

A227158


Secondorder 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
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OFFSET

0,1


COMMENTS

K = A064533, the LandauRamanujan constant, is the firstorder 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 LandauRamanujan constants, p. 99.


LINKS

Table of n, a(n) for n=0..100.
D. Shanks, The secondorder term in the asymptotic expansion of B(x), Mathematics of Computation 18 (1964), pp. 7586.
Eric Weisstein's World of Mathematics, LandauRamanujan 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 (* JeanFrançois 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^81))); s+1/49/x^7+log(x)/7/x^7
1/2+log(2)/4Euler/4LL(1)/4ZZ(2)/4+LL(2)/4log(2)/12ZZ(4)/4+LL(4)/4log(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 JeanFrançois Alcover, Mar 19 2014 and again May 27 2014


STATUS

approved



