A unified strategy to compute some special functions of number-theoretic interest
نویسندگان
چکیده
We introduce an algorithm to compute the functions belonging a suitable set F defined as follows: f∈F means that f(s,x), s∈A⊂R being fixed and x>0, has power series expansion centred at x0=1 with convergence radius greater or equal than 1; moreover, it satisfies functional equation of step 1 Euler-Maclaurin summation formula can be applied f. Denoting Euler gamma-function Γ, we will show that, for logΓ(x), digamma function ψ(x), polygamma ψ(w)(x), w∈N, w≥1, and, s>1 fixed, Hurwitz ζ(s,x)-function its first partial derivative ∂ζ∂s(s,x) are in F. In all these cases coefficients involved depend on values ζ(u), u>1, where ζ is Riemann zeta-function. As by-product, also how Dirichlet L-functions L(s,χ) L′(s,χ), s>1, χ primitive character, by inserting reflection formulae ζ(s,x) into Fast Fourier Transform algorithm. Moreover, obtain some new algorithms β-function Catalan constant G. Finally, study case Bateman G-function alternating zeta-function, known η-function; even if they not F, our approach adapted handle them too. last section describe tests performance gain respect standard multiprecision implementation ∂ζ∂s(s,x), x>0.
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2023
ISSN: ['0022-314X', '1096-1658']
DOI: https://doi.org/10.1016/j.jnt.2022.12.003