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Quotients of values of the Dedekind Eta function

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Abstract

Inspired by Riemann’s work on certain quotients of the Dedekind Eta function, in this paper we investigate the value distribution of quotients of values of the Dedekind Eta function in the complex plane, using the form \({\frac{\eta(A_jz)} {\eta(A_{j-1}z)}}\) , where A j-1 and A j are matrices whose rows are the coordinates of consecutive visible lattice points in a dilation XΩ of a fixed region Ω in \({\mathbb{R}^2}\) , and z is a fixed complex number in the upper half plane. In particular, we show that the limiting distribution of these quotients depends heavily on the index of Farey fractions which was first introduced and studied by Hall and Shiu. The distribution of Farey fractions with respect to the value of the index dictates the universal limiting behavior of these quotients. Motivated by chains of these quotients, we show how to obtain a generalization, due to Zagier, of an important formula of Hall and Shiu on the sum of the index of Farey fractions.

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Authors and Affiliations

  1. Department of Mathematics, Koc University, Rumelifeneri Yolu, 34450, Sariyer, Istanbul, Turkey

    Emre Alkan

  2. Department of Mathematics, Eberly College of Science, University Park, Penn State University, State College, PA, 16802, USA

    Maosheng Xiong

  3. Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700, Bucharest, Romania

    Alexandru Zaharescu

  4. Department of Mathematics, University of Illinois at Urbana-Champaign, 273 Altgeld Hall, MC-382, 1049 W. Green Street, Urbana, IL, 61801, USA

    Alexandru Zaharescu

Authors
  1. Emre Alkan
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  2. Maosheng Xiong
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  3. Alexandru Zaharescu
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Corresponding author

Correspondence to Maosheng Xiong.

Additional information

A. Zaharescu is supported by National Science Foundation Grant DMS-0456615.

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Alkan, E., Xiong, M. & Zaharescu, A. Quotients of values of the Dedekind Eta function. Math. Ann. 342, 157–176 (2008). https://doi.org/10.1007/s00208-008-0228-1

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