×

Sums over primes. (English) Zbl 1489.11140

Summary: In this paper, we give explicit asymptotic formulas for some sums over primes involving generalized hyperharmonic numbers. Analogous results for numbers with \(k\)-prime factors will also be considered.

MSC:

11N05 Distribution of primes
11L20 Sums over primes

References:

[1] J. H. Conway and R.K. Guy, The Book of Numbers, Springer, New York, 1996. · Zbl 0866.00001
[2] C. de la Vallée Poussin, Recherches analytiques sur la théorie des nombres premiers, Ann. Soc. Sci. Bruxelles. 20 (1896), 183-256. · JFM 27.0155.03
[3] A. Dil, I. Mező and M. Cenkci, Evaluation of Euler-like sums via Hurwitz zeta values, Turkish J. Math. 41 (2017), no. 6, 1640-1655. · Zbl 1424.11130
[4] J. Gerard and L. C. Washington, Sums of powers of primes, Ramanujan J. 45 (2018), no. 1, 171-180. · Zbl 1428.11160
[5] A. Granville, Analytic Number Theory Revealed: The Distribution of Prime Numbers, book draft, unpublished.
[6] J. Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bull. Soc. Math. France 24 (1896), 199-220. · JFM 27.0154.01
[7] R. Jakimczuk, A note on sums of powers which have a fixed number of prime factors, JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), no. 2, Article 31, 4 pp. · Zbl 1149.11046
[8] R. Jakimczuk, Functions of slow increase and integer sequences, J. Integer Seq. 13 (2010), no. 1, Article 10.1.1, 14 pp. · Zbl 1210.26005
[9] E. Landau, Sur quelques problèmes relatifsà la distribution des nombres premiers, Bull. Soc. Math. France 28 (1900), 25-38. · JFM 31.0200.01
[10] R. Li, Euler sums of generalized hyperharmonic numbers, arXiv:2103.10622, submitted. · Zbl 1497.11054
[11] I. Mező and A. Dil, Hyperharmonic series involving Hurwitz zeta function, J. Number Theory 130 (2010), 360-369. · Zbl 1225.11032
[12] M. B. Nathanson, Additive Number Theory: The Classical Bases, Springer, 1996. · Zbl 0859.11002
[13] N.Ömür and S. Koparal, On the matrices with the generalized hyperharmonic numbers of order r, Asian-Eur. J. Math. 11 (2018), no. 3, 1850045, 9 pp. · Zbl 1391.11056
[14] T.Šalát andŠ. Znám, On sums of the prime powers, Acta Fac. Rerum Natur. Univ. Come-nian. Math. 21 (1968), 21-24 (1969). · Zbl 0194.35101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.