Abstract:

The distribution of primes is determined by the distribution of zeros
of Riemann zeta function, and indirectly by the distribution of magnitude of
this function on the critical line <s =
1
2
. Similarly, in order to consider the
distribution of primes in arithmetic progressions, Dirichlet introduced Lfunctions
as a generalization of Riemann zeta function. Generalized Riemann hypothesis,
the most important open problem in mathematics, predicts that all nontrivial
zeros of Dirichlet Lfunction are located on the critical line.
Therefore, one of the main goals in Analytic Number Theory is to consider the
moments of Dirichlet Lfunctions (according to a certain well defined family). The
relation with the characteristic polynomials of random unitary matrices is one of
the fundamental tools for heuristic understanding of Lfunctions and derivation
hypotheses about asymptotic formulae for their moments. Asymptotics for even
moments
1
T
Z
T
0
ζ
1
2
+ it
2k
dt,
as T → ∞, is still an open question (except for k = 1, 2), and it is related to the
Lindelöf Hypothesis.
In this dissertation we consider the sixth moment of Dirichlet Lfunctions
over rational function fields Fq(x), where Fq is a finite field. We will present
the asymptotic formula for the sixth moment with the triple average
X
Q monic
deg Q=d
X
χ (mod Q)
χ odd primitive
2π
Z
log q
0
L
1
2
+ it, χ
6
dt
2π
log q
as d → ∞. All additional averaging is currently necessary to obtain the
asymptotics. The summation over Dirichlet characters and their moduli is
motivated by BombieriVinogradov Theorem. Our result is a function field
analogue of the paper [25] for the corresponding family and averaging over field
Q. Also, our main term confirms the existing Random matrix theory predictions. 