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Harmonic and Probabilistic Approaches to Zeros of Riemann's Zeta Function |
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| Authors: | M M Rao |
| Institution: | 1. Department of Mathematics , Unversity of California , Riverside , California , USA rao@math.ucr.edu |
| Abstract: | Probability concepts and results are closely related to the study of zeros of the classical Riemann zeta function and its affinity to Gaussian and Gamma distributions. This is elaborated in obtaining the functional and integral equations for the zeta and in the determination first of the nonzero sets and then sets containing almost all (i.e., for the CLT probability measure) nontrivial zeros of the zeta function ζ(·). Also probability distributions determined by the zeta, based on the behavior of their finite dimensional distributions of the ζ(σ +it), σ > 0; as t varies and particularly the results of Denjoy, slightly sharpened, and also one of Salem are included. Several related opinions and comments are discussed. |
| Keywords: | Central limit theorem Extended Denjoy's approach Gamma Gaussian distributions Littlewood's criterion of equivalence of Riemann's hypothesis Null sets in the problem Probability methods Trivial zeros Zero free sets of zeta Zeros on the critical line Zeta function |