A321943 Decimal expansion of Ni_1 = (1/2)*(gamma - log(2*Pi)) + 1, where gamma is Euler's constant (or the Euler-Mascheroni constant).

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

Offset

0,1

Comments

This constant links Euler's constant and Pi to the values of the Riemann zeta function at positive integers (see formulas).

References

D. Suryanarayana, Sums of Riemann zeta function, Math. Student, 42 (1974), 141-143.

Links

Stefano Spezia, Table of n, a(n) for n = 0..10000

Bernard Candelpergher, Ramanujan summation of divergent series, HAL Id : hal-01150208; Lecture Notes in Math. Series (Springer), 2185, (2017), 93.

Junesang Choi, H. M. Srivastava, and J. R. Quine, Some series involving the zeta function, Bulletin of the Australian Mathematical Society, Vol. 51, No. 3 (1995), pp. 383-393. See eq. (2.29), p. 390.

Marc-Antoine Coppo, A note on some alternating series involving zeta and multiple zeta values, Journal of Mathematical Analysis and Applications Volume 475, Issue 2, 15 July 2019, Pages 1831-1841; Preprint, <hal-01735381v4>, 2018.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 378.

R. J. Singh and V. P. Verma, Some series involving Riemann zeta function, Yokohama Math. J. 31 (1983), 1-4.

H. M. Srivastava, Sums of certain series of the Riemann zeta function, J. Math. Anal. App. 134 (1988), 129-140.

Formula

Ni_1 = Sum_{k>=2} (-1)^k*zeta(k)/(k+1).

Ni_1 = Sum_{n>0} (Integral_{x=0..1} x^2*(1-x)_{n-1} dx)/(n*n!), where (z)_n = z*(z+1)*(z+2)*...*(z+n-1) is the Pochhammer symbol.

Ni_1 = Sum_{n>=0} A193546(n)/(A000290(n + 1)*A194506(n)).

Equals Sum_{k>=1} (1/(2*k) + k * log(1+1/k) - 1) (Shamos, 2007). - Amiram Eldar, Oct 10 2025

Example

0.369669299246093688522926308635583575659682194332178386585...

Maple

Digits := 100; evalf((1/2)*(gamma-ln(2*Pi))+1);

Mathematica

First[RealDigits[N[(1/2)*(EulerGamma-Log[2*Pi])+1, 100], 10]]

Prog

(PARI) (1/2)*(Euler-log(2*Pi))+1
(Python)
from mpmath import *
mp.dps = 100; mp.pretty = True
+(1/2)*(euler-log(2*pi))+1

Crossrefs

Cf. A001620 (Euler's constant), A000796 (Pi).

Cf. A193546, A000290, A194506, A131688.

Sequence in context: A067722 A382236 A049341 * A386691 A351189 A187082

Adjacent sequences: A321940 A321941 A321942 * A321944 A321945 A321946

Keyword

nonn,cons

Author

Stefano Spezia, Dec 12 2018

Status

approved