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.

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

Offset: 0

Author

Artur Jasinski, Feb 21 2023

Keywords

• nonn
• cons

Comments

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

Examples

			0.077783989961792964431079...
		

Links

• Kano Kono, Summary Dirichlet beta function, Alien's Mathematics, p. 5.

Crossrefs

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

Programs

Mathematica

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

PARI

4*log(gamma(3/4)) + Euler/2 + log(2) - 3*log(Pi)/2 \\ Michel Marcus, Mar 15 2023

Formula

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

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

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

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