cp's OEIS Frontend

A100554 Decimal expansion of the fractional part of Sum_{n>=1} cos((n + 1)*Pi)*zeta(2*n) = zeta(2) - zeta(4) + zeta(6) - zeta(8) + ..., where Zeta is the Riemann zeta function.

%I A100554 #25 Jan 20 2022 15:16:00
%S A100554 5,7,6,6,7,4,0,4,7,4,6,8,5,8,1,1,7,4,1,3,4,0,5,0,7,9,4,7,5,0,0,0,0,4,
%T A100554 9,0,4,4,5,6,5,6,2,6,6,4,0,3,8,1,6,6,6,5,5,7,5,0,6,2,4,8,4,3,9,0,1,5,
%U A100554 4,2,4,7,9,1,8,3,1,0,0,2,1,7,4,3,5,6,5,5,5,1,7,5,9,3,9,5,4,9,1,8,7,6,5,1,7
%N A100554 Decimal expansion of the fractional part of Sum_{n>=1} cos((n + 1)*Pi)*zeta(2*n) = zeta(2) - zeta(4) + zeta(6) - zeta(8) + ..., where Zeta is the Riemann zeta function.
%C A100554 For odd upper bounds, the sum converges to the given value p in (0,1) with no fractional part function necessary. For even upper bounds, the sum converges to p+1.
%C A100554 Decimal expansion of (psi(i)-psi(-i))/2/i-3/2 where psi is the digamma function. - _Benoit Cloitre_, Nov 28 2004
%F A100554 Equals Pi*(coth(Pi))/2 -1 where Pi = A000796. - _R. J. Mathar_, Apr 01 2010
%F A100554 Equals Sum_{k>=2} 1/(k^2 + 1). - _Amiram Eldar_, Aug 15 2020
%e A100554 0.576674047468581174134050794750000490...
%p A100554 evalf(Pi*coth(Pi)/2-1) ; # _R. J. Mathar_, Apr 01 2010
%t A100554 N[FractionalPart[Sum[Cos[(n + 1)*Pi]*Zeta[2*n], {n, 1000}]], 140]
%t A100554 RealDigits[Pi*Coth[Pi]/2 - 1, 10, 105] // First (* _Jean-François Alcover_, Jan 06 2014, after _R. J. Mathar_ *)
%o A100554 (PARI) (psi(I)-psi(-I))/2/I-3/2
%o A100554 (PARI) sumnumrat(1/(x^2+1),2) \\ _Charles R Greathouse IV_, Jan 20 2022
%o A100554 (PARI) sumnumrat(1/(x^2+4*x+5),0) \\ _Charles R Greathouse IV_, Jan 20 2022
%Y A100554 Cf. A000796.
%K A100554 cons,nonn
%O A100554 0,1
%A A100554 Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 27 2004