# zbMATH — the first resource for mathematics

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 $$G(n)= {\sigma(n)\over n\log\log n}$$ $$(n> 1)$$: RH is true if and only if $$n= 4$$ is the only composite number with the two properties:
(i) $$G(n)\geq G({n\over p})$$ for every prime factor $$p$$ of $$n$$;
(ii) $$G(n)\geq G(an)$$ for every positive integer $$a$$.
The proof is elementary and uses the results of Gronwall and Robin.

##### MSC:
 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11N64 Other results on the distribution of values or the characterization of arithmetic functions 11Y55 Calculation of integer sequences
##### Keywords:
Riemann hypothesis; Gronwall function; Robin’s theorem
Full Text: