Abstract
We study the variance of sums of the indicator function of square-full polynomials in both arithmetic progressions and short intervals. Our work is in the context of the ring mathbb {F}_q[T] of polynomials over a finite field mathbb {F}_q of q elements, in the limit qrightarrow infty . We use a recent equidistribution result due to N. Katz to express these variances in terms of triple matrix integrals over the unitary group, and evaluate them.
Highlights
We study the variance of sums of the indicator function of square-full polynomials in both arithmetic progressions and short intervals
Concerning the distribution of square full numbers in arithmetic progressions, the most recent result is due to Munsch [17]
The unitarized Frobenii χ for the family of even primitive characters mod T m+1 become equidistributed in the projective unitary group PU (m − 1) of size m − 1, as q goes to infinity
Summary
Let Fq be a finite field of an odd cardinality q, and let Mn be the set of all monic polynomials of degree n with coefficients in Fq. The generating function for the number of monic square-full polynomials of degree n, i.e. where Z(u) is the zeta function of Fq[T ] ( set ζq := Z(q−s)) , given by the following product over prime polynomials in Fq[T ]. The sum of α2 over all monic polynomials of degree n lying in the arithmetic progressions f = A mod Q is The average of this sum Sα2;n;Q(A) when we vary A over residue classes coprime to Q is. The conditions one needs to place on n in both Theorems 2.1 and 2.2 are not obvious to begin with They will follow eventually because we express the variance in both cases in terms of zeros of L-functions, which are known to be polynomials in the function field settings
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