An asymptotic formula for the Chebyshev theta function

Aditya Ghosh
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 4, Pages 1–7
DOI: 10.7546/nntdm.2019.25.4.1-7
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Aditya Ghosh
Indian Statistical Institute, Kolkata, India

Abstract

Let \{p_n\}_{n\ge 1} be the sequence of primes and \vartheta(x) = \sum_{p \leq x} \log p, where p runs over the primes not exceeding x, be the Chebyshev \vartheta-function. In this note, we derive lower and upper bounds for \vartheta(p_n)/n, by comparing it with \log p_{n+1} and deduce the asymptotic formula \vartheta(p_n)/n=\log p_{n+1}\left(1-\frac{1}{\log n}+\frac{\log\log n}{\log^2 n}\left(1+o(1)\right)\right).

Keywords

  • Chebyshev theta function
  • Geometric mean of first n primes
  • Prime numbers

2010 Mathematics Subject Classification

  • 11A41
  • 11A25

References

  1. Axler, C. (2018). On the arithmetic and geometric means of the first n prime numbers, Mediterr. J. Math., 15 (3), Art. 93, 21 pages.
  2. Bonse, H. (1907). Uber eine bekannte Eigenschaft der Zahl 30 und ihre Verallgemeinerung, Archiv Math. Phys., 3 (12), 292–295.
  3. Dusart, P. (1998). Autour de la fonction qui compte le nombre de nombres premiers, PhD Thesis, Limoges.
  4. Dusart, P. (1999). The k-th prime is greater than k(ln k + ln ln k − 1) for k ≥ 2, Math. Comp., 68 (225), 411–415.
  5. Hassani, M. (2005). Approximation of the product p1p2…pn, RGMIA Research Report Collection, 8 (2), Article 20.
  6. Massias, J.-P., & Robin, G. (1996). Bornes effectives pour certaines fonctions concernant les nombres premiers, J. Theor. Nombres de Bordeaux, 8, 213–238.
  7. Panaitopol, L. (2000). An inequality involving prime numbers, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 11, 33–35.
  8. Posa, L. (1960). Uber eine Eigenschaft der Primzahlen, Mat. Lapok, 11, 124–129.
  9. Robin, G. (1983). Estimation de la fonction de Tchebychef θ sur le k-ieme nombre premier et grandes valeurs de la fonction ω(n) nombre de diviseurs premiers de n, Acta Arith., 42 (4), 367–389..

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Cite this paper

Ghosh, A. (2019). An asymptotic formula for the Chebyshev theta function. Notes on Number Theory and Discrete Mathematics, 25(4), 1-7, DOI: 10.7546/nntdm.2019.25.4.1-7.

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