The product of primes
(1)
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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)
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(3)
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(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)
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and
(5)
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(Muñoz Garcia and Pérez-Marco 2003).
Mertens theorem states that
(6)
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where is the Euler-Mascheroni constant, and a closely related result is given by
(7)
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There are amazing infinite product formulas for primes given by
(8)
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(Ramanujan 1913-1914; Le Lionnais 1983, p. 46) and
(9)
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(OEIS A082020; Ramanujan 1913-1914).
More general formulas are given by
(10)
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where is the Riemann zeta function and by the Euler product
(11)
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Named prime products include Barban's constant
(12)
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(13)
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(OEIS A175640), the Feller-Tornier constant
(14)
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(15)
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(OEIS A065493), Heath-Brown-Moroz constant
(16)
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(17)
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(OEIS A118228), Murata's constant
(18)
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(19)
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(OEIS A065485), the quadratic class number constant
(20)
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(21)
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(OEIS A065465), Sarnak's constant
(22)
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(23)
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(OEIS A065476), and Taniguchi's constant
(24)
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(25)
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(OEIS A175639), where the product is over the primes .
Define the number theoretic character by
(26)
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then
(27)
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(28)
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(29)
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(30)
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(31)
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(32)
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(OEIS A060294; Oakes 2003). Similarly,
(33)
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(34)
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(Oakes 2004). This is equivalent to the formula due to Euler
(35)
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(36)
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(Blatner 1997).
Let be the number of consecutive numbers with such that and are both squarefree. Then is given asymptotically by
(37)
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(OEIS A065474), where is the th prime.