Euler’s function on products of primes in a fixed arithmetic progression

Akbary, Amir; Francis, Forrest J.

doi:10.1090/mcom/3463

Search citation statements

Order By: Relevance

Paper Sections

Select...

Introduction and Statement Of Results1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2021

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

(1 citation statement)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As usual, let (p k ) k≥1 denote the increasing sequence of prime numbers, and let N k be the primorial integer of index k, the product of its k first terms. The Riemann Hypothesis (RH) claims that the nontrivial zeros of zeta function ζ(s) = n≥1 n −s are located on the critical line R(s) = 1 2 . Several equivalent formulations of RH appeared, but the one which interests us here is that in terms of arithmetic functions, here we cite the first papers of Gronwall [8], Nicolas [11] and Robin [13], followed by, for instance, Akbary [1], Caveney et al [6] and Lagarias [10].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On Robin’s criterion for the Riemann Hypothesis

Aoudjit¹,

Berkane²,

Dusart³

2021

NNTDM

View full text Add to dashboard Cite

Robin’s criterion says that the Riemann Hypothesis is equivalent to \[\forall n\geq 5041, \ \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n,\] where \sigma(n) is the sum of the divisors of n, \gamma represents the Euler–Mascheroni constant, and \log_i denotes the i-fold iterated logarithm. In this note we get the following better effective estimates: \begin{equation*} \forall n\geq3, \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n+\frac{0.3741}{\log_2^2n}. \end{equation*} The idea employed will lead us to a possible new reformulation of the Riemann Hypothesis in terms of arithmetic functions.

show abstract