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Weighted Prime Number Theorem on Arithmetic Progressions with Refinements

Author

Listed:
  • Koji Shimada

    (Department of Mechanical Engineering, Toyo University, 2100 Kujirai, Saitama 350-8585, Japan)

  • Shin-ya Koyama

    (Department of Mechanical Engineering, Toyo University, 2100 Kujirai, Saitama 350-8585, Japan)

Abstract

We present an extension of the Dirichlet-type prime number theorem to weighted counting functions, the importance of which has recently been recognized for formulating Chebyshev’s bias. Moreover, we prove that their difference π w ( x ; q , a ) − π w ( x ; q , b ) ( 0 ≤ w < 1 / 2 ) changes its sign infinitely many times as x grows for any coprime a , b ( a ≠ b ) with q , under the assumption that Dirichlet L -functions have no real nontrivial zeros. This result gives a justification of the theory of Aoki–Koyama that Chebyshev’s bias is formulated by the asymptotic behavior of π w ( x ; q , a ) − π w ( x ; q , b ) at w = 1 / 2 .

Suggested Citation

  • Koji Shimada & Shin-ya Koyama, 2025. "Weighted Prime Number Theorem on Arithmetic Progressions with Refinements," Mathematics, MDPI, vol. 13(16), pages 1-10, August.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:16:p:2564-:d:1721725
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    1. Koji Shimada & Shin-ya Koyama, 2025. "Weighted Prime Number Theorem on Arithmetic Progressions with Refinements," Mathematics, MDPI, vol. 13(16), pages 1-10, August.
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    1. Koji Shimada & Shin-ya Koyama, 2025. "Weighted Prime Number Theorem on Arithmetic Progressions with Refinements," Mathematics, MDPI, vol. 13(16), pages 1-10, August.

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