arb/doc/source/acb_dirichlet.rst

.. _acb-dirichlet:

**acb_dirichlet.h** -- Dirichlet L-functions, zeta functions, and related functions
===================================================================================

This module will eventually allow working with Dirichlet L-functions and
possibly slightly more general Dirichlet series. At the moment, it contains
nothing interesting.

The code in other modules for computing the Riemann zeta function,
Hurwitz zeta function and polylogarithm will possibly be migrated to this
module in the future.

A Dirichlet L-function is the analytic continuation of an L-series

.. math ::

    L(s,\chi) = \sum_{k=1}^\infty \frac{\chi(k)}{k^s}

where `\chi(k)` is a Dirichlet character.

Euler products
-------------------------------------------------------------------------------

.. function:: void _acb_dirichlet_euler_product_real_ui(arb_t res, ulong s, const signed char * chi, int mod, int reciprocal, slong prec)

    Sets *res* to `L(s,\chi)` where `\chi` is a real Dirichlet character
    given by the explicit list *chi* of character values at
    0, 1, ..., *mod* - 1. If *reciprocal* is set, computes `1 / L(s,\chi)`
    (this is faster if the reciprocal can be used directly).

    This function uses the Euler product, and is only intended for use when
    *s* is large. An error bound is computed via :func:`mag_hurwitz_zeta_uiui`.
    Since

    .. math ::

        \frac{1}{L(s,\chi)} = \prod_{p} \left(1 - \frac{\chi(p)}{p^s}\right)
                = \sum_{k=1}^{\infty} \frac{\mu(k)\chi(k)}{k^s}

    and the truncated product gives all smooth-index terms in the series, we have

    .. math ::

        \left|\prod_{p < N} \left(1 - \frac{\chi(p)}{p^s}\right) - \frac{1}{L(s,\chi)}\right|
        \le \sum_{k=N}^{\infty} \frac{1}{k^s} = \zeta(s,N).

Simple functions
-------------------------------------------------------------------------------

.. function:: void acb_dirichlet_eta(acb_t res, const acb_t s, slong prec)

    Sets *res* to the Dirichlet eta function
    `\eta(s) = \sum_{k=1}^{\infty} (-1)^k / k^s = (1-2^{1-s}) \zeta(s)`,
    also known as the alternating zeta function.
    Note that the alternating character `\{1,-1\}` is not itself
    a Dirichlet character.