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Current File : /lib/python3/dist-packages/mpmath/functions/__pycache__/rszeta.cpython-312.pyc
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---------------------------------------------------------------------
.. sectionauthor:: Juan Arias de Reyna <arias@us.es>

This module implements zeta-related functions using the Riemann-Siegel
expansion: zeta_offline(s,k=0)

* coef(J, eps): Need in the computation of Rzeta(s,k)

* Rzeta_simul(s, der=0) computes Rzeta^(k)(s) and Rzeta^(k)(1-s) simultaneously
  for  0 <= k <= der. Used by zeta_offline and z_offline

* Rzeta_set(s, derivatives) computes Rzeta^(k)(s) for given derivatives, used by
  z_half(t,k) and zeta_half

* z_offline(w,k): Z(w) and its derivatives of order k <= 4
* z_half(t,k): Z(t) (Riemann Siegel function) and its derivatives of order k <= 4
* zeta_offline(s): zeta(s) and its derivatives of order k<= 4
* zeta_half(1/2+it,k):  zeta(s)  and its derivatives of order k<= 4

* rs_zeta(s,k=0) Computes zeta^(k)(s)   Unifies zeta_half and zeta_offline
* rs_z(w,k=0)    Computes Z^(k)(w)      Unifies z_offline and z_half
----------------------------------------------------------------------

This program uses Riemann-Siegel expansion even to compute
zeta(s) on points s = sigma + i t  with sigma arbitrary not
necessarily equal to 1/2.

It is founded on a new deduction of the formula, with rigorous
and sharp bounds for the  terms and rest of this expansion.

More information on the papers:

 J. Arias de Reyna, High Precision Computation of Riemann's
 Zeta Function by the Riemann-Siegel Formula I, II

 We refer to them as I, II.

 In them we shall find detailed explanation of all the
 procedure.

The program uses Riemann-Siegel expansion.
This  is useful when t is big, ( say  t > 10000 ).
The precision is limited, roughly it can compute zeta(sigma+it)
with an error less than exp(-c t) for some constant c depending
on sigma.  The program gives an error when the Riemann-Siegel
formula can not compute to the wanted precision.

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    Computes the coefficients  `c_n`  for `0\le n\le 2J` with error less than eps

    **Definition**

    The coefficients c_n are defined by

    .. math ::

        \begin{equation}
        F(z)=\frac{e^{\pi i
        \bigl(\frac{z^2}{2}+\frac38\bigr)}-i\sqrt{2}\cos\frac{\pi}{2}z}{2\cos\pi
        z}=\sum_{n=0}^\infty c_{2n} z^{2n}
        \end{equation}

    they are computed applying the relation

    .. math ::

        \begin{multline}
        c_{2n}=-\frac{i}{\sqrt{2}}\Bigl(\frac{\pi}{2}\Bigr)^{2n}
        \sum_{k=0}^n\frac{(-1)^k}{(2k)!}
        2^{2n-2k}\frac{(-1)^{n-k}E_{2n-2k}}{(2n-2k)!}+\\
        +e^{3\pi i/8}\sum_{j=0}^n(-1)^j\frac{
        E_{2j}}{(2j)!}\frac{i^{n-j}\pi^{n+j}}{(n-j)!2^{n-j+1}}.
        \end{multline}
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    Computes several derivatives of the auxiliary function of Riemann `R(s)`.

    **Definition**

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    .. math ::

        \begin{equation}
        {\mathop{\mathcal R }\nolimits}(s)=
        \int_{0\swarrow1}\frac{x^{-s} e^{\pi i x^2}}{e^{\pi i x}-
        e^{-\pi i x}}\,dx
        \end{equation}

    To this function we apply the Riemann-Siegel expansion.
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Youez - 2016 - github.com/yon3zu
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