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A086239
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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).
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4
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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; internal format)
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OFFSET
| 0,1
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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).
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REFERENCES
| S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98
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LINKS
| D. Broadhurst, post in primenumbers group, Oct 29 2009 [From M. F. Hasler (www.univ-ag.fr/~mhasler), Oct 29 2009]
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, (1991)
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
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EXAMPLE
| 0.33498132529999...
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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], .))} \\ [From M. F. Hasler (www.univ-ag.fr/~mhasler), Oct 29 2009]
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CROSSREFS
| Sequence in context: A197431 A197672 A183501 * A016605 A185395 A060372
Adjacent sequences: A086236 A086237 A086238 * A086240 A086241 A086242
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KEYWORD
| nonn,cons
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com), Jul 13, 2003
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Jun 10 2008
Corrected a(9) and example, added a(10)-a(104) following Broadhurst and Cohen. - M. F. Hasler, Oct 29 2009
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