A227158 Second-order term in the asymptotic expansion of B(x), the count of numbers up to x which are the sum of two squares.

5, 8, 1, 9, 4, 8, 6, 5, 9, 3, 1, 7, 2, 9, 0, 7, 9, 7, 9, 2, 8, 1, 4, 9, 8, 8, 4, 5, 0, 2, 3, 6, 7, 5, 5, 9, 3, 0, 4, 8, 3, 2, 8, 7, 3, 0, 7, 1, 7, 7, 2, 5, 2, 1, 8, 2, 3, 4, 2, 1, 2, 9, 9, 2, 6, 5, 2, 5, 1, 2, 3, 1, 5, 5, 5, 9, 5, 0, 3, 4, 6, 1, 4, 3, 0, 1, 2, 3, 6, 1, 3, 1, 4, 9, 2, 4, 1, 3, 4, 9, 6

Offset

0,1

Comments

K = A064533, the Landau-Ramanujan constant, is the first-order term. This constant is c = lim_{x->infinity} (B(x)*sqrt(log x)/(K*x) - 1)*log x. [Corrected by Alessandro Languasco, Sep 14 2022]

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constants, p. 99.

Links

Table of n, a(n) for n=0..100.

Alexandru Ciolan, Alessandro Languasco and Pieter Moree, Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms, section 10, 47500 digits are obtained, Journal of Mathematical Analysis and Applications, 2022; see also preprint on arXiv, arXiv:2109.03288 [math.NT], 2021.

David Hare, Landau-Ramanujan Constant, second order obtained about 5000 digits, 1996.

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

Example

0.58194865931729079777136487517474826173838317235153574360562....

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[f[m], 10, digits] // First (* Jean-François Alcover, May 27 2014 *)

Prog

(PARI) L(s)=sumalt(k=0, (-1)^k/(2*k+1)^s)
LL(s)=L'(s)/L(s)
ZZ(s)=zeta'(s)/zeta(s)
sm(x)=my(s); forprime(q=2, x, if(q%4==3, s+=log(q)/(q^8-1))); s+1/49/x^7+log(x)/7/x^7
1/2+log(2)/4-Euler/4-LL(1)/4-ZZ(2)/4+LL(2)/4-log(2)/12-ZZ(4)/4+LL(4)/4-log(2)/60+sm(1e5)/2

Crossrefs

Cf. A064533, A001481.

Sequence in context: A171709 A093157 A122998 * A098881 A185393 A073333

Adjacent sequences: A227155 A227156 A227157 * A227159 A227160 A227161

Keyword

nonn,cons

Author

Charles R Greathouse IV, Jul 03 2013

Extensions

Corrected and extended by Jean-François Alcover, Mar 19 2014 and again May 27 2014

Status

approved