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Artin’s conjecture, Turing’s method, and the Riemann hypothesis. (English) Zbl 1169.11019

A finite group \(G\) is called almost monomial if for each irreducible representation \(\rho\), if \(\text{Tr}\,\rho=\chi_1+\chi_2\) for virtual characters \(\chi_i\), such that \(<\chi,\sigma>\geq 0\) for all monomial \(\sigma\), then either \(\chi_1=0\) or \(\chi_2=0\). The groups \(\mathrm{SL}_2({\mathbb F}_3)\), \(S_5\) and \(A_5\) are almost monomial, and \(\mathrm{GL}_2({\mathbb F}_3)\), \(\mathrm{SL}_2({\mathbb F}_5)\) are not. Under the working hypothesis that that the zeros of different irreducible Artin \(L\)-functions are distinct and simple, the author checks Artin’s conjecture for almost monomial Galois groups. He discusses two methods for locating zeros of arbitrary \(L\)-functions: the first uses the explicit formula; the second generalizes that of Turing for verifying the Riemann hypothesis. Finally, he presents numerical results testing Artin’s conjecture for \(S_5\) representations, and the Riemann hypothesis for Dedekind zeta functions of \(S_5\) and \(A_5\) fields.

MSC:

11F80 Galois representations
11Y35 Analytic computations
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
20C30 Representations of finite symmetric groups

Citations:

Zbl 1147.11030

Software:

GAP; PARI/GP
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