Weighted Prime Number Theorem on Arithmetic Progressions with Refinements

Koji Shimada and Shin-ya Koyama ()
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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

Mathematics, 2025, vol. 13, issue 16, 1-10

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 .

Keywords: prime numbers; Chebyshev’s bias; zeta functions; Dirichlet L-functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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