On the zeros of the Ramanujan $\tau$-Dirichlet series in the critical strip
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- by J. B. Keiper PDF
- Math. Comp. 65 (1996), 1613-1619 Request permission
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 Mathematica™.References
- M. V. Berry and J. P. Keating, A new asymptotic representation for $\zeta (\frac 12+it)$ and quantum spectral determinants, Proc. Roy. Soc. London Ser. A 437 (1992), no. 1899, 151–173. MR 1177749, DOI 10.1098/rspa.1992.0053
- H. M. Edwards, Riemann’s zeta function, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0466039
- H. R. P. Ferguson, R. D. Major, K. E. Powell, and H. G. Throolin, On zeros of Mellin transforms of $\textrm {SL}_{2}(\textbf {Z})$ cusp forms, Math. Comp. 42 (1984), no. 165, 241–255. MR 726002, DOI 10.1090/S0025-5718-1984-0726002-2
- A. Guthmann, Ramanujans Tau-Funktion, unpublished, June, 1988.
- G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work. , Chelsea Publishing Co., New York, 1959. MR 0106147
- J. B. Keiper, Power series expansions of Riemann’s $\xi$ function, Math. Comp. 58 (1992), no. 198, 765–773. MR 1122072, DOI 10.1090/S0025-5718-1992-1122072-5
- A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273–308. MR 866115, DOI 10.1090/S0025-5718-1987-0866115-0
- A. M. Odlyzko, The $10^{20}$-th Zero of the Riemann Zeta function and 70 Million of its Neighbors, unpublished.
- Robert Spira, Calculation of the Ramanujan $\tau$-Dirichlet series, Math. Comp. 27 (1973), 379–385. MR 326995, DOI 10.1090/S0025-5718-1973-0326995-4
- Hiroyuki Yoshida, On calculations of zeros of $L$-functions related with Ramanujan’s discriminant function on the critical line, J. Ramanujan Math. Soc. 3 (1988), no. 1, 87–95. MR 975839
- H. Yoshida, On Calculations of Zeros of Various L-functions, to appear.
Additional Information
- 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
- © Copyright 1996 American Mathematical Society
- 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