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A227158 Second-order term in the asymptotic expansion of B(x). 4
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

K = A064533, the Landau-Ramanujan constant, is the first-order term. This constant is c = lim (B(x)*sqrt(log x)/x - 1)log x, where the limit is taken as x increases without bound. B(x) is the count of numbers up to x which are the sum of two squares.

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.

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

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Last modified December 11 12:33 EST 2019. Contains 329916 sequences. (Running on oeis4.)