2010/08/27 by Byungchul Cha, Cha, Byungchul · 1 citation
Mathematics · #11N56 (Primary) 11M50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11M50 #msc:11N56
paper · pdf · doi:10.48550/arxiv.1008.4711
16 pages. In this revision, an error in residue calculation in Section 2 is corrected, and a few bibliographic items are updated
openalex publication_date 2010/08/27 · arxiv created 2011/11/14 · arxiv updated 2011/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the growth rate of the summatory function of the Möbius function in the context of an algebraic curve over a finite field. Our work shows a strong resemblance to its number field counterpart, which was proved by Ng in 2004. We find an expression for a bound of the summatory function, which becomes sharp when the zeta zeros of the curve satisfy a certain linear independence property. Extending a result of Kowalski in 2008, we prove that most curves in the family of universal hyperelliptic curves satisfy this property. Then, we consider a certain geometric average of such bound in this family, using Katz and Sarnak's reformulation of the equidistribution theorem of Deligne. Lastly, we study an asymptotic behavior of this average as the family gets larger by evaluating the average values of powers of characteristic polynomials of random unitary symplectic matrices.