A242013 Decimal expansion of the Euler-Kronecker constant (as named by P. Moree) for hypotenuse numbers.
1, 6, 3, 8, 9, 7, 3, 1, 8, 6, 3, 4, 5, 8, 1, 5, 9, 5, 8, 5, 6, 2, 9, 9, 7, 6, 9, 0, 0, 4, 7, 3, 5, 1, 1, 8, 6, 0, 9, 6, 6, 5, 7, 4, 6, 1, 4, 3, 5, 4, 5, 0, 4, 3, 6, 4, 6, 8, 4, 2, 5, 9, 8, 5, 3, 0, 5, 0, 2, 4, 6, 3, 1, 1, 1, 9, 0, 0, 6, 9, 2, 2, 8, 6, 0, 2, 4, 7, 2, 2, 6, 2, 9, 8, 4, 8, 2, 6, 9, 9, 2
Offset: 0
Examples
-0.1638973186345815958562997690047351186096657461435450436468425985305...
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], 202-2024, p. 11.
- Alessandro Languasco, Programs and numerical results for the paper "Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms".
- Alessandro Languasco, Shanks' asymptotic constants for the number of positive integers less or equal than x that are the sum of two squares (Source code), gp script, 2024.
- Pieter Moree, Counting numbers in multiplicative sets: Landau versus Ramanujan, p. 13, arXiv:1110.0708v1 [math.NT] 4 Oct 2011.
- D. 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. A227158.
Programs
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Mathematica
digits = 101; 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], 10, digits] // First
Formula
1 - 2*A227158.
Comments