%I #33 Feb 16 2025 08:33:11
%S 1,6,5,0,1,8,6,7,4,7,0,0,0,0,6,8,1,8,9,3,6,6,8,2,8,7,8,5,1,2,4,5,6,4,
%T 2,6,2,2,0,0,2,4,6,1,9,2,4,4,9,2,2,9,5,1,8,9,1,9,7,9,4,2,1,1,5,4,7,7,
%U 7,1,5,6,7,2,8,1,1,5,8,8,9,3,7,5,1,0,0,3,6,8,9,7,0,1,9,6,6,5,4,6,0,7,5,1,3
%N Decimal expansion of Sum_{p prime} (Kronecker(-1/p)/p).
%C As a decimal number, this equals 0.5 - A086239. The Kronecker-Legendre-Jacobi symbol Kronecker(-1/p) equals -1 if p=3 (mod 4) and +1 for all other primes. The first ten digits have been computed using the numerical result from Finch for A086239.
%C Using results from Cohen's preprint, _David Broadhurst_ computed 300 decimals in less than 0.2 seconds (cf. link to primenumbers group). He reports that the method can give 70 correct decimals using only two odd primes, p=3 and p=5. - _M. F. Hasler_, Oct 29 2009
%C Contrary to an earlier comment, this constant is not hard to evaluate: see the link for the first 10000 decimal digits. - _David Broadhurst_, Oct 30 2009
%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%D Steven R. Finch, Meissel-Mertens Constants, Sect. 2.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, 2003, p. 96.
%H J. P. Benney and others, <a href="http://groups.yahoo.com/group/primenumbers/message/21075">Is this a convergent series and if so what is its sum?</a>, in primenumbers group, Oct 26 2009.
%H Julien Benney, David Broadhurst, Andrew J. Walker <a href="/A166509/a166509_1.txt">Is this a convergent series and if so what is its sum?</a>, digest of 9 messages in primenumbers Yahoo group, Oct 26 - Oct 30, 2009.
%H David Broadhurst, <a href="http://physics.open.ac.uk/~dbroadhu/cert/cohenbig.out">10000 digits</a> [From _David Broadhurst_, Oct 30 2009]
%H David Broadhurst, <a href="/A166509/a166509.txt">10000 digits</a> [Cached copy]
%H Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High Precision Computation of Hardy-Littlewood Constants</a>, preprint (1991).
%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>
%e 0.16501867470000681893668287851245642622... = 1/2 - 1/3 + 1/5 - 1/7 - 1/11 + 1/13 + 1/17 ... with negative sign iff p=3 (mod 4).
%o (PARI) p=1; sum( i=2,10^6, kronecker(-1,p=nextprime(p+2))/p,.5) /* the first 10^6 terms give 0.16502... */
%o (PARI) /* The given parameters NP,N yield the correct result for default(realprecision,300) in less than 0.5 seconds. Code based on _David Broadhurst_'s script posted on primenumbers group. */
%o A166509(NP=70,N=115)={ local( P=vector(NP, k, (2-prime(k)%4)/prime(k)));
%o sum( s=1,N, moebius(s)/s*log( prod( k=2, #P, 1-P[k]^s,
%o if( s%2, if( s==1, Pi/4, sumalt( k=0, (-1)^k/(2*k+1)^s)),zeta(s)*(1-1/2^s)) /* end if s%2 */
%o )), sum(k=2,#P, P[k], 1/2.))} \\ _M. F. Hasler_, Oct 29 2009
%Y Cf. A166510.
%K cons,easy,nonn,changed
%O 0,2
%A _M. F. Hasler_, Oct 28 2009
%E More terms, computed using _David Broadhurst_'s method, from _M. F. Hasler_, Oct 29 2009