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

%I A321943 #33 Dec 23 2024 22:02:35
%S A321943 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,
%T A321943 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,
%U A321943 8,1,6,7,4,6,1,5,7,1,9,3,7,7,7,3,7,3,0
%N A321943 Decimal expansion of Ni_1 = (1/2)*(gamma - log(2*Pi)) + 1, where gamma is Euler's constant (or the Euler-Mascheroni constant).
%C A321943 This constant links Euler's constant and Pi to the values of the Riemann zeta function at positive integers (see formulas).
%D A321943 D. Suryanarayana, Sums of Riemann zeta function, Math. Student, 42 (1974), 141-143.
%H A321943 Stefano Spezia, <a href="/A321943/b321943.txt">Table of n, a(n) for n = 0..10000</a>
%H A321943 B. Candelpergher, <a href="https://hal.archives-ouvertes.fr/hal-01150208">Ramanujan summation of divergent series</a>, HAL Id : hal-01150208; Lecture Notes in Math. Series (Springer), 2185, (2017), 93.
%H A321943 Marc-Antoine Coppo, <a href="https://doi.org/10.1016/j.jmaa.2019.03.057">A note on some alternating series involving zeta and multiple zeta values</a>, Journal of Mathematical Analysis and Applications Volume 475, Issue 2, 15 July 2019, Pages 1831-1841; <a href="https://hal.univ-cotedazur.fr/hal-01735381">Preprint</a>,  <hal-01735381v4>, 2018.
%H A321943 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2007). See p. 378.
%H A321943 R. J. Singh and V. P. Verma, <a href="https://ynu.repo.nii.ac.jp/?action=repository_action_common_download&amp;item_id=6701&amp;item_no=1&amp;attribute_id=20&amp;file_no=1">Some series involving Riemann zeta function</a>, Yokohama Math. J. 31 (1983), 1-4.
%H A321943 H. M. Srivastava, <a href="https://doi.org/10.1016/0022-247X(88)90013-3">Sums of certain series of the Riemann zeta function</a>, J. Math. Anal. App. 134 (1988), 129-140.
%F A321943 Ni_1 = Sum_{k>=2} (-1)^k*zeta(k)/(k+1).
%F A321943 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.
%F A321943 Ni_1 = Sum_{n>=0} A193546(n)/(A000290(n + 1)*A194506(n)).
%e A321943 0.369669299246093688522926308635583575659682194332178386585...
%p A321943 Digits := 100; evalf((1/2)*(gamma-ln(2*Pi))+1);
%t A321943 First[RealDigits[N[(1/2)*(EulerGamma-Log[2*Pi])+1, 100], 10]]
%o A321943 (PARI) (1/2)*(Euler-log(2*Pi))+1
%o A321943 (Python)
%o A321943 from mpmath import *
%o A321943 mp.dps = 100; mp.pretty = True
%o A321943 +(1/2)*(euler-log(2*pi))+1
%Y A321943 Cf. A001620 (Euler's constant), A000796 (Pi).
%Y A321943 Cf. A193546, A000290, A194506, A131688.
%K A321943 nonn,cons
%O A321943 0,1
%A A321943 _Stefano Spezia_, Dec 12 2018