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

Zbl 1147.11030

GAP; PARI/GP
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