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Manin’s conjecture for a singular sextic del Pezzo surface

Daniel Loughran (2010)

Journal de Théorie des Nombres de Bordeaux

We prove Manin’s conjecture for a del Pezzo surface of degree six which has one singularity of type A 2 . Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.

Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs

Ilwoo Cho, Palle E. T. Jorgensen (2015)

Special Matrices

In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we...

Mean value theorems for L-functions over prime polynomials for the rational function field

Julio C. Andrade, Jonathan P. Keating (2013)

Acta Arithmetica

The first and second moments are established for the family of quadratic Dirichlet L-functions over the rational function field at the central point s=1/2, where the character χ is defined by the Legendre symbol for polynomials over finite fields and runs over all monic irreducible polynomials P of a given odd degree. Asymptotic formulae are derived for fixed finite fields when the degree of P is large. The first moment obtained here is the function field analogue of a result due to Jutila in the...

Mean-periodicity and zeta functions

Ivan Fesenko, Guillaume Ricotta, Masatoshi Suzuki (2012)

Annales de l’institut Fourier

This paper establishes new bridges between zeta functions in number theory and modern harmonic analysis, namely between the class of complex functions, which contains the zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of mean-periodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and functional equation of the zeta function of an arithmetic scheme with its expected analytic shape is shown...

Minoration de la hauteur normalisée des hypersurfaces

Francesco Amoroso, Sinnou David (2000)

Acta Arithmetica

1. Introduction. Dans un article célèbre, D. H. Lehmer posait la question suivante (voir [Le], §13, page 476): «The following problem arises immediately. If ε is a positive quantity, to find a polynomial of the form: f ( x ) = x r + a 1 x r - 1 + ⋯ + a r where the a’s are integers, such that the absolute value of the product of those roots of f which lie outside the unit circle, lies between 1 and 1 + ε (...). Whether or not the problem has a solution for ε < 0.176 we do not know.» Cette question, toujours ouverte, est la source...

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