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).

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, 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, 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

Offset

1,1

References

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

Links

Table of n, a(n) for n=1..105.

Steven R. Finch, Errata and Addenda to Mathematical Constants. 2.32 p. 27.

Jeffrey C. Lagarias and David Montague, The Integral of the Riemann xi-function. arXiv:1106.4348 [math.NT], 2011.

Jeffrey C. Lagarias and David Montague, The Integral of the Riemann xi-function, Commentarii Mathematici Universitatis Sancti Pauli 60 (2011), No. 1-2, pp. 143-169.

Formula

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

Example

2.8066794017776921830509142738181545641549800285022563559424697...

Mathematica

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

Prog

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

Crossrefs

Sequence in context: A011055 A268813 A372719 * A195009 A337997 A372338

Adjacent sequences: A242053 A242054 A242055 * A242057 A242058 A242059

Keyword

nonn,cons

Author

Jean-François Alcover, Aug 13 2014

Status

approved