Gram's law and Selberg's conjecture on the distribution of zeros of the Riemann zeta function

Korolev, Maxim Aleksandrovich

doi:10.1070/im2010v074n04abeh002506

Cited by 17 publications

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“…The later remark made by Selberg on p. 355 in [5] shows that in the 1980s he had a proof of a much more precise statement that the differences Δ n , being appropriately normalized, have a Gaussian distribution. A partial reconstruction of Selberg's unpublished arguments that I made in [6,7] leads to the following results.…”

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confidence: 80%

On Karatsuba’s problem related to Gram’s law

Korolev

2012

Proc. Steklov Inst. Math.

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A number of new results related to Gram's law in the theory of the Riemann zetafunction are proved. In particular, a lower bound is obtained for the number of ordinates of the zeros of the zeta-function that lie in a given interval and satisfy Gram's law.Denote by ϑ(t) the increment of an arbitrary continuous branch of the argument of the function π −s/2 Γ s 2 under the variation of s along the straight-line segment connecting the points s = 1 2 and s = 1 2 + it. It can be shown that the function ϑ(t) monotonically increases for t > 7 and the following asymptotic formula holds as t → +∞:By a Gram point t n (n is an integer) we will mean a solution to the equation ϑ(t n ) = (n − 1)π. It follows from the above-listed properties of ϑ(t) that such a solution exists and is unique for any n ≥ 0 and that when n increases unboundedly, one has the equalitiesThus, the Gram points form a rather "regular" grid on the real axis, which can sometimes be assumed "almost uniform." Indeed, if the number N is large enough, 1 ≤ M ≤ N α , where 0 < α < 1, and n runs through the interval (N, N + M ], then the differences t n+1 − t n are virtually indistinguishable from 2π(ln N ) −1 , which is independent of n. A few first terms of the sequence of Gram points are shown in Table 1.Denote by n = β n + iγ n , n = 1, 2, 3, . . . , the zeros of the Riemann zeta-function ζ(s) that lie in the upper half-plane and are numbered in the order of increasing imaginary parts (if the imaginary parts coincide, the zeros are arranged arbitrarily): 0 < γ 1 < γ 2 < . . . ≤ γ n ≤ γ n+1 ≤ . . . . It is well known that the real parts β n of such zeros satisfy the inequalities 0 ≤ β n ≤ 1; the prime number theorem is equivalent to the validity of the inequalities 0 < β n < 1 for all n; the Riemann hypothesis states that β n = 1 2 for all n. The latter relation has been verified for all n ≤ 10 13 (Gourdon, 2004 [1]). Approximate values of the first 15 ordinates γ n are shown in Table 2.These values (with slightly lower accuracy) were found in 1903 by Gram [2]. He also noticed that for 1 ≤ n ≤ 15 the ordinate γ n lies between the neighboring Gram points t n−1 and t n : t n−1 < γ n ≤ t n . Various generalizations of the law discovered by Gram were later given the same name of "Gram's law"; a brief survey of the results related to different "kinds" of Gram's laws is presented in [3]. Below we will only speak of the interpretation of Grams' law offered by Selberg [4].

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confidence: 80%

On Karatsuba’s problem related to Gram’s law

Korolev

2012

Proc. Steklov Inst. Math.

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“…The approximate expression for the distribution function of discrete random quantity with the values δ n = π∆ n 2 L , N < n N + M, and the proof of the assertion that ∆ n = 0 for 'almost all' follow from Theorem 6 by standard technic (see, for example, Theorem 4 from [17]). §5.…”

Section: Let Us Consider the Summentioning

confidence: 99%

“…It follows from the remark on p.355 of [16] that Selberg found a proof of his own assumption a long before 1989, but he didn't publish it. For the reconstruction of such proof, see author's papers [17], [18].…”

Section: §1 Introductionmentioning

confidence: 99%

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Gram’s law and the argument of the Riemann zeta function

Korolev

2012

Publ Inst Math (Belgr)

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“…Given n, we indicate the unique number m = m(n) such that t m−1 < γ n t m . Following Selberg [18], we denote ∆ n = m − n. It is known (see [32, p. 355, remark 1] and [33]) that ∆ n = 0 for "almost all" n. Moreover, one can show that the number of indices n N satisfying the condition Proof. We precede the proof by some remarks.…”

Section: Theorem 2 Suppose That the Quantitymentioning

confidence: 99%