Gaps between zeros of the Riemann zeta-function
نویسندگان
چکیده
منابع مشابه
Large Gaps between the Zeros of the Riemann Zeta Function
If the Riemann hypothesis (RH) is true then the non-trivial zeros of the Riemann zeta function, ζ(s), satisfy 1/2+iγn with γn ∈ R. Riemann noted that the argument principle implies that number of zeros of ζ(s) in the box with vertices 0, 1, 1 + iT, and iT is N(T ) ∼ (T/2π) log (T/2πe). This implies that on average (γn+1 − γn) ≈ 2π/ log γn and hence the average spacing of the sequence γ̂n = γn lo...
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An important problem in number theory is to study the distribution of the non-trivial zeros of the Riemann zeta-function which, if one is willing to assume the Riemann Hypothesis, all lie on a vertical line. It is relatively easy to count how many of these zeros lie in a large interval, so the average spacing between consecutive zeros is easy to compute. However, it is a difficult and interesti...
متن کاملLarge gaps between consecutive zeros of the Riemann zeta-function
Combining the mollifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zeta-function. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist infinitely many consecutive gaps greater than 3.0155 times the average spacing.
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Feng and Wu introduced a new general coefficient sequence into Montgomery and Odlyzko’s method for exhibiting irregularity in the gaps between consecutive zeros of ζ (s) assuming the Riemann hypothesis. They used a special case of their sequence to improve upon earlier results on the gaps. In this paper we consider a general sequence related to that of Feng and Wu, and introduce a somewhat less...
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0 |ζ( 1 2 + it)| 4 dt ∼ T 2π2 log T (T → ∞), this means that |ζ( 1 2 + it)| is small “most of the time”. The problem, then, is to evaluate asymptotically the measure of the subset of [0, T ] where |ζ( 1 2 + it)| is “small”. There are several ways in which one can proceed, and a natural way is the following one. Let c > 0 be a given constant, let μ(·) denote measure, and let Ac(T ) := {0 < t ≤ T...
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ژورنال
عنوان ژورنال: The Quarterly Journal of Mathematics
سال: 2017
ISSN: 0033-5606,1464-3847
DOI: 10.1093/qmath/hax047