A368645 Decimal expansion of the Mertens constant M(4,1) arising in the formula for the sum of reciprocals of primes p == 1 (mod 4) (negated).

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

Offset: 0

Author

Amiram Eldar, Jan 02 2024

Keywords

• nonn
• cons

Comments

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

Examples

			-0.28674205622617519865394514143942385736420436624693...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 95.
• Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 205.

Links

• Alessandro Languasco and Alessandro Zaccagnini, Computing the Mertens and Meissel-Mertens constants for sums over arithmetic progressions, Experimental Mathematics, Vol. 19, No. 3 (2010), pp. 279-284; arXiv preprint, arXiv:0906.2132 [math.NT], 2009.
• Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens and Meissel-Mertens constants for sums over arithmetic progressions.

Crossrefs

Cf. A001620, A002144, A064533, A077761, A086239, A161529, A368644, A368646.

Formula

Equals A368646 - A086239.

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

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