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An $\Omega$-result related to $r_4(n)$.

Abstract : 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 Erd\"os 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
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Sukumar Das Adhikari, R Balasubramanian, A Sankaranarayanan. An $\Omega$-result related to $r_4(n)$.. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1989, Volume 12 - 1989, pp.29 - 30. ⟨10.46298/hrj.1989.113⟩. ⟨hal-01104372⟩

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