Mann's Theorem

Mann's theorem is a theorem widely circulated as the " conjecture" that was subsequently proven by Mann (1942). It states that if  and  are sets of integers each containing 0, then

Here,  denotes the direct sum, i.e., , and  is the Schnirelmann density.

Mann's theorem is optimal in the sense that  satisfies .

Mann's theorem implies Schnirelmann's theorem as follows. Let , then Mann's theorem proves that , so as more and more copies of the primes are included, the Schnirelmann density increases at least linearly, and so reaches 1 with at most  copies of the primes. Since the only sets with Schnirelmann density 1 are the sets containing all positive integers, Schnirelmann's theorem follows.

REFERENCES:

Garrison, B. K. "A Nontransformation Proof of Mann's Density Theorem." J. reine angew. Math. 245, 41-46, 1970.

Khinchin, A. Y. "The Landau-Schnirelmann Hypothesis and Mann's Theorem." Ch. 2 in Three Pearls of Number Theory. New York: Dover, pp. 18-36, 1998.

Mann, H. B. "A Proof of the Fundamental Theorem on the Density of Sets of Positive Integers." Ann. Math. 43, 523-527, 1942.