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Selberg's Formula


SelbergsFormula

Let x be a positive number, and define

lambda(d)=mu(d)[ln(x/d)]^2
(1)
f(n)=sum_(d)lambda(d),
(2)

where the sum extends over the divisors d of n, and mu(n) is the Möbius function. Then

 S(x)=sum_(n<=x)f(n)=2xlnx+o(xlnx)
(3)

(Nagell 1951, p. 286).

For x=1, 2, ..., nint(S(x)) is given by 0, 1, 3, 7, 11, 15, 20, 25, ... (OEIS A109507), where nint(x) is the nearest integer function


See also

Prime Number Theorem, Selberg Trace Formula

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References

Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976.Nagell, T. "Further Lemmata. Proofs of Selberg's Formula." §73 in Introduction to Number Theory. New York: Wiley, pp. 279-280 and 283-286, 1951.Selberg, A. "An Elementary Proof of the Prime Number Theorem." Ann. Math. 50, 305-313, 1949.Sloane, N. J. A. Sequence A109507 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Selberg's Formula

Cite this as:

Weisstein, Eric W. "Selberg's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SelbergsFormula.html

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