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Moments of real Dirichlet $L$-functions and multiple Dirichlet series
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abstract
We consider the multiple Dirichlet series associated to the $k$th moment of real Dirichlet $L$-functions, and prove that it has a meromorphic continuation to a specific region in $\mathbb{C}^{k+1}$, which is conditional under the generalized Lindel\"of hypothesis for $k\geq 5$. As a corollary, we obtain asymptotic formulas for the first three moments with a power-saving error term, and detect the 0- and 1-swap terms in related problems for any $k$ (conditionally under the Generalized Lindel\"of Hypothesis), recovering the recent results of Conrey and Rodgers on long Dirichlet polynomials. The advantage of our method is its simplicity, since we don't need to modify the multiple Dirichlet series to obtain its meromorphic continuation. As a result, we obtain the asymptotic formulas directly in the form as they appear in the recipe predictions of Conrey, Farmer, Keating, Rubinstein and Snaith.
Forward citations
Cited by 2 Pith papers
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First moment of quadratic Dirichlet $L$-functions with secondary terms
Conditionally on the Riemann hypothesis and the generalized Lindelöf hypothesis, the first moment of quadratic Dirichlet L-functions has error O(X^{1/4+ε}) and lower-order terms X^{1/3} log X.
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Multiple Dirichlet series predictions for moments of $L$-functions: unitary, symplectic and orthogonal examples
Multiple Dirichlet series heuristics reproduce, and for elliptic curve twists correct, the standard moment predictions for four families of L-functions, with each recipe term matched to a residue.
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