On the sum of the first n prime numbers
Christian Axler1
1 Institute of Mathematics Heinrich-Heine-University Düsseldorf Building 25.22, Room 02.42 Universitätsstraße 1 40225 Düsseldorf, Germany
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 293-311.

In this paper we establish an asymptotic formula for the sum of the first n prime numbers, more precise than the one given by Massias and Robin in 1996. Further we prove a series of results concerning Mandl’s inequality on the sum of the first n prime numbers. We use these results to find new explicit estimates for the sum of the first n prime numbers, which improve the currently best known estimates.

Dans cet article, nous établissons une formule asymptotique pour la somme des n premiers nombres premiers, plus précise que celle donnée par Massias et Robin en 1996. En outre, nous prouvons un certain nombre de résultats concernant l’inégalité de Mandl pour la somme des n premiers nombres premiers. Nous utilisons ces résultats pour établir de nouvelles estimations explicites de la somme des n premiers nombres premiers, qui améliorent les meilleures estimations actuellement connues.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1081
Classification: 11N05, 11A41
Keywords: Asymptotic expansion, Mandl’s inequality, Sum of prime numbers

Christian Axler 1

1 Institute of Mathematics Heinrich-Heine-University Düsseldorf Building 25.22, Room 02.42 Universitätsstraße 1 40225 Düsseldorf, Germany
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Christian Axler. On the sum of the first $n$ prime numbers. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 293-311. doi : 10.5802/jtnb.1081. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1081/

[1] Tom Apostol Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer, 1976 | Zbl

[2] Christian Axler Über die Primzahl-Zählfunktion, die n-te Primzahl und verallgemeinerte Ramanujan-Primzahlen, Ph. D. Thesis, Heinrich-Heine-Universität Düsseldorf (Germany) (2013) (https://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=26247)

[3] Christian Axler On a sequence involving prime numbers, J. Integer Seq., Volume 18 (2015) no. 7, 15.7.6, 13 pages | MR | Zbl

[4] Christian Axler New estimates for some functions defined over primes, Integers, Volume 18 (2018), A52, 21 pages | MR | Zbl

[5] Christian Axler Improving the Estimates for a Sequence Involving Prime Numbers, Notes Number Theory Discrete Math., Volume 25 (2019) no. 1, pp. 8-13 | DOI | Zbl

[6] Christian Axler New estimates for the nth prime number, J. Integer Seq., Volume 22 (2019) no. 4, 19.4.2, 30 pages | Zbl

[7] Michele Cipolla La determinazione assintotica dell’ n i mo numero primo, Napoli Rend., Volume 8 (1902), pp. 132-166 | Zbl

[8] Marc Deléglise; Jean-Louis Nicolas Maximal product of primes whose sum is bounded, Proc. Steklov Inst. Math., Volume 282 (2013) no. 1, pp. 73-102 | DOI | MR | Zbl

[9] Marc Deléglise; Jean-Louis Nicolas An arithmetic equivalence of the Riemann hypothesis, J. Aust. Math. Soc., Volume 106 (2019) no. 2, pp. 235-273 | DOI | MR | Zbl

[10] Pierre Dusart Autour de la fonction qui compte le nombre de nombres premiers, Ph. D. Thesis, Université de Limoges (France) (1998) (http://www.theses.fr/1998LIMO0007)

[11] Pierre Dusart The kth prime is greater than k(lnk+lnlnk-1) for k≥2, Math. Comput., Volume 68 (1999) no. 225, pp. 411-415 | DOI | MR | Zbl

[12] Pierre Dusart Explicit estimates of some functions over primes, Ramanujan J., Volume 45 (2018) no. 1, pp. 227-251 | DOI | MR | Zbl

[13] Jacques Hadamard Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bull. Soc. Math. Fr., Volume 24 (1896), pp. 199-220 | DOI | Zbl

[14] Medhi Hassani On the ratio of the arithmetic and geometric means of the prime numbers and the number e, Int. J. Number Theory, Volume 9 (2013) no. 6, pp. 1593-1603 | DOI | Zbl

[15] Jean-Pierre Massias; Jean-Louis Nicolas; Guy Robin Effective bounds for the maximal order of an element in the symmetric group, Math. Comput., Volume 53 (1989) no. 188, pp. 665-678 | DOI | MR | Zbl

[16] Jean-Pierre Massias; Guy Robin Bornes effectives pour certaines fonctions concernant les nombres premiers, J. Théor. Nombres Bordeaux, Volume 8 (1996) no. 1, pp. 213-238 | MR | Zbl

[17] Guy Robin Permanence de relations de récurrence dans certains développements asymptotiques, Publ. Inst. Math., Nouv. Sér., Volume 43 (1988), pp. 17-25 | MR | Zbl

[18] J. Barkley Rosser The n-th prime is greater than nlogn, Proc. Lond. Math. Soc., Volume 45 (1939), pp. 21-44 | DOI | MR | Zbl

[19] J. Barkley Rosser; Lowell Schoenfeld Approximate formulas for some functions of prime numbers, Ill. J. Math., Volume 6 (1962), pp. 64-94 | DOI | MR | Zbl

[20] J. Barkley Rosser; Lowell Schoenfeld Sharper bounds for the Chebyshev functions θ(x) and ψ(x), Math. Comput., Volume 29 (1975), pp. 243-269 | MR | Zbl

[21] Bruno Salvy Fast computation of some asymptotic functional inverses, J. Symb. Comput., Volume 17 (1994) no. 3, pp. 227-236 | DOI | MR | Zbl

[22] Nilotpal K. Sinha On the asymptotic expansion of the sum of the first n primes (2015) (https://arxiv.org/abs/1011.1667)

[23] Zhi-Wei Sun Some new inequalities for primes (2012) (https://arxiv.org/abs/1209.3729)

[24] M. Szalay On the maximal order in S n and S n * , Acta Arith., Volume 37 (1980), pp. 321-331 | DOI | Zbl

[25] Charles-Jean de la Vallée Poussin Recherches analytiques la théorie des nombres premiers, Ann. Soc. scient. Bruxelles, Volume 20 (1896), pp. 183-256 | Zbl

[26] Charles-Jean de la Vallée Poussin Sur la fonction ζ(s) de Riemann et le nombre des nombres premiers inférieurs à une limite donnée, Mem. Couronnés de l’Acad. Roy. Sci. Bruxelles, Volume 59 (1899), pp. 1-74 | Zbl

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