On the zeros of the Ramanujan -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 -Dirichlet series of the form in the region is simple and lies on the line . The failures of Gram's law in this region are also noted. The first zeros and successive zeros beginning with the 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 -Dirichlet series and evaluated using .

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-96-00734-X

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