On the Product of the Primes

Abstract

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 )<4 n , 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(3.37) n , and Selfridge (unpublished) proved A ( n )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.

Locations

• Canadian Mathematical Bulletin - View - PDF