Strong multiplicity one for the Selberg class (Q2740999)

The authors show that the so-called strong multiplicity one property holds unconditionally for a subclass of the Selberg \(\mathcal S\) class of Dirichlet series [see the authors' comprehensive article for the relevant definitions and properties of \(\mathcal S\), in: Number Theory in Progress, Györy K. et al. (eds.), Proc. Conf. in Honor of A. Schinzel, de Gruyter, 953-992 (1999; Zbl 0929.11028)]. Their work builds on the paper of \textit{M. R. Murty} and \textit{V. K. Murty} [C. R. Acad. Sci., Paris, Sér. I 319, 315-320 (1994; Zbl 0823.11049)]. Every \(F(s) \in \mathcal S\) for \(\sigma > 1\) may be written in the form (\(p\) denotes primes) NEWLINE\[NEWLINE F(s) = \sum_{n=1}^\infty a_F(n)n^{-s} = \prod_p F_p(s),\quad \log F_p(s) = \sum_{m=1}^\infty b_F(p^m)p^{-ms} NEWLINE\]NEWLINE with \(b_F(n) \ll n^\theta (\theta < 1/2)\). If one defines, for \(F(s) \in \mathcal S\), NEWLINE\[NEWLINE \mu_F = \liminf_{\sigma\to 1+} (\sigma-1)\sum_p|a_F(p^2)|^2p^{-\sigma}\log p, NEWLINE\]NEWLINEthen the authors' main result is as follows: Let \(F,G \in \mathcal S\) satisfy \(\mu_F,\mu_G < \infty\), and assume that \(a_F(p) = a_G(p)\) for all but finitely many primes \(p\). Then \(F(s) = G(s)\) identically. The proof uses a general Lemma on zeros and poles of Dirichlet series, the functional equation with gamma-factors (key property of \(\mathcal S\)), plus some results on H. Bohr's almost periodic functions.