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Mathematics of Computation

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On the zeros of the Ramanujan $\tau $-Dirichlet series in the critical strip

Author: J. B. Keiper
Journal: Math. Comp. 65 (1996), 1613-1619
MSC (1991): Primary 11M41, 65A05
MathSciNet review: 1344615
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Abstract: We describe computations which show that each of the first 12069 zeros of the Ramanujan $\tau $-Dirichlet series of the form $\sigma + i t$ in the region $0 < t < 6397$ is simple and lies on the line $\sigma = 6$. The failures of Gram's law in this region are also noted. The first $5018$ zeros and $2228$ successive zeros beginning with the $20001$st 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 comptuations are done with a Berry-Keating formula for the $\tau $-Dirichlet series and evaluated using $\text{\it Mathematica}{}^{\scriptstyle\mathrm{TM}}$.

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Keywords: Riemann hypothesis, L-functions, Ramanujan conjecture, Ramanujan $\tau$-Dirichlet series
Received by editor(s): August 26, 1991
Received by editor(s) in revised form: January 8, 1993, and January 10, 1995
Additional Notes: $^{*}$Deceased January 19, 1995
Article copyright: © Copyright 1996 American Mathematical Society