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Explicit Formula for the Symmetric Coherence Kernel Associated with Dirichlet L-Functions
2026-07-21
We introduce the symmetric coherence kernel K_χ(σ, t) associated with a primitive Dirichlet character χ modulo q. The kernel is defined via an absolutely convergent integral of the character theta function weighted by 1/cosh(u/2). The main result is Theorem 3.1, which provides an explicit formula for the kernel on the critical line σ = 1/2: K_χ(1/2, t) = (τ(χ)/√q) · (π/cosh(πt)) · Λ(1/2 + it, χ), where τ(χ) is the Gauss sum, and Λ(s, χ) = (q/π)^(s/2) Γ(s/2) L(s, χ) is the completed Dirichlet L-function. The proof employs the Mellin transform, contour shift, residue computation, and the functional equation of the L-function. A corollary establishes that the squared modulus of the kernel is proportional to |L(1/2+it, χ)|² with a positive factor, so that the zeros of the kernel coincide exactly with the critical zeros of the L-function.
Ссылка для цитирования:
Тишков В. В. 2026. Explicit Formula for the Symmetric Coherence Kernel Associated with Dirichlet L-Functions. PREPRINTS.RU. https://doi.org/10.24108/preprints-3115945
Список литературы
1. Edwards, H. M. (1974). Riemann's Zeta Function. Academic Press. (Reprint: Dover, 2001).
2. Davenport, H. (2000). Multiplicative Number Theory (3rd ed.). Springer-Verlag.
3. Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-Function (2nd ed.). Oxford University Press.
4. Ivić, A. (1985). The Riemann Zeta-Function: Theory and Applications. Wiley. (Reprint: Dover, 2003).
5. Siegel, C. L. (1932). Über Riemanns Nachlass zur analytischen Zahlentheorie. Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abt. B: Studien, Bd. 2, 45-80.
6. Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie, 671-680.
7. Iwaniec, H., & Kowalski, E. (2004). Analytic Number Theory. American Mathematical Society.
8. Iwaniec, H. (1997). Topics in Classical Automorphic Forms. American Mathematical Society.
9. Whittaker, E. T., & Watson, G. N. (1927). A Course of Modern Analysis (4th ed.). Cambridge University Press. (Reprint: Cambridge Mathematical Library, 1996).
10. Gradshteyn, I. S., & Ryzhik, I. M. (2007). Table of Integrals, Series, and Products (7th ed.). Academic Press.
11. Karatsuba, A. A., & Voronin, S. M. (1992). The Riemann Zeta-Function. Walter de Gruyter.
12. Neukirch, J. (1999). Algebraic Number Theory. Springer-Verlag.
13. LMFDB Collaboration. (2025). The L-functions and Modular Forms Database. http://www.lmfdb.org (Accessed July 2026).
14. Miller, W. Jr., Post, S., & Winternitz, P. (2013). Classical and quantum superintegrability with applications. J. Phys. A: Math. Theor., 46, 423001.
