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Prime Products


The product of primes

 p_n#=product_(k=1)^np_k,
(1)

with p_n the nth prime, is called the primorial function, by analogy with the factorial function. Its logarithm is closely related to the Chebyshev function theta(x).

The zeta-regularized product over all primes is given by

p_infty#=product_(k=1)^^^inftyp_k
(2)
=4pi^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

 product_(k=1)^^^inftyp_k^s=(2pi)^(2s)
(4)

and

 product_(k=1)^^^infty(p_k^s-1)=((2pi)^(2s))/(zeta(s))
(5)

(Muñoz Garcia and Pérez-Marco 2003).

Mertens theorem states that

 lim_(n->infty)1/(lnp_n)product_(k=1)^n1/(1-1/(p_k))=e^gamma,
(6)

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

 lim_(n->infty)lnp_nproduct_(k=1)^n1/(1+1/(p_k))=(pi^2)/(6e^gamma).
(7)

There are amazing infinite product formulas for primes given by

 product_(k=1)^infty(p_k^2+1)/(p_k^2-1)=5/2.
(8)

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

 product_(k=1)^infty(1+1/(p_k^2))=(15)/(pi^2)=1.519817...
(9)

(OEIS A082020; Ramanujan 1913-1914).

More general formulas are given by

 product_(k=1)^infty(1+1/(p_k^s))=(zeta(s))/(zeta(2s)),
(10)

where zeta(s) is the Riemann zeta function and by the Euler product

 product_(k=1)^infty(1-1/(p_k^s))=1/(zeta(s)).
(11)

Named prime products include Barban's constant

C_(Barban)=product_(p)[1+(3p^2-1)/(p(p+1)(p^2-1))]
(12)
=2.596536...
(13)

(OEIS A175640), the Feller-Tornier constant

C_(Feller-Tornier)=1/2+1/2product_(n=1)^(infty)(1-2/(p_n^2))
(14)
=0.6613170494...
(15)

(OEIS A065493), Heath-Brown-Moroz constant

C_(Heath-Brown-Moroz)=product_(p)(1-1/p)^7(1+(7p+1)/(p^2))
(16)
=0.00131764115...
(17)

(OEIS A118228), Murata's constant

C_(Murata)=product_(p)[1+1/((p-1)^2)]
(18)
=2.82641999...
(19)

(OEIS A065485), the quadratic class number constant

Q=product_(p)[1-1/(p^2(p+1))]
(20)
=0.88151383972...
(21)

(OEIS A065465), Sarnak's constant

C_(Sarnak)=product_(p>=3)(1-(p+2)/(p^3))
(22)
=0.7236484022...
(23)

(OEIS A065476), and Taniguchi's constant

C_(Taniguchi)=product_(p)[1-3/(p^3)+2/(p^4)+1/(p^5)-1/(p^6)]
(24)
=0.6782344...
(25)

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

Define the number theoretic character chi(p) by

 chi(p)={+1   if p=1 (mod 4); -1   if p=3 (mod 4),
(26)

then

product_(k=2)^(infty)[1+(chi(p_k))/(p_k)]=product_(k=2)^(infty)(1-1/(p_k^2))/(1-(chi(p_k))/(p_k))
(27)
=(4/3product_(k=1)^(infty)1-1/(p_k^2))/(product_(k=2)^(infty)1-(chi(p_k))/(p_k))
(28)
=4/3[zeta(2)]^(-1)L(chi,1)
(29)
=8/(pi^2)pi/4
(30)
=2/pi
(31)
=0.636619...
(32)

(OEIS A060294; Oakes 2003). Similarly,

product_(k=2)^(infty)[1-(chi(p_k))/(p_k)]=4/pi
(33)
=1.273239...
(34)

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

pi/2=product_(n=1)^(infty)[1+(sin(1/2pip_n))/(p_n)]^(-1)
(35)
=product_(n=2)^(infty)[1+((-1)^((p_n-1)/2))/(p_n)]^(-1)
(36)

(Blatner 1997).

Let Q_2(n) be the number of consecutive numbers (k,k+1) with k<=n such that k and k+1 are both squarefree. Then Q_2(n)/n is given asymptotically by

 product_(n=1)^infty(1-2/(p_n^2))=0.3226340989...
(37)

(OEIS A065474), where p_n is the nth prime.


See also

Artin's Constant, Chebyshev Functions, Euler Product, Feller-Tornier Constant, Heath-Brown-Moroz Constant, Infinite Product, Mertens Theorem, Murata's Constant, Prime Constellation, Prime Formulas, Prime Number, Prime Sums, Primorial, Primorial Prime, Quadratic Class Number Constant, Sarnak's Constant, Stephens' Constant, Totient Summatory Function, Twin Primes Constant

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References

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 4pi^2." 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 4pi^2." Commun. Math. Phys. 277, 69-81, 2008.Niklasch, G. "Some Number-Theoretical Constants Arising as Products of Rational Functions of p 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.Ramanujan, S. "Modular Equations and Approximations to pi." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.Sloane, N. J. A. Sequences A065465, A065474, A065485, A065493, A082020, A118228, A175639, and A175640 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.

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Prime Products

Cite this as:

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

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