The summatory function of the Möbius function in function fields

Byungchul Cha¹

¹ Department of Mathematics and Computer Science Muhlenberg College 2400 Chew St. Allentown, PA 18104, U.S.A.

Acta Arithmetica, Tome 179 (2017) no. 4, pp. 375-395

Résumé

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, as described 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 2008 result of Kowalski, we prove that most curves in the family of universal hyperelliptic curves have 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 the 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.

Détail

DOI : 10.4064/aa8590-1-2017

Keywords: study growth rate summatory function bius function context algebraic curve finite field work shows strong resemblance its number field counterpart described expression bound summatory function which becomes sharp zeta zeros curve satisfy certain linear independence property extending result kowalski prove curves family universal hyperelliptic curves have property consider certain geometric average bound family using katz sarnak reformulation equidistribution theorem deligne lastly study asymptotic behavior average family gets larger evaluating average values powers characteristic polynomials random unitary symplectic matrices