Taniguchi's Constant

Taniguchi's constant is defined as

			(1)
			(2)

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

			(3)
			(4)

where  is the prime zeta function and the s are rational numbers given as the coefficients of  in the series

(5)

REFERENCES:

Finch, S. "Class Number Theory." https://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." https://arxiv.org/abs/math/0410531. 3 Jul 2006.