) - \tfrac {1}{{12}}} \right \} = o({x^{1/4}}).} } \]


References [Enhancements On Off] (What's this?)

  • S. Chowla and H. Walum, On the divisor problem, Norske Vid. Selsk. Forh. (Trondheim) 36 (1963), 127–134. MR 160761
  • S. Chowla, The Riemann hypothesis and Hilbert’s tenth problem, Mathematics and its Applications, Vol. 4, Gordon and Breach Science Publishers, New York-London-Paris, 1965. MR 0177943
  • G. A. Kolesnik, An estimate for certain trigonometric sums, Acta Arith. 25 (1973/74), 7–30. (errata insert) (Russian). MR 332676
  • Sanford L. Segal, A note on the average order of number-theoretic error terms, Duke Math. J. 32 (1965), 279–284. MR 179140

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Keywords: Dirichlet’s divisor problem, average order of number-theoretic error terms
Article copyright: © Copyright 1976 American Mathematical Society