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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a conjecture of S. Chowla
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by D. Suryanarayana PDF
Proc. Amer. Math. Soc. 56 (1976), 27-33 Request permission

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
  • 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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Additional Information
  • © Copyright 1976 American Mathematical Society
  • 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