×

zbMATH — the first resource for mathematics

Prime numbers, sums of divisors and the Riemann Hypothesis. (Primzahlen, Teilersummen und die Riemannsche Vermutung.) (German) Zbl 0982.11052
One of the central problems in number theory is the location of the complex zeros of the Riemann zeta-function \(\zeta(s)\). The famous yet unproved Riemann hypothesis states that all complex zeros of \(\zeta(s)\) lie on the so-called critical line \({\mathfrak R}s=1/2\). Recently, J. C. Lagarias [An elementary problem equivalent to the Riemann hypothesis, preprint (2000), arXiv:math.NT/0008177] discovered that Riemann’s hypothesis is true if, and only if, for all positive integers \(n\) the inequality \[ \sigma(n)\leq h(n)+\exp(h(n))\log(h(n)) \] holds; here \(\sigma(n):=\sum_{d|n}d\) is the sum of divisors function and \(h(n):=\sum_{j=1}^n{1\over j}\) is the \(n\)th harmonic number. First steps towards Lagarias’ result were done by G. Robin [J. Math. Pures Appl., IX. Sér. 63, 187-213 (1984; Zbl 0539.10036, resp. Zbl 0516.10036)], who showed that Riemann’s hypothesis is equivalent to the inequality \[ \sigma(n)<e^\gamma n\log\log n\qquad\text{for all}\quad n\geq 5041, \] where \(\gamma\) is the Euler-Mascheroni constant. In the nicely written paper under review the author sketches the main ideas of Robin and Lagarias as well as the relation between prime number distribution and the zeros of \(\zeta(s)\).

MSC:
11N37 Asymptotic results on arithmetic functions
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
PDF BibTeX XML Cite
Full Text: DOI