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Abstract:
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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 L-functions
as a generalization of Riemann zeta function. Generalized Riemann hypothesis,
the most important open problem in mathematics, predicts that all nontrivial
zeros of Dirichlet L-function are located on the critical line.
Therefore, one of the main goals in Analytic Number Theory is to consider the
moments of Dirichlet L-functions (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 L-functions 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 L-functions
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 Bombieri-Vinogradov 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. |