A368646 - OEIS

%I #11 Sep 27 2024 08:05:07

%S 4,8,2,3,9,2,6,9,0,7,3,8,1,7,9,8,2,4,0,9,3,7,1,9,8,0,0,4,8,1,1,9,7,1,

%T 6,4,1,5,7,7,1,0,1,4,5,0,8,1,3,8,1,2,7,1,5,0,0,4,2,3,9,6,3,8,5,0,7,3,

%U 6,7,7,0,8,4,6,2,6,1,6,0,1,6,9,5,2,1,1,3,1,2,3,2,0,2,4,9,9,8,2,0,3,5,5,8,9

%N Decimal expansion of the Mertens constant M(4,3) arising in the formula for the sum of reciprocals of primes p == 1 (mod 4).

%C Data were taken from Languasco and Zaccagnini's web site.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 95.

%D Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 205.

%H Alessandro Languasco and Alessandro Zaccagnini, <a href="https://doi.org/10.1080/10586458.2010.10390624">Computing the Mertens and Meissel-Mertens constants for sums over arithmetic progressions</a>, Experimental Mathematics, Vol. 19, No. 3 (2010), pp. 279-284; <a href="https://arxiv.org/abs/0906.2132">arXiv preprint</a>, arXiv:0906.2132 [math.NT], 2009.

%H Alessandro Languasco and Alessandro Zaccagnini, <a href="https://www.dei.unipd.it/~languasco/Mertens-comput.html">Computation of the Mertens and Meissel-Mertens constants for sums over arithmetic progressions</a>.

%F Equals A086239 + A368645.

%F Equals lim_{x->oo} (Sum_{primes p == 3 (mod 4), p <= x} 1/p - log(log(x))/2).

%F Equals gamma/2 - log(sqrt(P)/(2*K_1)) + Sum_{prime p == 3 (mod 4)} (log(1-1/p) + 1/p), where gamma is Euler's constant (A001620) and K_1 is Landau-Ramanujan constant (A064533).

%e 0.048239269073817982409371980048119716415771014508138...

%Y Cf. A001620, A002145, A064533, A077761, A086239, A161529, A368644, A368645.

%K nonn,cons

%O -1,1

%A _Amiram Eldar_, Jan 02 2024