cp's OEIS Frontend

A242056 Decimal expansion of 2*Pi*phi(0), a constant appearing in connection with a study of zeros of the integral of xi(z), where phi(t) and xi(z) are functions related to Riemann's zeta function (see Finch reference for the definition of these functions).

%I A242056 #32 Jan 17 2020 05:42:49
%S A242056 2,8,0,6,6,7,9,4,0,1,7,7,7,6,9,2,1,8,3,0,5,0,9,1,4,2,7,3,8,1,8,1,5,4,
%T A242056 5,6,4,1,5,4,9,8,0,0,2,8,5,0,2,2,5,6,3,5,5,9,4,2,4,6,9,7,1,2,7,0,6,9,
%U A242056 9,2,2,6,5,6,0,1,3,8,3,0,2,1,8,2,2,4,4,8,9,6,6,2,3,0,3,6,2,6,6,0,9,6,6,5,3
%N A242056 Decimal expansion of 2*Pi*phi(0), a constant appearing in connection with a study of zeros of the integral of xi(z), where phi(t) and xi(z) are functions related to Riemann's zeta function (see Finch reference for the definition of these functions).
%D A242056 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.32 De Bruijn-Newman constant, p. 203.
%H A242056 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants.</a> 2.32 p. 27.
%H A242056 Jeffrey C. Lagarias and David Montague, <a href="http://arxiv.org/abs/1106.4348">The Integral of the Riemann xi-function.</a> arXiv:1106.4348 [math.NT], 2011.
%H A242056 Jeffrey C. Lagarias and David Montague, <a href="http://doi.org/10.14992/00008623 ">The Integral of the Riemann xi-function</a>, Commentarii Mathematici Universitatis Sancti Pauli 60 (2011), No. 1-2, pp. 143-169.
%F A242056 Equals 2*Pi*sum_{n>=1} (Pi*n^2*(2*Pi*n^2-3))/e^(Pi*n^2).
%e A242056 2.8066794017776921830509142738181545641549800285022563559424697...
%t A242056 digits = 105; 2*Pi*NSum[(Pi*n^2*(2*Pi*n^2-3))/E^(Pi*n^2), {n, 1, Infinity}, WorkingPrecision -> digits+5] // RealDigits[#, 10, digits]& // First
%o A242056 (PARI) 2*Pi*suminf(n=1, t=Pi*n^2; t*(2*t-3)/exp(t)) \\ _Charles R Greathouse IV_, Mar 10 2016
%K A242056 nonn,cons
%O A242056 1,1
%A A242056 _Jean-François Alcover_, Aug 13 2014