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A368646
Decimal expansion of the Mertens constant M(4,3) arising in the formula for the sum of reciprocals of primes p == 1 (mod 4).
3
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, 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, 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
OFFSET
-1,1
COMMENTS
Data were taken from Languasco and Zaccagnini's web site.
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.
FORMULA
Equals A086239 + A368645.
Equals lim_{x->oo} (Sum_{primes p == 3 (mod 4), p <= x} 1/p - log(log(x))/2).
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).
EXAMPLE
0.048239269073817982409371980048119716415771014508138...
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jan 02 2024
STATUS
approved