Gaps between primes, and the pair correlation of zeros of the zeta-function

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The pair correlation of zeros of the Riemann zeta function and distribution of primes

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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...

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ژورنال

عنوان ژورنال: Acta Arithmetica

سال: 1982

ISSN: 0065-1036,1730-6264

DOI: 10.4064/aa-41-1-85-99