# Moments of the Dedekind zeta function

Heap, Winston (2013) Moments of the Dedekind zeta function. PhD thesis, University of York.

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We study analytic aspects of the Dedekind zeta function of a Galois extension. Specifically, we are interested in its mean values. In the first part of this thesis we give a formula for the second moment of the Dedekind zeta function of a quadratic field times an arbitrary Dirichlet polynomial of length $T^{1/11-\epsilon}$. In the second part, we derive a hybrid Euler-Hadamard product for the Dedekind zeta function of an arbitrary number field. We rigorously calculate the $2k$th moment of the Euler product part as well as conjecture the $2k$th moment of the Hadamard product using random matrix theory. In both instances we are restricted to Galois extensions. We then conjecture that the $2k$th moment of the Dedekind zeta function of a Galois extension is given by the product of the two. By using our results from the first part of this thesis we are able to prove both conjectures in the case $k=1$ for quadratic fields. We also re-derive our conjecture for the $2k$th moment of quadratic Dedekind zeta functions by using a modification of the moments recipe. Finally, we apply our methods to general non-primitive $L$-functions and gain a conjecture regarding their moments. Our main idea is that, to leading order, the moment of a product of distinct $L$-functions should be the product of the individual moments of the constituent $L$-functions.