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On a class of functions that satisfies explicit formulae involving the Möbius function

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Abstract

A class of functions that satisfies intriguing explicit formulae of Ramanujan and Titchmarsh involving the zeros of an \(L\)-function in the reduced Selberg class of degree one and its associated Möbius function is studied. Moreover, a sufficient and necessary condition for the truth of the Riemann hypothesis due to Riesz is generalized.

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Notes

  1. Two functions \(f(x)\) and \(g(x)\) are cosine reciprocal if

    $$\begin{aligned} \frac{{\sqrt{\pi }}}{2}f(x) = \int \limits _0^\infty {g(u)\cos (2ux){ d}u} ,\quad \frac{{\sqrt{\pi }}}{2}g(x) = \int \limits _0^\infty {f(u)\cos (2ux){ d}u}. \end{aligned}$$

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Acknowledgments

The authors wish to acknowledge the helpful comments of the referee.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zurich, Switzerland

    Patrick Kühn & Nicolas Robles

  2. Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL, 61801, USA

    Arindam Roy

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  1. Patrick Kühn
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  2. Nicolas Robles
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  3. Arindam Roy
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Correspondence to Nicolas Robles.

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The authors were partially supported by the SNF Grants PP00P2_138906 and 200020_149150\({\backslash }1\).

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Kühn, P., Robles, N. & Roy, A. On a class of functions that satisfies explicit formulae involving the Möbius function. Ramanujan J 38, 383–422 (2015). https://doi.org/10.1007/s11139-014-9608-1

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