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On the Zeros of the Ramanujan tau-Dirichlet Series in the Critical Strip

Jerry B. Keiper
Organization: Wolfram Research, Inc.
Journal / Anthology

Mathematics of Computation
Year: 1996
Volume: 65
Issue: 216
Page range: 1613-1619

We describe computations which show that each of the first 12069 zeros of the Ramanujan tau-Dirichlet series of the form s + it in the region 0 < t <6397 is simple and lies on the line s = 6. The failures of Gram's law in this region are also noted. The fist 5018 zeros and 2228 successive zeros beginning with the 20001st zero are also calculated. The distribution of the normalized spacing of the zeros is examined and it appears to be that of the eigenvalues of random matrices. These computations are done with a Berry-Keating formula for the tau-Dirichlet series and evaluated using Mathematica.

*Mathematics > Calculus and Analysis > Series
*Mathematics > Number Theory