A247042 - OEIS

A247042

Decimal expansion of delta_2 (negated), a constant associated with a certain two-dimensional lattice sum.

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

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OFFSET

1,1

COMMENTS

This constant is named sigma(1/2) in the Borwein reference.

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 79.

LINKS

Table of n, a(n) for n=1..102.

D. Borwein, J. M. Borwein and R. Shail, Analysis of Certain Lattice Sums, Journal of Mathematical Analysis and Applications, Volume 143, Issue 1, October 1989, Pages 126-137.

Eric Weisstein's MathWorld, Lattice Sum

Eric Weisstein's MathWorld, Madelung Constants

FORMULA

delta_2 = 2*zeta(1/2)*(zeta(1/2, 1/4) - zeta(1/2, 3/4)), where zeta(s,a) gives the generalized Riemann zeta function.

EXAMPLE

-3.900264920001955882845475336604973219209047856477537388...

MATHEMATICA

delta2 = 2*Zeta[1/2]*(Zeta[1/2, 1/4] - Zeta[1/2, 3/4]); RealDigits[delta2, 10, 102] // First

PROG

(PARI) 2*zeta(1/2)*(zetahurwitz(1/2, 1/4)-zetahurwitz(1/2, 3/4)) \\ Charles R Greathouse IV, Jan 31 2018

CROSSREFS

Cf. A088537, A085469, A090734, A247040.