A086239 Decimal expansion of Sum_{k>=2} c(k)/prime(k), where c(k) = -1 if p == 1 (mod 4) and c(k) = +1 if p == 3 (mod 4).

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

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.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 94-98.

Links

Table of n, a(n) for n=0..104.

Julien Benney, Mark Underwood, Andrew J. Walker and 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]

David Broadhurst, post in primenumbers group, Oct 29 2009. [Broken link]

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; arXiv preprint, arXiv:1411.4594 [math.NT], 2014.

Richard J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, variable S(m=4,r=2,s=1) Section 3.1.

Eric Weisstein's World of Mathematics, Prime Sums.

Formula

Equals A368646 - A368645. - Amiram Eldar, Jan 02 2024

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 *)
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums); $MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[-S[4, 2, 1], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 22 2021 *)

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, A368645, A368646.

Sequence in context: A348884 A284115 A183501 * A016605 A374628 A185395

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