Discretization of prime counting functions, convexity and the Riemann hypothesis

Emre Alkan

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 1, page 15-48
  • ISSN: 0011-4642

Abstract

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We study tails of prime counting functions. Our approach leads to representations with a main term and an error term for the asymptotic size of each tail. It is further shown that the main term is of a specific shape and can be written discretely as a sum involving probabilities of certain events belonging to a perturbed binomial distribution. The limitations of the error term in our representation give us equivalent conditions for various forms of the Riemann hypothesis, for classical type zero-free regions in the case of the Riemann zeta function and the size of semigroups of integers in the sense of Beurling. Inspired by the works of Panaitopol, asymptotic companions pertaining to the magnitude of specific prime counting functions are obtained in terms of harmonic numbers, hyperharmonic numbers and the number of indecomposable permutations. By introducing the notion of asymptotic convexity and fusing it with a nice generalization of an inequality of Ramanujan due to Hassani, we arrive at a curious asymptotic inequality for the classical prime counting function at any convex combination of its arguments and further show that quotients arising from prime counting functions of progressions furnish examples of asymptotically convex, but not convex functions.

How to cite

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Alkan, Emre. "Discretization of prime counting functions, convexity and the Riemann hypothesis." Czechoslovak Mathematical Journal 73.1 (2023): 15-48. <http://eudml.org/doc/299422>.

@article{Alkan2023,
abstract = {We study tails of prime counting functions. Our approach leads to representations with a main term and an error term for the asymptotic size of each tail. It is further shown that the main term is of a specific shape and can be written discretely as a sum involving probabilities of certain events belonging to a perturbed binomial distribution. The limitations of the error term in our representation give us equivalent conditions for various forms of the Riemann hypothesis, for classical type zero-free regions in the case of the Riemann zeta function and the size of semigroups of integers in the sense of Beurling. Inspired by the works of Panaitopol, asymptotic companions pertaining to the magnitude of specific prime counting functions are obtained in terms of harmonic numbers, hyperharmonic numbers and the number of indecomposable permutations. By introducing the notion of asymptotic convexity and fusing it with a nice generalization of an inequality of Ramanujan due to Hassani, we arrive at a curious asymptotic inequality for the classical prime counting function at any convex combination of its arguments and further show that quotients arising from prime counting functions of progressions furnish examples of asymptotically convex, but not convex functions.},
author = {Alkan, Emre},
journal = {Czechoslovak Mathematical Journal},
keywords = {prime counting function; discretization; Riemann hypothesis; harmonic number; indecomposable permutation; asymptotic convexity},
language = {eng},
number = {1},
pages = {15-48},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Discretization of prime counting functions, convexity and the Riemann hypothesis},
url = {http://eudml.org/doc/299422},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Alkan, Emre
TI - Discretization of prime counting functions, convexity and the Riemann hypothesis
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 1
SP - 15
EP - 48
AB - We study tails of prime counting functions. Our approach leads to representations with a main term and an error term for the asymptotic size of each tail. It is further shown that the main term is of a specific shape and can be written discretely as a sum involving probabilities of certain events belonging to a perturbed binomial distribution. The limitations of the error term in our representation give us equivalent conditions for various forms of the Riemann hypothesis, for classical type zero-free regions in the case of the Riemann zeta function and the size of semigroups of integers in the sense of Beurling. Inspired by the works of Panaitopol, asymptotic companions pertaining to the magnitude of specific prime counting functions are obtained in terms of harmonic numbers, hyperharmonic numbers and the number of indecomposable permutations. By introducing the notion of asymptotic convexity and fusing it with a nice generalization of an inequality of Ramanujan due to Hassani, we arrive at a curious asymptotic inequality for the classical prime counting function at any convex combination of its arguments and further show that quotients arising from prime counting functions of progressions furnish examples of asymptotically convex, but not convex functions.
LA - eng
KW - prime counting function; discretization; Riemann hypothesis; harmonic number; indecomposable permutation; asymptotic convexity
UR - http://eudml.org/doc/299422
ER -

References

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