Prime Products

Prime products are products taken over primes. They include the primorial function

(1)

where is the th prime and is a positive integer. The name is an analogy with the factorial function. Its logarithm is closely related to the Chebyshev function .

Using the extension of zeta-regularized product defined by Muñoz Garcia and Pérez-Marco (2008, pp. 72-75), the regularized product over all primes is given by

			(2)
			(3)

(Muñoz Garcia and Pérez-Marco 2008, pp. 78-79), answering the question posed by Soulé et al. (1992, p. 101). The ordinary definition does not apply because the prime zeta function has no analytic continuation to a neighborhood of 0. Together with the Euler product, the extension gives, for real ,

(4)

and

(5)

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 1913-1914; Le Lionnais 1983, p. 46) and

(9)

(OEIS A082020; Ramanujan 1913-1914).

For , more general formulas are given by

(10)

where is the Riemann zeta function and by the Euler product

(11)

Named prime products include Barban's constant

			(12)
			(13)

(OEIS A175640), the Feller-Tornier constant

			(14)
			(15)

(OEIS A065493), Heath-Brown-Moroz constant

			(16)
			(17)

(OEIS A118228), Murata's constant

			(18)
			(19)

(OEIS A065485), the quadratic class number constant

			(20)
			(21)

(OEIS A065465), Sarnak's constant

			(22)
			(23)

(OEIS A065476), and Taniguchi's constant

			(24)
			(25)

(OEIS A175639), where the product is over the primes .

For an odd prime , define the number theoretic character by

(26)

then

			(27)
			(28)
			(29)
			(30)
			(31)
			(32)

(OEIS A060294; Oakes 2003). Similarly,

			(33)
			(34)

(Oakes 2004). This is equivalent to the formula due to Euler

			(35)
			(36)

(Blatner 1997).

For a positive integer , let count the pairs of consecutive positive integers with such that and are both squarefree. Then is given asymptotically by

(37)

(OEIS A065474), where is the th prime.