.. _sphx_glr_api_gallery_number_theory_prime_distribution:

Distribution of primes
----------------------

The prime number theorem and primes in arithmetic progressions.


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    <div class="sphx-glr-thumbcontainer" tooltip="Compares the prime-counting function \pi(x) with Gauss&#x27;s logarithmic integral \operatorname{li}(x) and Legendre-style x/\ln x up to ten million. Both ratios tend to 1, as Hadamard and de la Vallée Poussin proved in 1896, but \operatorname{li}(x) is far more accurate.">

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  .. image:: /api/gallery/number_theory/prime_distribution/images/thumb/sphx_glr_plot_01_prime_number_theorem_thumb.png
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  :doc:`/api/gallery/number_theory/prime_distribution/plot_01_prime_number_theorem`

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      <div class="sphx-glr-thumbnail-title">The prime number theorem</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Dirichlet proved in 1837 that every progression a, a+q, a+2q, \dots with \gcd(a, q) = 1 contains infinitely many primes, and the classes share the primes equally in the limit. Equal in the limit is not equal at every point: the race between primes \equiv 3 and \equiv 1 \pmod 4 is led by the 3s almost all the time (&quot;Chebyshev&#x27;s bias&quot;, 1853).">

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  .. image:: /api/gallery/number_theory/prime_distribution/images/thumb/sphx_glr_plot_02_dirichlet_prime_races_thumb.png
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  :doc:`/api/gallery/number_theory/prime_distribution/plot_02_dirichlet_prime_races`

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      <div class="sphx-glr-thumbnail-title">Dirichlet's theorem and prime races</div>
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.. toctree::
   :hidden:

   /api/gallery/number_theory/prime_distribution/plot_01_prime_number_theorem
   /api/gallery/number_theory/prime_distribution/plot_02_dirichlet_prime_races