10.46298/hrj.1989.113
https://hrj.episciences.org/113
Adhikari, Sukumar Das
Sukumar Das
Adhikari
Balasubramanian, R
R
Balasubramanian
Sankaranarayanan, A
A
Sankaranarayanan
An $\Omega$-result related to $r_4(n)$.
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
episciences.org
Omega results of the error terms
arithmetical functions
[MATH] Mathematics [math]
2015-06-12
1989-01-01
1989-01-01
en
journal article
https://hal.archives-ouvertes.fr/hal-01104372v1
2804-7370
https://hrj.episciences.org/113/pdf
VoR
application/pdf
Hardy-Ramanujan Journal
Volume 12 - 1989
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