Selberg's Formula

Let  be a positive number, and define

			(1)
			(2)

where the sum extends over the divisors  of , and  is the Möbius function. Then

(3)

(Nagell 1951, p. 286).

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

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."