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Mann,s Theorem

المؤلف:  Khinchin, A. Y.

المصدر:  "The Landau-Schnirelmann Hypothesis and Mann,s Theorem." Ch. 2 in Three Pearls of Number Theory. New York: Dover

الجزء والصفحة:  ...

19-9-2020

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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

{1,sigma(A)+sigma(B)}. " src="https://mathworld.wolfram.com/images/equations/MannsTheorem/NumberedEquation1.gif" style="height:15px; width:198px" />

Here,  denotes the direct sum, i.e., {a+b:a in A,b in B}" src="https://mathworld.wolfram.com/images/equations/MannsTheorem/Inline5.gif" style="height:15px; width:169px" />, and  is the Schnirelmann density.

Mann's theorem is optimal in the sense that {0,1,11,12,13,...}" src="https://mathworld.wolfram.com/images/equations/MannsTheorem/Inline7.gif" style="height:15px; width:172px" /> satisfies .

Mann's theorem implies Schnirelmann's theorem as follows. Let {0,1} union {p:p prime}" src="https://mathworld.wolfram.com/images/equations/MannsTheorem/Inline9.gif" style="height:15px; width:150px" />, 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.

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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.