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

Author

Stefano Spezia, Dec 12 2018

Keywords

• nonn
• cons

Comments

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

Examples

			0.369669299246093688522926308635583575659682194332178386585...
		

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

Crossrefs

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

Cf. A193546, A000290, A194506, A131688.

Programs

Maple

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

Mathematica

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

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

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)).