On Some Weighted Average Values of L-functions

2008/02/18 by Igor E. Shparlinski, Igor Shparlinski, Shparlinski, Igor
Mathematics · #11M06 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT) #math.NT #msc:11M06

paper · pdf · doi:10.48550/arxiv.0802.2378

Bull. Aust. Math. Soc. (to appear)

openalex publication_date 2008/02/18 · arxiv created 2008/07/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let q≥ 2 and N≥ 1 be integers. W. Zhang (2008) has shown that for any fixed ε> 0, and q^ε ≤ N ≤ q^{1/2 -ε}, ∑_{χ≠ χ0} |∑_n=1^N χ(n)|² |L(1, χ)|² = (1 + o(1)) α_q q N where the sum is take over all nonprincipal characters χ modulo q, L(s, χ) is the L-functions L(1, χ) corresponding to χ and α_q = q^o(1) is some explicit function of q. Here we show that the same formula holds in the range q^ε ≤ N ≤ q^{1 -ε}.

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