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Robin’s theorem, primes, and a new elementary reformulation of the Riemann hypothesis. (English) Zbl 1235.11082

There are equivalent formulations of the Riemann hypothesis (RH) by G. Robin (1984, using Euler-constant $\gamma$) and J. C. Lagarias [Am. Math. Mon. 109, No. 6, 534–543 (2002; Zbl 1098.11005), using harmonic numbers). The authors give another elementary one, using Gronwall’s function $G\left(n\right)=\frac{\sigma \left(n\right)}{nloglogn}$ $\left(n>1\right)$: RH is true if and only if $n=4$ is the only composite number with the two properties:

(i) $G\left(n\right)\ge G\left(\frac{n}{p}\right)$ for every prime factor $p$ of $n$;

(ii) $G\left(n\right)\ge G\left(an\right)$ for every positive integer $a$.

The proof is elementary and uses the results of Gronwall and Robin.

##### MSC:
 11M26 Nonreal zeros of $\zeta \left(s\right)$ and $L\left(s,\chi \right)$; Riemann and other hypotheses 11N64 Characterization of arithmetic functions 11Y55 Calculation of integer sequences
##### Keywords:
Riemann hypothesis; Gronwall function; Robin’s theorem