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Taniguchi's Constant


Taniguchi's constant is defined as

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

(OEIS A175639), where the product is over the primes p. Taking the logarithm, expand the sum about infinity, and then summing the terms gives a "closed" form as

C_(Taniguchi)=exp[sum_(n=3)^(infty)c_nP(n)]
(3)
=exp[-3P(3)+2P(4)+P(5)-(11)/2P(6)+6P(7)+...],
(4)

where P(n) is the prime zeta function and the c_ns are rational numbers given as the coefficients of p^(-1) in the series

 ln(1-1/(p^6)+1/(p^5)+2/(p^4)-3/(p^3))=-3/(p^3)+2/(p^4)+1/(p^5)+....
(5)

See also

Artin's Constant, Barban's Constant, Feller-Tornier Constant, Heath-Brown-Moroz Constant, Murata's Constant, Prime Products, Prime Zeta Function, Quadratic Class Number Constant, Sarnak's Constant, Twin Primes Constant

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References

Finch, S. "Class Number Theory." http://algo.inria.fr/csolve/clss.pdf. May 6, 2005.Sloane, N. J. A. Sequence A175639 in "The On-Line Encyclopedia of Integer Sequences."Taniguchi, T. "A Mean Value Theorem for the Square of Class Number Times Regulator of Quadratic Extensions." http://arxiv.org/abs/math/0410531. 3 Jul 2006.

Cite this as:

Weisstein, Eric W. "Taniguchi's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TaniguchisConstant.html

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