The Riemann Hypothesis, simple zeros and the asymptotic convergence degree of improper Riemann sums
نویسندگان
چکیده
منابع مشابه
Simple Zeros of the Riemann Zeta-function
Assuming the Riemann Hypothesis, Montgomery and Taylor showed that at least 67.25% of the zeros of the Riemann zeta-function are simple. Using Montgomery and Taylor's argument together with an elementary combinatorial argument, we prove that assuming the Riemann Hypothesis at least 67.275% of the zeros are simple.
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We write ρX = βX + iγX for a typical zero of FX(s). The number of these up to height T we denote by NX(T ), and the number of these with βX ≥ σ by NX(σ, T ). We follow the convention that if T is the ordinate of a zero, then NX(T ), say, is defined as limǫ→0+ NX(T + ǫ). There are two natural ways to pose questions about NX(T ), NX(σ, T ), and the distribution of the zeros generally. We can fix ...
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Convergence in probability of random Riemann sums of a Lebesgue integrable function on [0, 1) to the integral has been proved. In this article we generalize the result to an abstract probability space under some natural conditions and we show L1convergence rather than convergence in probability..
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We investigate the distribution of simple zeros of the Riemann zeta-function. Let H ≤ T and L = log T . We calculate in a new way (following old ideas of Atkinson and new ideas of Jutila and Motohashi) the mean square of the product of F (s) = ζ(s) + 1 Lζ ′(s) and a certain Dirichlet polynomial A(s) = ∑ n≤M a(n) ns of length M = T θ with θ < 38 near the critical line: if R is a positive constan...
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Associated to classical semi-simple groups and their maximal parabolics are genuine zeta functions. Naturally related to Riemann’s zeta and governed by symmetries, including that of Weyl, these zetas are expected to satisfy the Riemann hypothesis. For simplicity, G here denotes a classical semi-simple algebraic group defined over the field Q of rationals. With a fixed Borel, as usual, ∆0 stands...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1998
ISSN: 0002-9939,1088-6826
DOI: 10.1090/s0002-9939-98-04607-3