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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
DOI: https://doi.org/10.1090/S0025-5718-96-00734-X
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 $\text{\it Mathematica}{}^{\scriptstyle\mathrm{TM}}$.


References [Enhancements On Off] (What's this?)

  • 1. M. B. Berry and J. P. Keating, A new asymptotic representation for $\zeta (\frac {1}{2} + i t)$ and quantum spectral determinants, Proc. R. Soc. Lond. A 437, (1992), 151-173. MR 93j:11057
  • 2. H. M. Edwards, Riemann's zeta function, Academic Press, New York, 1974. MR 57:5922
  • 3. H. R. P. Ferguson, R. D. Major, K. E. Powell, and H. G. Throolin, On Zeros of Mellin Transforms of $SL_2(% \mathbf {Z})$ Cusp Forms, Math. Comp. 42 (1984), 241-255. MR 85e:11036
  • 4. A. Guthmann, Ramanujans Tau-Funktion, unpublished, June, 1988.
  • 5. G. H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Chelsea, New York, 1959. MR 21:4881
  • 6. J. B. Keiper, Power series expansions of Riemann's $\xi $ function, Math. Comp. 58 (1992), 765-773. MR 92f:11116
  • 7. A. M. Odlyzko, On the Distribution of Spacings Between Zeros of the Zeta Function, Math. Comp. 48 (1987), 273-308. MR 88d:11082
  • 8. A. M. Odlyzko, The $10^{20}$-th Zero of the Riemann Zeta function and 70 Million of its Neighbors, unpublished.
  • 9. R. Spira, Calculation of the Ramanujan $\tau $-Dirichlet Series, Math. Comp. 27 (1973), 379-385. MR 48:5337
  • 10. H. Yoshida, On Calculations of Zeros of L-functions Related With Ramanujan's Discriminant Function on the Critical Line, J. Ramanujan Math. Soc. 3 (1988), 87-95. MR 90b:11044
  • 11. H. Yoshida, On Calculations of Zeros of Various L-functions, to appear.

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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

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