Czechoslovak Mathematical Journal, first online, pp. 1-34


Discretization of prime counting functions, convexity and the Riemann hypothesis

Emre Alkan

Received August 8, 2021.   Published online March 18, 2022.

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.
Keywords:  prime counting function; discretization; Riemann hypothesis; harmonic number; indecomposable permutation; asymptotic convexity
Classification MSC:  11N05, 11N37, 11A41
DOI:  10.21136/CMJ.2022.0280-21

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References:
[1] E. Alkan: On Dirichlet $L$-functions with periodic coefficients and Eisenstein series. Monatsh. Math. 163 (2011), 249-280. DOI 10.1007/s00605-010-0211-2 | MR 2805873 | Zbl 1278.11081
[2] E. Alkan: Distribution of averages of Ramanujan sums. Ramanujan J. 29 (2012), 385-408. DOI 10.1007/s11139-012-9424-4 | MR 2994108 | Zbl 1297.11092
[3] E. Alkan: Ramanujan sums and the Burgess zeta function. Int. J. Number Theory 8 (2012), 2069-2092. DOI 10.1142/S1793042112501187 | MR 2978857 | Zbl 1323.11062
[4] E. Alkan: Biased behavior of weighted Mertens sums. Int. J. Number Theory 16 (2020), 547-577. DOI 10.1142/S1793042120500281 | MR 4079395 | Zbl 1437.11128
[5] E. Alkan: Inequalities between sums over prime numbers in progressions. Res. Number Theory 6 (2020), Article ID 36, 29 pages. DOI 10.1007/s40993-020-00211-3 | MR 4150473 | Zbl 07304531
[6] E. Alkan: Variations on criteria of Pólya and Turán for the Riemann hypothesis. J. Number Theory 225 (2021), 90-124. DOI 10.1016/j.jnt.2021.01.004 | MR 4231545 | Zbl 07355818
[7] E. Alkan, A. Zaharescu: $B$-free numbers in short arithmetic progressions. J. Number Theory 113 (2005), 226-243. DOI 10.1016/j.jnt.2004.10.003 | MR 2153277 | Zbl 1138.11339
[8] A. T. Benjamin, D. Gaebler, R. Gaebler: A combinatorial approach to hyperharmonic numbers. Integers 3 (2003), Article ID A15, 9 pages. MR 2036481 | Zbl 1128.11309
[9] B. C. Berndt: Ramanujan's Notebooks. Part IV. Springer, New York (1994). DOI 10.1007/978-1-4612-0879-2 | MR 1261634 | Zbl 0785.11001
[10] C. Cobeli, L. Panaitopol, M. Vâjâitu, A. Zaharescu: Some asymptotic formulas involving primes in arithmetic progressions. Comment. Math. Univ. St. Pauli 53 (2004), 23-35. MR 2084357 | Zbl 1065.11074
[11] L. Comtet: 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. (In French.) MR 0302457 | Zbl 0246.05003
[12] L. Comtet: Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel, Dordrecht (1974). DOI 10.1007/978-94-010-2196-8 | MR 0460128 | Zbl 0283.05001
[13] J. H. Conway, R. K. Guy: The Book of Numbers. Springer, Berlin (1996). DOI 10.1007/978-1-4612-4072-3 | MR 1411676 | Zbl 0866.00001
[14] H. Davenport: Multiplicative Number Theory. Graduate Texts in Mathematics 74. Springer, New York (2000). DOI 10.1007/978-1-4757-5927-3 | MR 1790423 | Zbl 1002.11001
[15] A. P. de Camargo, P. A. Martin: 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 10.1007/s00574-021-00267-4
[16] K. Ford: Vinogradov's integral and bounds for the Riemann zeta function. Proc. Lond. Math. Soc., III. Ser. 85 (2002), 565-633. DOI 10.1112/S0024611502013655 | MR 1936814 | Zbl 1034.11044
[17] É. Grosswald: Sur l'ordre de grandeur des différences $\psi(x)-x$ et $\pi(x)-{ li} x$. C. R. Acad. Sci., Paris 260 (1965), 3813-3816. (In French.) MR 0179146 | Zbl 0127.02005
[18] M. Hassani: Approximation of $\pi(x)$ by $\Psi(x)$. JIPAM, J. Inequal. Pure Appl. Math. 7 (2006), Articles ID 7, 7 pages. MR 2217170 | Zbl 1137.11009
