A360807 - OEIS

A360807

Decimal expansion of Sum_{m>=1} 1/(1/4 + z(m)^2) where z(m) is the imaginary part of the m-th nontrivial zero of the Dirichlet beta function whose real part is 1/2.

%I #28 Apr 09 2023 02:36:14

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

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

%U 4,1,0,3,3,0,7,1,8,6,3,3,6,8,8,0,8,2,6

%N Decimal expansion of Sum_{m>=1} 1/(1/4 + z(m)^2) where z(m) is the imaginary part of the m-th nontrivial zero of the Dirichlet beta function whose real part is 1/2.

%C Conjecture: Nontrivial zeros whose real part is not 1/2 do not exist.

%H Kano Kono, <a href="https://fractional-calculus.com/summary_dirichlet_beta.pdf">Summary Dirichlet beta function</a>, Alien's Mathematics, p. 5.

%F Equals 4*log(Gamma(3/4)) + A001620/2 + log(2) - 3*log(Pi)/2.

%F Equals A074760 - 1 + log(4) - log(Pi) + 4*(log(Gamma(3/4)).

%F Equals 1 - A074760 + A001620 - 2*log(Pi) + 4*(log(Gamma(3/4)).

%F Equals 3* A074760 - 3 + A001620 + 4*log(2) + 4*(log(Gamma(3/4)).

%e 0.077783989961792964431079...

%t kk = RealDigits[N[4 Log[Gamma[3/4]] + EulerGamma/2 + Log[2] - 3 Log[Pi]/2, 115]][[1]]; Prepend[kk, 0]

%o (PARI) 4*log(gamma(3/4)) + Euler/2 + log(2) - 3*log(Pi)/2 \\ _Michel Marcus_, Mar 15 2023

%Y Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360.

%K nonn,cons

%O 0,2

%A _Artur Jasinski_, Feb 21 2023