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Proceedings of the American Mathematical Society

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

 

 

On $ \sum \sb{n}\leq \sb{x}(\sigma \sp{\ast} (n))$ and $ \sum \sb{n}\leq \sb{x}(\phi\sp{\ast} (n))$


Authors: R. Sitaramachandrarao and D. Suryanarayana
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
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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

$\displaystyle 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

$\displaystyle F(x) = O(x{\log ^{5/3}}x{(\log \log x)^{4/3}}).$


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DOI: https://doi.org/10.1090/S0002-9939-1973-0319922-9
Keywords: Unitary divisor, $ a$ is semiprime to $ b$, abscissa of absolute convergence, partial summation
Article copyright: © Copyright 1973 American Mathematical Society