

A086239


Decimal expansion of sum(c[k]/prime[k], k=2..infinity), where c[k]=1 if p==1 (mod 4) and c[k]=+1 if p==3 (mod 4).


5



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

0,1


COMMENTS

This is sum_{p prime, p>=3} (4/p)/p where (4/.) is the Legendre symbol and is equal to  L(1,(4/.)) plus an absolutely convergent sum (and therefore converges).


REFERENCES

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208209.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 9498


LINKS

Table of n, a(n) for n=0..104.
Julien Benney, Mark Underwood, Andrew J. Walker, David Broadhurst, Is this a convergent series and if so what is its sum?, digest of 12 messages in primenumbers Yahoo group, Oct 26  Oct 30, 2009. [Cached copy]
D. Broadhurst, post in primenumbers group, Oct 29 2009
Henri Cohen, High Precision Computation of HardyLittlewood Constants, (1991)
Henri Cohen, Highprecision computation of HardyLittlewood constants. [pdf copy, with permission]
David Dummit, Andrew Granville, and Hershy Kisilevsky, Big biases amongst products of two primes, Mathematika 62 (2016), pp. 502507.
R. J. Mathar, Table of Dirichlet Lseries and prime zeta modulo functions for small moduli, arXiv:1008.2537, variable S(m=4,r=2,s=1) Section 3.1
Eric Weisstein's World of Mathematics, PrimeSums


EXAMPLE

0.33498132529999...


MATHEMATICA

Do[Print[N[Log[2]/2 + Sum[Log[2^(4*n)*(2^(2*n + 1) + 1)*(2^(2*n + 3)  4)*(Zeta[4*n + 2] / (Zeta[2*n + 1, 1/4]  Zeta[2*n + 1, 3/4])^2)] * MoebiusMu[2*n + 1]/(4*n + 2), {n, 1, m}], 120]], {m, 20, 200, 20}] (* Vaclav Kotesovec, Jun 28 2020 *)


PROG

(PARI) /* the given number of primes and terms in the sum yield over 105 correct digits */ P=vector(15, k, (2prime(k)%4)/prime(k)); sum(s=1, 60, moebius(s)/s*log( prod( k=2, #P, 1P[k]^s, if(s%2, if(s==1, Pi/4, sumalt(k=0, (1)^k/(2*k+1)^s)), zeta(s)*(11/2^s) ))), sum(k=2, #P, P[k], .)) \\ M. F. Hasler, Oct 29 2009


CROSSREFS

Cf. A166509.
Sequence in context: A197672 A284115 A183501 * A016605 A185395 A060372
Adjacent sequences: A086236 A086237 A086238 * A086240 A086241 A086242


KEYWORD

nonn,cons


AUTHOR

Eric W. Weisstein, Jul 13 2003


EXTENSIONS

Edited by N. J. A. Sloane, Jun 10 2008
Corrected a(9) and example, added a(10)a(104) following Broadhurst and Cohen.  M. F. Hasler, Oct 29 2009


STATUS

approved



