On the zeros of the Ramanujan 𝜏-Dirichlet series in the critical strip

Keiper, J.

doi:10.1090/S0025-5718-96-00734-X

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™.

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