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

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Cite this paper

APA

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.

Chicago

Ghosh, Aditya. “An Asymptotic Formula for the Chebyshev Theta Function.” Notes on Number Theory and Discrete Mathematics 25, no. 4 (2019): 1-7, doi: 10.7546/nntdm.2019.25.4.1-7.

MLA

Ghosh, Aditya. “An Asymptotic Formula for the Chebyshev Theta Function.” Notes on Number Theory and Discrete Mathematics 25.4 (2019): 1-7. Print, doi: 10.7546/nntdm.2019.25.4.1-7.

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