Sums over primes. II. (English) Zbl 07842622
The generalized alternating hyperharmonic numbers of types I, II, and III defined respectively, as
\begin{align*}
&H_n^{(p,r,1)}:=\sum_{k=1}^n (-1)^{k-1} H_{k}^{(p,r-1,1)}\quad (H_n^{(p,1,1)}=H_n^{(p)}),\\
&H_n^{(p,r,2)}:=\sum_{k=1}^n H_{k}^{(p,r-1,2)} \quad (H_n^{(p,1,2)}=\overline{H}_n^{(p)}:=\sum_{j=1}^n (-1)^{j-1}/j^p),\\
&H_n^{(p,r,3)}:=\sum_{k=1}^n (-1)^{k-1} H_{k}^{(p,r-1,3)} \quad (H_n^{(p,1,3)}=\overline{H}_n^{(p)}).
\end{align*}
In the paper under review, the author obtains explicit asymptotic formulas for some sums over primes involving three types of generalized alternating hyperharmonic numbers of types I, II, and III. This paper is a continuation of the previous paper of the author, Part I [Integers 21, Paper A94, 7 p. (2021; Zbl 1489.11140)].
Reviewer: Mehdi Hassani (Zanjan)
MSC:
| 11N05 | Distribution of primes |
| 11L20 | Sums over primes |
| 11N25 | Distribution of integers with specified multiplicative constraints |
| 11N37 | Asymptotic results on arithmetic functions |
Keywords:
sum over primes; generalized alternating hyperharmonic number; asymptotic formula; number with \(k\) prime factorsCitations:
Zbl 1489.11140Software:
OEISReferences:
| [1] | J. H. Conway and R. K. Guy, The Book of Numbers, Springer, 1996. · Zbl 0866.00001 |
| [2] | A. Dil, I. Mező, and M. Cenkci. Evaluation of Euler-like sums via Hurwitz zeta values, Turkish J. Math. 41 (2017), 1640-1655. · Zbl 1424.11130 |
| [3] | J. Gerard and L. C. Washington, Sums of powers of primes, Ramanujan J. 45 (2018), 171-180. · Zbl 1428.11160 |
| [4] | A. Granville, Analytic Number Theory Revealed: The Distribution of Prime Numbers, book draft, unpublished. |
| [5] | 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 |
| [6] | G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th edition, Oxford University Press, 2008. |
| [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), Article 31. · Zbl 1149.11046 |
| [8] | R. Jakimczuk, Functions of slow increase and integer sequences, J. Integer Sequences 13 (2010), Article 10.1.1. · Zbl 1210.26005 |
| [9] | E. Landau, Sur quelques problèmes relatifs à la distribution des nombres premiers, Bull. Soc. Math. France 28 (1900), 25-38. · JFM 31.0200.01 |
| [10] | R. Li, Euler sums of generalized alternating hyperharmonic numbers, Rocky Mountain J. Math. 51 (2021), 1299-1313. · Zbl 1497.11054 |
| [11] | R. Li, Sums over primes, Integers 21 (2021), Paper No. A94. · Zbl 1489.11140 |
| [12] | R. Li, Euler sums of generalized hyperharmonic numbers, Funct. Approximatio, Com-ment. Math. 66 (2022), 179-189. · Zbl 1536.11045 |
| [13] | I. Mező and A. Dil, Hyperharmonic series involving Hurwitz zeta function, J. Number Theory 130 (2010), 360-369. · Zbl 1225.11032 |
| [14] | N. Ömür and S. Koparal, On the matrices with the generalized hyperharmonic numbers of order r, Asian-Eur. J. Math. 11 (2018), 1850045. · Zbl 1391.11056 |
| [15] | T. Šalát and Š. Znám, On sums of the prime powers, Acta Fac. Rerum Natur. Univ. Comenian. Math. 21 (1968), 21-24 (1969). · Zbl 0194.35101 |
| [16] | N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, 2023. Available at https://oeis.org. |
| [17] | C. D. L. Vallée Poussin, Recherches analytiques sur la théorie des nombres premiers, Ann. Soc. Sci. Bruxelles. 20 (1896), 183-256. |
| [18] | 2020 Mathematics Subject Classification: Primary 11B83; Secondary 11L20, 11N25, 11N37. Keywords: sum over primes, generalized alternating hyperharmonic number, asymptotic formula, number with k prime factors. (Concerned with sequence A000040.) |
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.
