The product of primes

(1)

with the th prime, is called the primorial function, by analogy with the factorial function. Its logarithm is closely related to the Chebyshev function .

The zeta-regularized product over all primes is given by

			(2)
			(3)

(Muñoz Garcia and Pérez-Marco 2003, 2008), answering the question posed by Soulé et al. (1992, p. 101). A derivation proceeds by algebraic manipulation of the prime zeta function and gives the more general results

(4)

and

(5)

(Muñoz Garcia and Pérez-Marco 2003).

Mertens theorem states that

(6)

where is the Euler-Mascheroni constant, and a closely related result is given by

(7)

There are amazing infinite product formulas for primes given by

(8)

(Ramanujan; Le Lionnais 1983, p. 46) and

(9)

(Sloane's A082020; Ramanujan 1913).

More general formulas are given by

(10)

where is the Riemann zeta function and by the Euler product

(11)

Named prime products include the Feller-Tornier constant

			(12)
			(13)

(Sloane's A065493), Heath-Brown-Moroz constant

			(14)
			(15)
			(16)

(Sloane's A118228), Murata's constant

			(17)
			(18)

(Sloane's A065485), the quadratic class number constant

			(19)
			(20)

(Sloane's A065465), and Sarnak's constant

			(21)
			(22)

(Sloane's A065476).

Define the number theoretic character by

(23)

then

			(24)
			(25)
			(26)
			(27)
			(28)
			(29)

(Sloane's A060294; Oakes 2003a). Similarly,

			(30)
			(31)

(Oakes 2003b). This is equivalent to the formula due to Euler

			(32)
			(33)

(Blatner 1997).

Let be the number of consecutive numbers with such that and are both squarefree. Then is given asymptotically by

(34)

(Sloane's A065474), where is the th prime.

Blatner, D. The Joy of Pi. New York: Walker, p. 110, 1997.

Grosswald, E. "Some Number Theoretical Products." Rev. Columbiana Mat. 21 231-242, 1987.

Guy, R. K. "Products Taken over Primes." §B87 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 102-103, 1994.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1983.

Muñoz García, E. and Pérez Marco, R. "The Product Over All Primes is ." Preprint IHES/M/03/34. May 2003. http://inc.web.ihes.fr/prepub/PREPRINTS/M03/Resu/resu-M03-34.html.

Muñoz García, E. and Pérez Marco, R. "The Product Over All Primes is ." Commun. Math. Phys. 277, 69-81, 2008.

Niklasch, G. "Some Number-Theoretical Constants Arising as Products of Rational Functions of over the Primes." http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml.

Oakes, M. "Re: [PrimeNumbers] pi=(2/1) (3/2) (5/6) (7/6) (11/10) (13/14) (17/18) (19/18)...." Dec. 21, 2003. http://groups.yahoo.com/group/primenumbers/message/14257.

Oakes, M. "Re: primes and pi." Jan. 29, 2004. http://groups.yahoo.com/group/primenumbers/message/14486.

Sloane, N. J. A. Sequences A065465, A065474, A065485, A065493, A082020, and A118228 in "The On-Line Encyclopedia of Integer Sequences."

Soulé, C.; Abramovich, D.; Burnois, J. F.; and Kramer, J. Lectures on Arakelov Geometry. Cambridge, England: Cambridge University Press, 1992.

Uchiyama, S. "On Some Products Involving Primes." Proc. Amer. Math. Soc. 28, 629-630, 1971.

CITE THIS AS:

Weisstein, Eric W. "Prime Products." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PrimeProducts.html