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
Examples
2.8066794017776921830509142738181545641549800285022563559424697...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.32 De Bruijn-Newman constant, p. 203.
Links
- 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.
Programs
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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 -
PARI
2*Pi*suminf(n=1, t=Pi*n^2; t*(2*t-3)/exp(t)) \\ Charles R Greathouse IV, Mar 10 2016
Formula
Equals 2*Pi*sum_{n>=1} (Pi*n^2*(2*Pi*n^2-3))/e^(Pi*n^2).