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
782
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
![sigma(A direct sum B)>=min<span style=]() {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 direct sum B=<span style=]()
{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 ![A=B=<span style=]()
{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 ![P=<span style=]()
{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.
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.
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