.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/zeta/plot_01_riemann_zeta.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_special_functions_zeta_plot_01_riemann_zeta.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_special_functions_zeta_plot_01_riemann_zeta.py:


The Riemann zeta function and Euler's product
==================================================

Plots zeta(s) on the real line, including its analytic continuation to
s < 1 with trivial zeros at the negative even integers, and shows
Euler's product over primes converging to zeta(2) = pi^2/6.

.. GENERATED FROM PYTHON SOURCE LINES 11-18

.. code-block:: Python

    import math

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import euler_product, riemann_zeta








.. GENERATED FROM PYTHON SOURCE LINES 19-21

zeta on the real line
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-36

.. code-block:: Python


    s = np.concatenate([np.linspace(-9, 0.95, 500), np.linspace(1.05, 6, 200)])
    values = riemann_zeta(s)
    fig, ax = plt.subplots()
    ax.plot(s[s < 1], values[s < 1], "C0")
    ax.plot(s[s > 1], values[s > 1], "C0")
    ax.plot([-2, -4, -6, -8], [0, 0, 0, 0], "ro", label="trivial zeros")
    ax.axhline(0, color="gray", lw=0.5)
    ax.set_ylim(-2, 4)
    ax.set_xlabel("s")
    ax.set_ylabel(r"$\zeta(s)$")
    ax.legend()
    print(f"zeta(2) = {riemann_zeta(2.0):.12f}, pi^2/6 = {math.pi**2 / 6:.12f}")
    print(f"zeta(0) = {riemann_zeta(0.0):.6f}, zeta(-1) = {riemann_zeta(-1.0):.6f} (= -1/12)")




.. image-sg:: /api/gallery/special_functions/zeta/images/sphx_glr_plot_01_riemann_zeta_001.png
   :alt: plot 01 riemann zeta
   :srcset: /api/gallery/special_functions/zeta/images/sphx_glr_plot_01_riemann_zeta_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    zeta(2) = 1.644934066848, pi^2/6 = 1.644934066848
    zeta(0) = -0.500000, zeta(-1) = -0.083333 (= -1/12)




.. GENERATED FROM PYTHON SOURCE LINES 37-39

Euler's product over primes
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-43

.. code-block:: Python


    for bound in (10, 100, 1000, 10000):
        product = euler_product(2.0, bound)
        print(f"primes <= {bound:5d}: product = {product:.8f}, error = {abs(product - math.pi**2 / 6):.1e}")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    primes <=    10: product = 1.59505208, error = 5.0e-02
    primes <=   100: product = 1.64194520, error = 3.0e-03
    primes <=  1000: product = 1.64472519, error = 2.1e-04
    primes <= 10000: product = 1.64491792, error = 1.6e-05





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.020 seconds)


.. _sphx_glr_download_api_gallery_special_functions_zeta_plot_01_riemann_zeta.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_riemann_zeta.ipynb <plot_01_riemann_zeta.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_riemann_zeta.py <plot_01_riemann_zeta.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_riemann_zeta.zip <plot_01_riemann_zeta.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_