Discretization of prime counting functions, convexity and the Riemann hypothesis

Alkan, Emre

Czechoslovak Mathematical Journal, Tome 73 (2023) no. 1, pp. 15-48

Abstract (VO)

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.

Détail

DOI : 10.21136/CMJ.2022.0280-21

Classification : 11A41, 11N05, 11N37
Keywords: prime counting function; discretization; Riemann hypothesis; harmonic number; indecomposable permutation; asymptotic convexity

Bibliographie

[1] Alkan, E.: On Dirichlet $L$-functions with periodic coefficients and Eisenstein series. Monatsh. Math. 163 (2011), 249-280. | DOI | MR | JFM

[2] Alkan, E.: Distribution of averages of Ramanujan sums. Ramanujan J. 29 (2012), 385-408. | DOI | MR | JFM

[3] Alkan, E.: Ramanujan sums and the Burgess zeta function. Int. J. Number Theory 8 (2012), 2069-2092. | DOI | MR | JFM

[4] Alkan, E.: Biased behavior of weighted Mertens sums. Int. J. Number Theory 16 (2020), 547-577. | DOI | MR | JFM

[5] Alkan, E.: Inequalities between sums over prime numbers in progressions. Res. Number Theory 6 (2020), Article ID 36, 29 pages. | DOI | MR | JFM

[6] Alkan, E.: Variations on criteria of Pólya and Turán for the Riemann hypothesis. J. Number Theory 225 (2021), 90-124. | DOI | MR | JFM

[7] Alkan, E., Zaharescu, A.: $B$-free numbers in short arithmetic progressions. J. Number Theory 113 (2005), 226-243. | DOI | MR | JFM

[8] Benjamin, A. T., Gaebler, D., Gaebler, R.: A combinatorial approach to hyperharmonic numbers. Integers 3 (2003), Article ID A15, 9 pages. | MR | JFM

[9] Berndt, B. C.: Ramanujan's Notebooks. Part IV. Springer, New York (1994). | DOI | MR | JFM

[10] Cobeli, C., Panaitopol, L., Vâjâitu, M., Zaharescu, A.: Some asymptotic formulas involving primes in arithmetic progressions. Comment. Math. Univ. St. Pauli 53 (2004), 23-35. | MR | JFM

[11] Comtet, L.: Sur les coefficients de l'inverse de la série formelle $\sum n!t^n$. C. R. Acad. Sci., Paris, Sér. A 275 (1972), 569-572 French. | MR | JFM

[12] Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel, Dordrecht (1974). | DOI | MR | JFM

[13] Conway, J. H., Guy, R. K.: The Book of Numbers. Springer, Berlin (1996). | DOI | MR | JFM

[14] Davenport, H.: Multiplicative Number Theory. Graduate Texts in Mathematics 74. Springer, New York (2000). | DOI | MR | JFM

[15] Camargo, A. P. de, Martin, P. A.: Constant components of the Mertens function and its connections with Tschebyschef's Theory for counting prime numbers. (to appear) in Bull. Braz. Math. Soc. (N.S.). | DOI | MR

[16] Ford, K.: Vinogradov's integral and bounds for the Riemann zeta function. Proc. Lond. Math. Soc., III. Ser. 85 (2002), 565-633. | DOI | MR | JFM

[17] Grosswald, É.: Sur l'ordre de grandeur des différences $\psi(x)-x$ et $\pi(x)-{ li} x$. C. R. Acad. Sci., Paris 260 (1965), 3813-3816 French. | MR | JFM

[18] Hassani, M.: Approximation of $\pi(x)$ by $\Psi(x)$. JIPAM, J. Inequal. Pure Appl. Math. 7 (2006), Articles ID 7, 7 pages. | MR | JFM

[19] Hassani, M.: On an inequality of Ramanujan concerning the prime counting function. Ramanujan J. 28 (2012), 435-442. | DOI | MR | JFM

[20] Hassani, M.: Generalizations of an inequality of Ramanujan concerning prime counting function. Appl. Math. E-Notes 13 (2013), 148-154. | MR | JFM

[21] Hassani, M.: Remarks on Ramanujan's inequality concerning the prime counting function. Commun. Math. 29 (2021), 473-482. | DOI | MR | JFM

[22] Ingham, A. E.: The Distribution of Prime Numbers. Cambridge Tracts in Mathematics and Mathematical Physics 30. Cambridge University Press, London (1932). | MR | JFM

[23] King, A.: Generating indecomposable permutations. Discrete Math. 306 (2006), 508-518. | DOI | MR | JFM

[24] Littlewood, J. E.: Sur la distribution des nombres premiers. C. R. Acad. Sci., Paris 158 (1914), 1869-1872 French \99999JFM99999 45.0305.01.

[25] Malliavin, P.: Sur le reste de la loi asymptotique de répartition des nombres premiers généralisés de Beurling. Acta Math. 106 (1961), 281-298 French. | DOI | MR | JFM

[26] Mincu, G., Panaitopol, L.: Properties of some functions connected to prime numbers. JIPAM, J. Inequal. Pure Appl. Math. 9 (2008), Article ID 12, 10 pages. | MR | JFM

[27] Mititica, G., Panaitopol, L.: Series involving the least and the greatest prime factor of a natural number. Math. Inequal. Appl. 13 (2010), 197-201. | DOI | MR | JFM

[28] Panaitopol, L.: Several approximations of $\pi(x)$. Math. Inequal. Appl. 2 (1999), 317-324. | DOI | MR | JFM

[29] Panaitopol, L.: A formula for $\pi(x)$ applied to a result of Koninck-Ivić. Nieuw Arch. Wiskd. 5 (2000), 55-56. | MR | JFM

[30] Panaitopol, L.: Inequalities concerning the function $\pi(x)$: Applications. Acta Arith. 94 (2000), 373-381. | DOI | MR | JFM

[31] Panaitopol, L.: Asymptotic formulas involving $\pi(x)$. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 44 (2001), 91-96. | MR | JFM

[32] Panaitopol, L.: Inequalities involving prime numbers. Math. Rep., Bucur 3(53) (2001), 251-256. | MR | JFM

[33] Panaitopol, L.: A special case of the Hardy-Littlewood conjecture. Math. Rep., Bucur 4(54) (2002), 265-268. | MR | JFM

[34] Riemann, B.: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte Königlichen Preußischen Akademie der Wissenschaften Berlin (1859), 671-680 German. | DOI

[35] Schmidt, E.: Über die Anzahl der Primzahlen unter gegebener Grenze. Math. Ann. 57 (1903), 195-204 German \99999JFM99999 34.0230.02. | DOI | MR

[36] Schoenfeld, L.: Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II. Math. Comput. 30 (1976), 337-360. | DOI | MR | JFM

[37] Turán, P.: On the remainder-term of the prime-number formula. II. Acta Math. Acad. Sci. Hung. 1 (1950), 155-166. | DOI | MR | JFM

[38] Walfisz, A.: Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte 15. VEB Deutscher Verlag der Wissenschaften, Berlin (1963), German. | MR | JFM