Entire (generalised) Bernoulli function#438
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Parcly-Taxel wants to merge 1 commit intoflintlib:masterfrom
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Entire (generalised) Bernoulli function#438Parcly-Taxel wants to merge 1 commit intoflintlib:masterfrom
Parcly-Taxel wants to merge 1 commit intoflintlib:masterfrom
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In Peter Luschny's "An introduction to the Bernoulli function" (https://arxiv.org/abs/2009.06743) a generalised Bernoulli function is defined: ⎧ 1 for s = 0 B(s, a) = ⎨ ⎩-s⋅ζ(1 - s, a) otherwise When s is a nonnegative integer this function reproduces the Bernoulli polynomials; when in addition a = 1 (the default choice for the Hurwitz zeta function's second argument, yielding Riemann's zeta) the Bernoulli numbers are obtained. Hence B(1) = +1/2, which is how Jakob Bernoulli originally defined his numbers in Ars Conjectandi (1713); this choice gives the simpler form/wider range of validity than B(1) = -1/2 for very many equations like Faulhaber's formula and the Euler–Maclaurin summation formula.
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This PR is like sympy/sympy#23984, but for Arb. It implements the generalised Bernoulli function from @PeterLuschny's "An introduction to the Bernoulli function":
as well as the ordinary version$B(s)=B(s,1)$ . These functions, like the Riemann/Hurwitz zeta functions, come in
arbandacbversions too.Note that the integer-only Bernoulli number functions (
arb_bernoulli_ui, etc.) are not touched, and that this PR is inapplicable to FLINT since it has no Hurwitz zeta function.It might be possible to improve accuracy near the zeta pole (i.e. the origin,$s=0$ ) by directly using the Maclaurin series expansion of the Bernoulli function (which is entire for fixed $a$ ), but this is not implemented here.
Example of usage:
Output: