Statistical properties of zeta functions’ zeros

Kargin, Vladislav

doi:10.1214/13-ps214

Cited by 3 publications

(1 citation statement)

References 58 publications

(89 reference statements)

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“…So the most interesting part of this formula for rhythms is the early part. There are studies which show that for high t the spacing of the zeta function takes on a random aspect relating to the eigenvalues of certain random matrices studied in physics [21,22]; in a related publication, Michael Berry has sonified various categories of random matrix, including in comparison to Riemann zero data, using frequency modulation [23].…”

Section: Spacings By Zeroesmentioning

confidence: 99%

Sonification of the Riemann Zeta Function

Collins

2019

Proceedings of the 25th International Conference on Auditory Display (ICAD 2019)

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The Riemann zeta function is one of the great wonders of mathematics, with a deep and still not fully solved connection to the prime numbers. It is defined via an infinite sum analogous to Fourier additive synthesis, and can be calculated in various ways. It was Riemann who extended the consideration of the series to complex number arguments, and the famous Riemann hypothesis states that the non-trivial zeroes of the function all occur on the critical line 0:5 + ti, and what is more, hold a deep correspondence with the prime numbers. For the purposes of sonification, the rich set of mathematical ideas to analyse the zeta function provide strong resources for sonic experimentation. The positions of the zeroes on the critical line can be directly sonified, as can values of the zeta function in the complex plane, approximations to the prime spectrum of prime powers and the Riemann spectrum of the zeroes rendered; more abstract ideas concerning the function also provide interesting scope.

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Section: Spacings By Zeroesmentioning

confidence: 99%

Sonification of the Riemann Zeta Function

Collins

2019

Proceedings of the 25th International Conference on Auditory Display (ICAD 2019)

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Aperiodic crystals, Riemann zeta function, and primes

Madison

Kozyrev

2022

Struct Chem

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Fractional calculus, zeta functions and Shannon entropy

Guariglia

2021

Open Mathematics

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This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz ζ \zeta function. All the results are based on the complex generalization of the Grünwald-Letnikov fractional derivative. We state and prove the functional equation together with an integral representation by Bernoulli numbers. Moreover, we treat an application in terms of Shannon entropy.

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Statistical properties of zeta functions’ zeros

Cited by 3 publications

References 58 publications

Sonification of the Riemann Zeta Function

Sonification of the Riemann Zeta Function

Aperiodic crystals, Riemann zeta function, and primes

Fractional calculus, zeta functions and Shannon entropy

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