On a Result of Hardy and Ramanujan

Authors

  • Mohammad Avalin Charsooghi Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
  • Yousof Azizi Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
  • Mehdi Hassani Department of Mathematics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
  • Laleh Mollazadeh-Beidokhti Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran

DOI:

https://doi.org/10.5644/SJM.04.2.01

Keywords:

Factorial function, prime number, inequality

Abstract

In this paper, we introduce some explicit approximations for the summation $\sum_{k\leq n}\Omega(k)$, where $\Omega(k)$ is the total number of prime factors of $k$.

 

2000 Mathematics Subject Classification. 05A10, 11A41, 26D15, 26D20

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References

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Dover Publications, 1972.

Apostol, Mathematical Analysis, Addison-Wesley Publishing Company, Inc., 1957.

P. Dusart, In´egalit´es explicites pour $psi(X)$, $theta(X)$, $pi(X)$ et les nombres premiers, C. R. Math. Acad. Sci. Soc. R. Can., 21 (2) (1999), 53–59.

G. Hardy and S. Ramanujan, The normal number of prime factors of a number n, Quart. J. Math., 48 (1917), 76–92.

M. Hassani, Equations and inequalities involving vp(n!), J. Inequal. Pure Appl. Math. (JIPAM), 6 (2) (2005), Article 29.

M. Hassani, On the decomposition of n! into primes, arXiv:math/0606316v6 [math.-NT].

Melvyn B. Nathanson, Elementary Methods in Number Theory, Springer, 2000.

J. S´andor, D. S. Mitrinovi´c and B. Crstici, Handbook of Number Theory I, Second printing of the 1996 original, Springer, Dordrecht, 2006.

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Published

11.06.2024

How to Cite

Charsooghi, M. A., Azizi, Y., Hassani, M., & Mollazadeh-Beidokhti, L. (2024). On a Result of Hardy and Ramanujan. Sarajevo Journal of Mathematics, 4(2), 147–153. https://doi.org/10.5644/SJM.04.2.01

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Articles