login
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).
1

%I #32 Jan 17 2020 05:42:49

%S 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 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 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 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 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.32 De Bruijn-Newman constant, p. 203.

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants.</a> 2.32 p. 27.

%H 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 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 Equals 2*Pi*sum_{n>=1} (Pi*n^2*(2*Pi*n^2-3))/e^(Pi*n^2).

%e 2.8066794017776921830509142738181545641549800285022563559424697...

%t 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 (PARI) 2*Pi*suminf(n=1, t=Pi*n^2; t*(2*t-3)/exp(t)) \\ _Charles R Greathouse IV_, Mar 10 2016

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Aug 13 2014