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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 ). A note by Moser  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(3.37)n, and Selfridge (unpublished) proved A(n)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  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  and  considerably.