TY - JOUR

T1 - Quotients of values of the Dedekind Eta function

AU - Alkan, Emre

AU - Xiong, Maosheng

AU - Zaharescu, Alexandru

PY - 2008/9

Y1 - 2008/9

N2 - 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 n(Ajz)/n(Aj-1z), 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 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.

AB - 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 n(Ajz)/n(Aj-1z), 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 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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U2 - 10.1007/s00208-008-0228-1

DO - 10.1007/s00208-008-0228-1

M3 - Article

AN - SCOPUS:46649087321

SN - 0025-5831

VL - 342

SP - 157

EP - 176

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1

ER -