login
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.
4

%I #25 Jan 20 2022 15:16:00

%S 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 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 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 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 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 Decimal expansion of (psi(i)-psi(-i))/2/i-3/2 where psi is the digamma function. - _Benoit Cloitre_, Nov 28 2004

%F Equals Pi*(coth(Pi))/2 -1 where Pi = A000796. - _R. J. Mathar_, Apr 01 2010

%F Equals Sum_{k>=2} 1/(k^2 + 1). - _Amiram Eldar_, Aug 15 2020

%e 0.576674047468581174134050794750000490...

%p evalf(Pi*coth(Pi)/2-1) ; # _R. J. Mathar_, Apr 01 2010

%t N[FractionalPart[Sum[Cos[(n + 1)*Pi]*Zeta[2*n], {n, 1000}]], 140]

%t RealDigits[Pi*Coth[Pi]/2 - 1, 10, 105] // First (* _Jean-François Alcover_, Jan 06 2014, after _R. J. Mathar_ *)

%o (PARI) (psi(I)-psi(-I))/2/I-3/2

%o (PARI) sumnumrat(1/(x^2+1),2) \\ _Charles R Greathouse IV_, Jan 20 2022

%o (PARI) sumnumrat(1/(x^2+4*x+5),0) \\ _Charles R Greathouse IV_, Jan 20 2022

%Y Cf. A000796.

%K cons,nonn

%O 0,1

%A Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 27 2004