Bounding $$S(t)$$ and $$S_1(t)$$ on the Riemann hypothesis

Carneiro, Emanuel; Chandee, Vorrapan; Milinovich, Micah B.

doi:10.1007/s00208-012-0876-z

Cited by 53 publications

(94 citation statements)

References 27 publications

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“…This is Theorem 2 of E. Carneiro, V. Chandee and M. Milinovich [3]. It improves the previous bound of D.A.…”

Section: Introductionsupporting

confidence: 83%

Some remarks on the differences between ordinates of consecutive zeta zeros

Ivić¹

2018

Funct. Approx. Comment. Math.

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Abstract. If 0 < γ 1 γ 2 γ 3 . . . denote ordinates of complex zeros of the Riemann zeta-function, then several results involving the maximal order of γ n+1 −γ n and the sumare proved. IntroductionLet 0 < γ 1 γ 2 γ 3 . . . denote ordinates of complex zeros of the Riemann zeta-functionFor ℜs 1 one defines ζ(s) by analytic continuation (see the monographs of A. Ivić [11] and E.C. Titchmarsh [19] for the properties of ζ(s)). Here the Riemann Hypothesis (RH), that all complex zeros of ζ(s) satisfy ℜs = 1 2 , is not assumed. Thus if equality among the γ n 's occurs for some n, it does not necessarily mean that the zero ρ n = β n + iγ n is not simple, i.e., ζ(ρ n ) = 0 and ζ ′ (ρ n ) = 0. Namely one could have γ n = γ n+1 , ρ n = β n + iγ n , ρ n+1 = β n+1 + iγ n+1 with β n = β n+1 , and both ρ n and ρ n+1 simple. Although all numerical evidence points to the simplicity of all zeta zeros, proving this is an open and difficult question. In fact, it seems that the simplicity of zeta-zeros and the RH are independent statements in the sense that, as far as it is known, both statements could be true or false, or one true and the other one false.Problems involving γ n+1 − γ n , the difference between consecutive ordinates of the zeros (if the zeros are arranged according to the size of their imaginary parts)1991 Mathematics Subject Classification. 11M06.

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“…This is Theorem 2 of E. Carneiro, V. Chandee and M. Milinovich [3]. It improves the previous bound of D.A.…”

Section: Introductionsupporting

confidence: 83%

Some remarks on the differences between ordinates of consecutive zeta zeros

Ivić¹

2018

Funct. Approx. Comment. Math.

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“…These are functions F of prescribed exponential type that minimize the L 1 (R, µ)-distance (µ is some non-negative measure) from a given function g and such that F lies below or above g on R. These functions have the special property that they interpolate the target function g (an its derivative) at a certain sequence of real points and have several special properties that are very useful in applications to analytic number theory, being the key to provide sharp (or improved) estimates. For instance, in connection to: large sieve inequalities [24,28], Erdös-Turán inequalities [15,28], Hilbert-type inequalities [12,14,15,22,28], Tauberian theorems [22] and bounds in the theory of the Riemann zeta-function and general L-functions [7,8,9,11,16,18,19]. Further constructions and applications can also be found in [6,10,13,21,25].…”

Section: Theorem a ([20 Theorem 1])mentioning

confidence: 99%

Interpolation formulas with derivatives in de Branges spaces II

Gonçalves

Littmann

2018

Journal of Mathematical Analysis and Applications

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“…As in the work of Goldston and Gonek mentioned above, we use the explicit formula in conjunction with the classical majorants and minorants of exponential type for characteristic functions of intervals that were constructed by Beurling and Selberg. In [5], two proofs of Theorem 1 are given. The more direct of these proofs proceeds as follows.…”

Section: Theorem 1 Assume the Riemann Hypothesis Thenmentioning

confidence: 99%

“…The more direct of these proofs proceeds as follows. Assuming the Riemann hypothesis, for t not corresponding to an ordinate of a zero of ζ(s), it is shown in [5] that…”

Section: Theorem 1 Assume the Riemann Hypothesis Thenmentioning

confidence: 99%

A note on the zeros of zeta and L-functions

2015

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Let π S(t) denote the argument of the Riemann zeta-function at the point s = 1 2 + it. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for S(t). We discuss a generalization of this bound for a large class of L-functions including those which arise from cuspidal automorphic representations of GL(m) over a number field. We also prove a number of related results including bounding the order of vanishing of an L-function at the central point and bounding the height of the lowest zero of an L-function.

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Bounding $$S(t)$$ and $$S_1(t)$$ on the Riemann hypothesis

Cited by 53 publications

References 27 publications

Some remarks on the differences between ordinates of consecutive zeta zeros

Some remarks on the differences between ordinates of consecutive zeta zeros

Interpolation formulas with derivatives in de Branges spaces II

A note on the zeros of zeta and L-functions

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