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A344966
Decimal expansion of the sum of the reciprocals of the fourth powers of the zeros of the digamma function.
4
1, 5, 9, 0, 1, 8, 4, 7, 0, 3, 3, 2, 2, 3, 4, 9, 1, 5, 6, 9, 7, 2, 0, 8, 4, 5, 5, 7, 3, 5, 8, 4, 2, 5, 1, 7, 6, 5, 1, 9, 2, 5, 6, 6, 7, 2, 6, 4, 3, 4, 0, 2, 0, 4, 1, 0, 5, 7, 5, 7, 1, 6, 7, 9, 6, 5, 2, 1, 0, 5, 3, 8, 3, 8, 8, 6, 4, 6, 8, 5, 7, 8, 8, 9, 3, 2, 4
OFFSET
2,2
COMMENTS
The sum is Sum_{k>=0} 1/x_k^4, where x_k is the k-th zero of the digamma function, i.e., root of psi(x) = 0: x_0 = 1.461632... (A030169) is the only positive root, x_1 = -0.504083... (A175472), etc.
LINKS
István Mező and Michael E. Hoffman, Zeros of the digamma function and its Barnes G-function analogue, Integral Transforms and Special Functions, Vol. 28, No. 11 (2017), pp. 846-858.
Wikipedia, Digamma function.
FORMULA
Equals Pi^4/9 + 2*gamma^2*Pi^2/3 + 4*gamma*zeta(3) + gamma^4, where gamma is Euler's constant (A001620).
EXAMPLE
15.90184703322349156972084557358425176519256672643402...
MATHEMATICA
RealDigits[Pi^4/9 + 2*EulerGamma^2*Pi^2/3 + 4*EulerGamma*Zeta[3] + EulerGamma^4, 10, 100][[1]]
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jun 03 2021
STATUS
approved