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

Author

Jean-François Alcover, Aug 11 2014

Keywords

• nonn
• cons

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

Examples

			-0.40950690341189576824511639518379763704319952909847166323489...
		

References

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

Links

• 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.

Crossrefs

Cf. A242013, A244662.

Programs

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

Formula

Equals 1 - 2*A244662.