A229495 -id:A229495 - cp's OEIS frontend

A110544 Decimal expansion of -Integral {x=1..2} log gamma(x) dx.

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

Offset: 0

Robert G. Wilson v, Jul 25 2005

			0.081061466795327258219670263594382360138602526362216587182848459...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.21, p. 168.

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. 2, the constant C in eq. (2.1).
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), p. 39, eq. (1.8.1).
Eric Weisstein's World of Mathematics, Log Gamma Function.
Eric Weisstein's World of Mathematics, Gamma Function.

Cf. A001620, A075700, A110543, A229495.

RealDigits[ N[ -Integrate[ Log[ Gamma[ x]], {x, 1, 2}], 128], 10, 128]
RealDigits[ 1/2*Log[2*Pi]-1, 10, 105] // First // Prepend[#, 0]& (* Jean-François Alcover, Jun 10 2013 *)

-intnum(x=1, 2, log(gamma(x))) \\ Michel Marcus, Jul 05 2020

Equals zeta'(0)+1 = -1/2*log(2*Pi)+1. - Jean-François Alcover, Jun 10 2013
From Amiram Eldar, Jul 05 2020: (Start)
Equals Sum_{k>=2} (1/(k + 1) - 1/(2*k))*(zeta(k)-1).
Equals Integral_{x=0..1} (1/2 - 1/(1 - x) - 1/log(x)) dx/log(x). (End)
Equals -Integral_{x=1..oo} ({x}-1/2)/x dx, where {.} is the fractional part [Nahin]. - R. J. Mathar, May 16 2024
Equals 1 - A075700 = log(A229495). - Hugo Pfoertner, Sep 05 2024
Equals log(2) - (gamma+1)/2 - Sum_{k>=2} (-1)^k*(zeta(k)-1)/(k+1), where gamma is Euler's constant (A001620) (Jakimczuk, 2025). - Amiram Eldar, May 30 2025

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.

Original entry on oeis.org

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

Amiram Eldar, May 30 2025

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))).

			-0.01253463419149670110397060725717716746329257223331...

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.

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

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

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).

cp's OEIS Frontend

A110544 Decimal expansion of -Integral {x=1..2} log gamma(x) dx.

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