A384455 Decimal expansion of Sum_{k>=2} (-1)^k*P(k)/(k+1) - M/2 (negated), where P(s) is the prime zeta function and M is Mertens's constant.

0, 1, 2, 5, 3, 4, 6, 3, 4, 1, 9, 1, 4, 9, 6, 7, 0, 1, 1, 0, 3, 9, 7, 0, 6, 0, 7, 2, 5, 7, 1, 7, 7, 1, 6, 7, 4, 6, 3, 2, 9, 2, 5, 7, 2, 2, 3, 3, 3, 1, 0, 5, 1, 7, 2, 2, 6, 5, 1, 5, 2, 1, 5, 7, 3, 1, 6, 3, 0, 0, 7, 1, 0, 5, 9, 1, 8, 9, 1, 8, 1, 6, 1, 8, 2, 9, 1, 6, 4, 1, 7, 2, 3, 3, 8, 6, 1, 7, 0, 9, 2, 9, 9, 0, 9, 0

Offset: 0

Author

Amiram Eldar, May 30 2025

Keywords

• nonn
• cons

Comments

The constant C in Theorem 2.2 in Jakimczuk (2025): Product_{p prime <= x} (1 + 1/p)^p = exp(PrimePi(x) + C)/sqrt(log(x)) + O(exp(PrimePi(x))/(sqrt(log(x))*exp(a*sqrt(log(x))))), where a is a positive constant, and Product_{k=1..n} (1 + 1/prime(k))^prime(k) = (exp(n + C)/sqrt(log(n))) * (1 - log(log(n))/(2*log(n)) + o(log(log(n))/log(n))).

Examples

			-0.01253463419149670110397060725717716746329257223331...
		

Links

• Rafael Jakimczuk, Two Topics in Number Theory: Products Related with the e Number and Sum of Subscripts in Prime Numbers, ResearchGate, May 2025. See p. 3.

Crossrefs

Cf. A000720 (PrimePi), A001620, A077761 (Mertens's constant), A110544 (analogous with product over positive integers), A229495.

Programs

PARI

suminf(k = 2, (1/(2*k) + (-1)^k/(k+1)) * sumeulerrat(1/p^k)) - Euler/2

Formula

Equals -gamma/2 + Sum_{k>=2} (1/(2*k) + (-1)^k/(k+1)) * P(k), where P(s) is the prime zeta function and gamma is Euler's constant (A001620).