Arithmetic correlations over large finite fields

The auto-correlations of arithmetic functions, such as the von Mangoldt function, the M\"obius function and the divisor function, are the subject of classical problems in analytic number theory. The function field analogues of these problems have recently been resolved in the limit of large finite field size $q$. However, in this limit the correlations disappear: the arithmetic functions become uncorrelated. We compute averages of terms of lower order in $q$ which detect correlations. Our results show that there is considerable cancellation in the averaging and have implications for the rate at which correlations disappear when $q \rightarrow\infty$; in particular one cannot expect remainder terms that are of the order of the square-root of the main term in this context.Comment: The paper has been accepted by IMR

Variance of sums in arithmetic progressions of divisor functions associated with higher degree -Functions in q[]

We compute the variances of sums in arithmetic progressions of generalised -divisor functions related to certain -functions in q[], in the limit as q → ∞. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when q → ∞, in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual -divisor function, when the -function in question has degree one. They illustrate the role played by the degree of the -functions; in particular, we find qualitatively new behaviour when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over q[], and we illustrate them by examining in some detail the generalised -divisor functions associated with the Legendre curve

Variance of arithmetic sums and L-functions in Fq[t]

We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain L-functions of degree 2 and higher in F q [t], in the limit as q →∞. This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-1 L-functions (i.e., situations considered previously using this approach). They correspond to expressions found recently in the number field setting assuming a generalization of the pair correlation conjecture. Our calculations apply, for example, to elliptic curves defined over F q [t]

Variance of arithmetic sums and L-functions in Fq[t]

We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain L-functions of degree 2 and higher in F q [t], in the limit as q →∞. This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-1 L-functions (i.e., situations considered previously using this approach). They correspond to expressions found recently in the number field setting assuming a generalization of the pair correlation conjecture. Our calculations apply, for example, to elliptic curves defined over F q [t]

Moments of quadratic twists of elliptic curve L-functions over function fields

We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over Fq[x], asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving derivatives of L-functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves

Sums of divisor functions in Fq[t] and matrix integrals

We study the mean square of sums of the kth divisor function dk(n) over short intervals and arithmetic progressions for the rational function field over a finite field of q elements. In the limit as q→∞ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of dk(n) in terms of a lattice point count. This lattice point count can in turn be calculated in terms of a certain piecewise polynomial function, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.</p