TOPICS
Search

Selberg's Formula

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
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

Explore with Wolfram|Alpha

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

Subject classifications