[19] M. Hassani: On an inequality of Ramanujan concerning the prime counting function. Ramanujan J. 28 (2012), 435-442. DOI 10.1007/s11139-011-9362-6 | MR 2950516 | Zbl 1286.11010
[20] M. Hassani: Generalizations of an inequality of Ramanujan concerning prime counting function. Appl. Math. E-Notes 13 (2013), 148-154. MR 3141823 | Zbl 1286.11011
[21] M. Hassani: Remarks on Ramanujan's inequality concerning the prime counting function. Commun. Math. 29 (2021), 473-482. DOI 10.2478/cm-2021-0014 | MR 4355420 | Zbl 07484381
[22] A. E. Ingham: The Distribution of Prime Numbers. Cambridge Tracts in Mathematics and Mathematical Physics 30. Cambridge University Press, London (1932). MR 0184920 | Zbl 0006.39701
[23] A. King: Generating indecomposable permutations. Discrete Math. 306 (2006), 508-518. DOI 10.1016/j.disc.2006.01.005 | MR 2212519 | Zbl 1086.05004
[24] J. E. Littlewood: Sur la distribution des nombres premiers. C. R. Acad. Sci., Paris 158 (1914), 1869-1872. (In French.) JFM 45.0305.01
[25] P. Malliavin: Sur le reste de la loi asymptotique de répartition des nombres premiers généralisés de Beurling. Acta Math. 106 (1961), 281-298. (In French.) DOI 10.1007/BF02545789 | MR 0142518 | Zbl 0102.28204
[26] G. Mincu, L. Panaitopol: Properties of some functions connected to prime numbers. JIPAM, J. Inequal. Pure Appl. Math. 9 (2008), Article ID 12, 10 pages. MR 2391279 | Zbl 1196.11124
[27] G. Mititica, L. Panaitopol: Series involving the least and the greatest prime factor of a natural number. Math. Inequal. Appl. 13 (2010), 197-201. DOI 10.7153/mia-13-15 | MR 2648241 | Zbl 1214.11103
[28] L. Panaitopol: Several approximations of $\pi(x)$. Math. Inequal. Appl. 2 (1999), 317-324. DOI 10.7153/mia-02-29 | MR 1698377 | Zbl 0932.11005
[29] L. Panaitopol: A formula for $\pi(x)$ applied to a result of Koninck-Ivić. Nieuw Arch. Wiskd. 5 (2000), 55-56. MR 1760776 | Zbl 0982.11003
[30] L. Panaitopol: Inequalities concerning the function $\pi(x)$: Applications. Acta Arith. 94 (2000), 373-381. DOI 10.4064/aa-94-4-373-381 | MR 1779949 | Zbl 0963.11050
[31] L. Panaitopol: Asymptotic formulas involving $\pi(x)$. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 44 (2001), 91-96. MR 2013429 | Zbl 1047.11007
[32] L. Panaitopol: Inequalities involving prime numbers. Math. Rep., Bucur 3(53) (2001), 251-256. MR 1929536 | Zbl 1059.11007
[33] L. Panaitopol: A special case of the Hardy-Littlewood conjecture. Math. Rep., Bucur 4(54) (2002), 265-268. MR 2067638 | Zbl 1070.11044
[34] B. Riemann: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte Königlichen Preussischen Akademie der Wissenschaften Berlin (1859), 671-680. (In German.) DOI 10.1017/CBO9781139568050.008
[35] E. Schmidt: Über die Anzahl der Primzahlen unter gegebener Grenze. Math. Ann. 57 (1903), 195-204. (In German.) DOI 10.1007/BF01444344 | MR 1511206 | JFM 34.0230.02
[36] L. Schoenfeld: Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II. Math. Comput. 30 (1976), 337-360. DOI 10.2307/2005976 | MR 0457374 | Zbl 0326.10037
[37] P. Turán: On the remainder-term of the prime-number formula. II. Acta Math. Acad. Sci. Hung. 1 (1950), 155-166. DOI 10.1007/BF02021308 | MR 0049219 | Zbl 0041.37102
[38] A. Walfisz: Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte 15. VEB Deutscher Verlag der Wissenschaften, Berlin (1963). (In German.) MR 0220685 | Zbl 0146.06003

Affiliations:   Emre Alkan, Department of Mathematics, Koç University, Rumelifeneri Yolu, 34450, Sariyer, Istanbul, Turkey, e-mail: ealkan@ku.edu.tr


 
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