OFFSET
0,2
COMMENTS
130000 digits are available (see Links). - Alessandro Languasco, Mar 27 2024
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constants, p. 99.
LINKS
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 11.
Alessandro Languasco, Programs and numerical results for the paper "Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms".
Alessandro Languasco, Shanks' asymptotic constants for the number of positive integers less or equal than x that are the sum of two squares (Source code), gp script, 2024.
Pieter Moree, Counting numbers in multiplicative sets: Landau versus Ramanujan. p. 13, arXiv:1110.0708v1 [math.NT] 4 Oct 2011
D. Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation 18 (1964), pp. 75-86.
Eric Weisstein's World of Mathematics, Landau-Ramanujan Constant.
FORMULA
1 - 2*A227158.
EXAMPLE
-0.1638973186345815958562997690047351186096657461435450436468425985305...
MATHEMATICA
digits = 101; m0 = 5; dm = 5; beta[x_] := 1/4^x*(Zeta[x, 1/4] - Zeta[x, 3/4]); L = Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2; Clear[f]; f[m_] := f[m] = 1/2*(1 - Log[Pi*E^EulerGamma/(2*L)]) - 1/4*NSum[Zeta'[2^k]/Zeta[2^k] - beta'[2^k]/beta[2^k] + Log[2]/(2^(2^k) - 1), {k, 1, m}, WorkingPrecision -> digits + 10]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], m = m + dm]; RealDigits[1 - 2*f[m], 10, digits] // First
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Aug 11 2014
STATUS
approved