Abstract
AbstractLet $$\sigma (n)$$
σ
(
n
)
denote the sum of divisors function of a positive integer n. Robin proved that the Riemann hypothesis is true if and only if the inequality $$\sigma (n) < \textrm{e}^{\gamma }n \log \log n$$
σ
(
n
)
<
e
γ
n
log
log
n
holds for every integer $$n > 5040$$
n
>
5040
, where $$\gamma $$
γ
is the Euler–Mascheroni constant. In this paper we establish a new family of integers for which Robin’s inequality $$\sigma (n) < \textrm{e}^{\gamma }n \log \log n$$
σ
(
n
)
<
e
γ
n
log
log
n
hold. Further, we establish a new unconditional upper bound for the sum of divisors function. For this purpose, we use an approximation for Chebyshev’s $$\vartheta $$
ϑ
-function and for some product defined over prime numbers.
Funder
Heinrich-Heine-Universität Düsseldorf
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
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