This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A227158 #98 Feb 16 2025 08:33:20
%S A227158 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,
%T A227158 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,
%U A227158 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
%N 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.
%C A227158 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]
%C A227158 130000 digits are available, see link to web page. - _Alessandro Languasco_, Mar 27 2024
%D A227158 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constants, p. 99.
%H A227158 Bruce C. Berndt and Pieter Moree, <a href="https://arxiv.org/abs/2409.03428">Sums of two squares and the tau-function: Ramanujan's trail</a>, arXiv:2409.03428 [math.NT], 2024. See p. 32.
%H A227158 Alexandru Ciolan, Alessandro Languasco and Pieter Moree, <a href="https://doi.org/10.1016/j.jmaa.2022.126854">Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms</a>, section 10, 47500 digits are obtained, Journal of Mathematical Analysis and Applications, 2022; see also preprint on <a href="https://arxiv.org/abs/2109.03288">arXiv</a>, arXiv:2109.03288 [math.NT], 2021.
%H A227158 Alessandro Languasco, <a href="https://www.dei.unipd.it/~languasco/CLM.html">Programs and numerical results</a>, providing 130000 digits. [Note: information ancillary to above link.]
%H A227158 Alessandro Languasco and Pieter Moree, <a href="https://arxiv.org/abs/2406.16547">Euler constants from primes in arithmetic progression</a>, arXiv:2406.16547 [math.NT], 2024. See p. 9.
%H A227158 David Hare, <a href="http://web.archive.org/web/20001210182300/http://www.lacim.uqam.ca/piDATA/landau.txt">Landau-Ramanujan Constant, second order</a> obtained about 5000 digits, 1996.
%H A227158 Daniel Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1964-0159174-9">The second-order term in the asymptotic expansion of B(x)</a>, Mathematics of Computation 18 (1964), pp. 75-86.
%H A227158 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Landau-RamanujanConstant.html">Landau-Ramanujan Constant</a>.
%e A227158 0.58194865931729079777136487517474826173838317235153574360562...
%t A227158 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 *)
%o A227158 (PARI) L(s)=sumalt(k=0,(-1)^k/(2*k+1)^s)
%o A227158 LL(s)=L'(s)/L(s)
%o A227158 ZZ(s)=zeta'(s)/zeta(s)
%o A227158 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
%o A227158 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
%Y A227158 Cf. A064533, A001481.
%K A227158 nonn,cons
%O A227158 0,1
%A A227158 _Charles R Greathouse IV_, Jul 03 2013
%E A227158 Corrected and extended by _Jean-François Alcover_, Mar 19 2014 and again May 27 2014