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Euler Product

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For s>1, the Riemann zeta function is given by

zeta(s)=sum_(n=1)^(infty)1/(n^s)
(1)
=product_(k=1)^(infty)1/(1-1/(p_k^s)),
(2)

where p_k is the kth prime. This is Euler's product (Whittaker and Watson 1990), called by Havil (2003, p. 61) the "all-important formula" and by Derbyshire (2004, pp. 104-106) the "golden key."

This can be proved by expanding the product, writing each term as a geometric series, expanding, multiplying, and rearranging terms,

product_(k=1)^infty1/(1-1/(p_k^s))=1/(1-1/(p_1^s))1/(1-1/(p_2^s))1/(1-1/(p_3^s))... =[sum_(k=0)^infty(1/(p_1^s))^k][sum_(k=0)^infty(1/(p_2^s))^k][sum_(k=0)^infty(1/(p_3^s))^k]... =(1+1/(p_1^s)+1/(p_1^(2s))+1/(p_1^(3s))+...)(1+1/(p_2^s)+1/(p_2^(2s))+1/(p_2^(3s))+...)... =1+sum_(1<=i)1/(p_i^s)+sum_(1<=i<=j)1/(p_i^sp_j^s)+sum_(1<=i<=j<=k)1/(p_i^sp_j^sp_k^s)+... =1+1/(2^s)+1/(3^s)+1/(4^s)+1/(5^s)+... =sum_(n=1)^infty1/(n^s) =zeta(s).
(3)

Here, the rearrangement leading to equation (1) follows from the fundamental theorem of arithmetic, since each product of prime powers appears in exactly one denominator and each positive integer equals exactly one product of prime powers.

This product is related to the Möbius function mu(n) via

1/(zeta(s))=sum_(n=1)^infty(mu(n))/(n^s),
(4)

which can be seen by expanding the product to obtain

1/(zeta(s))=product_(k=1)^(infty)(1-1/(p_k^s))
(5)
=(1-1/(p_1^s))(1-1/(p_2^s))(1-1/(p_3^s))...
(6)
=1-(1/(p_1^s)+1/(p_2^s)+1/(p_3^s)+...)+(1/(p_1^sp_2^s)+...+1/(p_1^sp_3^s)+1/(p_2^sp_3^s)+...)-...
(7)
=1-sum_(0<i)1/(p_i^s)+sum_(0<i<j)1/(p_i^sp_j^s)-sum_(0<i<j<k)1/(p_i^sp_j^sp_k^s)+...
(8)
=sum_(n=1)^(infty)(mu(n))/(n^s).
(9)

zeta(1)=infty, but the finite product exists, giving

P(n)=product_(k=1)^n1/(1-1/(p_k)).
(10)

For upper limits n=0, 1, 2, ..., the products are given by 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, ... (OEIS A060753 and A038110). Premultiplying by 1/lnp_n and letting n->infty gives a beautiful result known as the Mertens theorem.

The Euler product appears briefly in a pan of John Nash's (played by Russell Crowe) blackboard scribblings in Ron Howard's 2001 film A Beautiful Mind.

See also

Dedekind Function, Dirichlet L-Series, Euler-Mascheroni Constant, Infinite Product, Mertens Theorem, Prime Products, Riemann Zeta Function, Stieltjes Constants

Portions of this entry contributed by Jonathan Sondow (author's link)

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References

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.Edwards, H. M. "The Euler Product Formula." §1.2 in Riemann's Zeta Function. New York: Dover, pp. 6-7, 2001.Euler, L. "Variae observationes circa series infinitas." St. Petersburg Acad., 1737.Hardy, G. H. and Wright, E. M. "The Zeta Function." §17.2 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 245-247, 1979.Havil, J. "The All-Important Formula." §7.1 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 61-62, 2003.Ribenboim, P. The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, p. 216, 1996.Shimura, G. Euler Products and Eisenstein Series. Providence, RI: Amer. Math. Soc., 1997.Sloane, N. J. A. Sequences A038110 and A060753 in "The On-Line Encyclopedia of Integer Sequences."Whittaker, E. T. and Watson, G. N. "Euler's Product for zeta(s)." §13.3 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 271-272, 1990.

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Euler Product

Cite this as:

Sondow, Jonathan and Weisstein, Eric W. "Euler Product." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerProduct.html

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