Riv. Mat. Univ. Parma, Vol. 11, No. 1, 2020

Sergio Venturini [a]

Non vanishing of Dirichlet series of completely multiplicative functions

Pages: 153-180
Received: 7 January 2019
Accepted in revised form: 30 September 2019
Mathematics Subject Classification (2010): Primary 11M06, 11M20; Secondary 30B40, 30B50.
Keywords: Riemann zeta function, Dirichlet series, Absolutely/completely monotone functions.
Authors address:
[a]: University of Bologna, Dipartimento di Matematica, Piazza di Porta S. Donato 5, 40127 Bologna, Italy

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Abstract: Let \begin{equation*} L(s)=\sum_{n=1}^{+\infty}\dfrac{a(n)}{n^s} \end{equation*} be a Dirichlet series where \(a(n)\) is a bounded completely multiplicative function. We prove that if \(L(s)\) extends to a holomorphic function on the open half space \(\mathop{\rm Re} s >1-\delta\), \(\delta>0\) and \(L(1)=0\) then such a half space is a zero free region of the Riemann zeta function \(\zeta(s)\). Similar results are proven for completely multiplicative functions defined on the space of the ideals of the ring of the algebraic integers of a number field of finite degree.

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