A Remark on the Factorization of Factorials

Document Type : Original Scientific Paper

Authors

Department of Mathematics, University of Zanjan, University Blvd., 45371-38791 Zanjan, I. R. Iran

Abstract

The subject of this paper is to study distribution of the prime factors p and their exponents, which we denote by vp (n!), in standard factorization of n! into primes. We show that for each θ > 0 the primes p not exceeding nθ eventually assume almost all value of the sum ∑p⩽nθ vp(n!). Also, we introduce the notion of θ-truncated factorial, defined by n!θ =∏p⩽nθ  pvp (n!) and we show that the growth of log n!1/2 is almost half of growth of log n!1.

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References

[1] G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n, Q. J. Math. 48 (1917) 76 − 92.
[2] E. G. H. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 3rd edn. Chelsea Publishing Company, New York, 1974.
[3] J. Lee, The second central moment of additive functions, Proc. Amer. Math. Soc. 114 (4) (1992) 887 − 895.
[4] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962) 64 − 94.
Mathematics Interdisciplinary Research
Volume 6, Issue 3
September 2021
Pages 235-242

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