Hardy Ramanujan Journal |

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 ErdÃ¶s 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

Source : oai:HAL:hal-01104372v1

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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