A242015 Decimal expansion of the Euler-Kronecker constant (as named by P. Moree) for non-hypotenuse numbers.

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

Offset

0,1

Comments

130000 digits are available for this constant and the related one A244662; for links to the Languasco et al. article and the corresponding programs see A242013. - Alessandro Languasco, Apr 25 2024

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constants, p. 99.

Links

Table of n, a(n) for n=0..102.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022, p. 11.

Pieter Moree, Counting numbers in multiplicative sets: Landau versus Ramanujan, arXiv:1110.0708v1 [math.NT], 4 Oct 2011, p. 13.

Daniel Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation 18 (1964), pp. 75-86.

Eric Weisstein's World of Mathematics, Landau-Ramanujan Constant.

Formula

Equals 1 - 2*A244662.

Example

-0.40950690341189576824511639518379763704319952909847166323489...

Mathematica

digits = 103; m0 = 5; dm = 5; beta[x_] := 1/4^x*(Zeta[x, 1/4] - Zeta[x, 3/4]); L = Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2; Clear[f]; f[m_] := f[m] = 1/2*(1 - Log[Pi*E^EulerGamma/(2*L)]) - 1/4*NSum[Zeta'[2^k]/Zeta[2^k] - beta'[2^k]/beta[2^k] + Log[2]/(2^(2^k) - 1), {k, 1, m}, WorkingPrecision -> digits + 10]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], m = m + dm]; RealDigits[1 - 2*f[m] - EulerGamma + Log[Pi] - 4*Log[Gamma[3/4]], 10, digits] // First

Crossrefs

Cf. A242013, A244662.

Sequence in context: A100074 A330422 A035102 * A187507 A187857 A215499

Adjacent sequences: A242012 A242013 A242014 * A242016 A242017 A242018

Keyword

nonn,cons

Author

Jean-François Alcover, Aug 11 2014

Status

approved