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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

$\displaystyle \sum\limits_{n \leqslant \surd x} {\left\{ {{\psi ^2}\left( {\tfr... ...2}\left( {\tfrac{x}{n}} \right) - \tfrac{1}{{12}}} \right\} = o({x^{1/4}}).} } $


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

  • [1] S. Chowla and H. Walum, On the divisor problem, Norske Vid. Selsk. Forh. (Trondheim) 36 (1963), 127–134. MR 0160761
  • [2] 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
  • [3] G. A. Kolesnik, An estimate for certain trigonometric sums, Acta Arith. 25 (1973/74), 7–30. (errata insert) (Russian). MR 0332676
  • [4] Sanford L. Segal, A note on the average order of number-theoretic error terms, Duke Math. J. 32 (1965), 279–284. MR 0179140

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0401682-7
Keywords: Dirichlet's divisor problem, average order of number-theoretic error terms
Article copyright: © Copyright 1976 American Mathematical Society