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On the Product of the Primes

  • Denis Hanson (a1)

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In recent years several attempts have been made to obtain estimates for the product of the primes less than or equal to a given integer n. Denote by the above-mentioned product and define as usual

Analysis of binomial and multinomial coefficients has led to results such as A(n)<4n, due to Erdôs and Kalmar (see [2]). A note by Moser [3] gave an inductive proof of A(n)<(3.37)n, and Selfridge (unpublished) proved A(n)<(3.05)n More accurate results are known, in particular those in a paper of Rosser and Schoenfeld [4] in which they prove Θ(n)< 1.01624n; however their methods are considerably deeper and involve the theory of a complex variable as well as heavy computations. Using only elementary methods we will prove the following theorem, which improves the results of [2] and [3] considerably.

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References

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1. Appel, K. I. and Rosser, J. B., Tables for estimating functions of primes, Comm. Research Div. Technical No. 4, Von Neumann Hall, Princeton, N.J. (Sept. 1961).
2. Hardy, G. H. and Wright, E. M., The theory of numbers, Oxford Univ. Press, London, Ch. XXII, 4th éd., 1959.
3. Moser, L., On the product of the primes not exceeding n , Canad. Math. Bull. (2) 2 (1959), 119-121.
4. Rosser, J. B. and Schoenfeld, L., Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94.
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