On $\sum _{n}\leq _{x}(\sigma ^{\ast } (n))$ and $\sum _{n}\leq _{x}(\phi ^{\ast } (n))$
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- by R. Sitaramachandrarao and D. Suryanarayana PDF
- Proc. Amer. Math. Soc. 41 (1973), 61-66 Request permission
Abstract:
Let ${\sigma ^ \ast }(n)$ and ${\varphi ^ \ast }(n)$ be the unitary analogues of $\sigma (n)$ and $\varphi (n)$ respectively. It is known that $E(x) = \sum \nolimits _{n \leqq x} {{\sigma ^ \ast }} (n) - ({\pi ^2}{x^2}/12\zeta (3)) = O(x{\log ^2}x)$ and \[ F(x) = \sum \limits _{n \leqq x} {{\varphi ^ \ast }(n) - \tfrac {1}{2}\alpha {x^2} = O(x{{\log }^2}x),} \] where $\alpha$ is a positive constant. In this paper we improve the order estimates of $E(x)$ and $F(x)$ to $E(x) = O(x{\log ^{5/3}}x)$ and \[ F(x) = O(x{\log ^{5/3}}x{(\log \log x)^{4/3}}).\]References
- Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Z. 74 (1960), 66–80. MR 112861, DOI 10.1007/BF01180473 G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, London, 1960.
- D. Suryanarayana, The number of $k$-ary divisors of an integer, Monatsh. Math. 72 (1968), 445–450. MR 236130, DOI 10.1007/BF01300368
- Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 (German). MR 0220685
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 41 (1973), 61-66
- MSC: Primary 10H25; Secondary 10A20
- DOI: https://doi.org/10.1090/S0002-9939-1973-0319922-9
- MathSciNet review: 0319922