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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


%S 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,

%T 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,

%U 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

%N 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).

%C 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).

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

%H Julien Benney, Mark Underwood, Andrew J. Walker, David Broadhurst, <a href="/A086239/a086239.txt">Is this a convergent series and if so what is its sum?</a>, digest of 12 messages in primenumbers Yahoo group, Oct 26 - Oct 30, 2009. [Cached copy]

%H D. Broadhurst, <a href="http://groups.yahoo.com/group/primenumbers/message/21083">post in primenumbers group</a>, Oct 29 2009

%H Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High Precision Computation of Hardy-Littlewood Constants</a>, (1991)

%H David Dummit, Andrew Granville, and Hershy Kisilevsky, <a href="https://arxiv.org/abs/1411.4594">Big biases amongst products of two primes</a>, Mathematika 62 (2016), pp. 502-507.

%H R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and prime zeta modulo functions for small moduli</a>, arXiv:1008.2537, variable S(m=4,r=2,s=1) Section 3.1

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">PrimeSums</a>

%e 0.33498132529999...

%o (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

%Y Cf. A166509.

%K nonn,cons,changed

%O 0,1

%A _Eric W. Weisstein_, Jul 13 2003

%E Edited by _N. J. A. Sloane_, Jun 10 2008

%E Corrected a(9) and example, added a(10)-a(104) following Broadhurst and Cohen. - _M. F. Hasler_, Oct 29 2009

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Last modified November 20 02:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)