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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).
1
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
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
KEYWORD
nonn,cons
AUTHOR
STATUS
approved