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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 (list; constant; graph; refs; listen; history; text; internal format)
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. 208-209.

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98

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 Hardy-Littlewood Constants, (1991)

Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]

David Dummit, Andrew Granville, and Hershy Kisilevsky, Big biases amongst products of two primes, Mathematika 62 (2016), pp. 502-507.

R. J. Mathar, Table of Dirichlet L-series 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, (2-prime(k)%4)/prime(k)); -sum(s=1, 60, moebius(s)/s*log( prod( k=2, #P, 1-P[k]^s, if(s%2, if(s==1, Pi/4, sumalt(k=0, (-1)^k/(2*k+1)^s)), zeta(s)*(1-1/2^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

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Last modified November 30 14:29 EST 2020. Contains 338802 sequences. (Running on oeis4.)