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

Author

Charles R Greathouse IV, Jul 03 2013

Keywords

• nonn
• cons

Comments

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

130000 digits are available, see link to web page. - Alessandro Languasco, Mar 27 2024

Examples

			0.58194865931729079777136487517474826173838317235153574360562...
		

References

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

Links

• Bruce C. Berndt and Pieter Moree, Sums of two squares and the tau-function: Ramanujan's trail, arXiv:2409.03428 [math.NT], 2024. See p. 32.
• 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.
• Alessandro Languasco, Programs and numerical results, providing 130000 digits. [Note: information ancillary to above link.]
• Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 9.
• David Hare, Landau-Ramanujan Constant, second order obtained about 5000 digits, 1996.
• Daniel 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.

Crossrefs

Cf. A064533, A001481.

Programs

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 *)

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

Extensions

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