Sukumar Das Adhikari ; R Balasubramanian ; A Sankaranarayanan - An $\Omega$-result related to $r_4(n)$.

hrj:113 - Hardy-Ramanujan Journal, January 1, 1989, Volume 12 -
An $\Omega$-result related to $r_4(n)$.

Authors: Sukumar Das Adhikari ; R Balasubramanian ; A Sankaranarayanan

Let $r_4(n)$ be the number of ways of writing $n$ as the sum of four squares. Set $P_4(x)= \sum \limits_{n\le x} r_4(n)-\frac {1}{2}\pi^2 x^2$, the error term for the average order of this arithmetical function. In this paper, following the ideas of Erdös and Shapiro, a new elementary method is developed which yields the slightly stronger result $P_4(x)= \Omega_{+}(x \log \log x)$. We also apply our method to give an upper bound for a quantity involving the Euler $\varphi$-function. This second result gives an elementary proof of a theorem of H. L. Montgomery

Volume: Volume 12
Published on: January 1, 1989
Submitted on: March 3, 2015
Keywords: Omega results of the error terms,arithmetical functions,[MATH] Mathematics [math]


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