login
Decimal expansion of Sum_{k>=1} (-1)^(k+1) * zeta(2k+1)/(2k+1).
4

%I #29 Jul 21 2022 05:59:20

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

%T 9,0,3,8,2,2,5,9,5,6,1,9,9,6,1,5,8,3,6,1,8,6,5,0,2,3,2,2,9,5,0,1,8,0,

%U 7,1,4,5,3,3,8,1,6,1,7,1,7,5,3,7,0,4,8,8,3,6,3,1,8,9,5,1,7,3,7

%N Decimal expansion of Sum_{k>=1} (-1)^(k+1) * zeta(2k+1)/(2k+1).

%C Is there a closed-form formula for this constant as for A352527?

%D Bernard Candelpergher, Ramanujan Summation of Divergent Series, Springer, 2017, p. 35.

%H Cornel Ioan Vălean, <a href="https://gaceta.rsme.es/abrir.php?id=1450">Problema 327</a>, La Gaceta de la Real Sociedad Matemática Española, Vol. 21, No. 2 (2018), pp. 331-343.

%F Equals gamma + arg(i!) (see Vălean).

%F Equals A001620 - A212880.

%F Equals Sum_{k>=1} (1/k - arctan(1/k)). - _Amiram Eldar_, Jul 21 2022

%e 0.2755753444339996627189...

%p evalf(gamma + argument(I!),100);

%t RealDigits[EulerGamma + Arg[Gamma[1 + I]], 10, 100][[1]] (* _Amiram Eldar_, Mar 24 2022 *)

%o (PARI) Euler + arg(I*gamma(I)) \\ _Michel Marcus_, Mar 25 2022

%Y Cf. A001620, A212880, A352527.

%K nonn,cons

%O 0,1

%A _Bernard Schott_, Mar 24 2022