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A Mean Value Theorem for Dirichlet Polynomials Associated With Primitive Dirichlet L -Functions

https://doi.org/10.1093/imrn/rnaf010

Baluyot, Siegfred; Turnage-Butterbaugh, Caroline L (February 2025, International Mathematics Research Notices)

Consider the family of Dirichlet $$L$$-functions of all even primitive characters of conductor at most $$Q$$, where $$Q$$ is a parameter tending to infinity. For $$X=Q^{\eta }$$ with $$1<\eta <2$$, we examine Dirichlet polynomials of length $$X$$ with coefficients those of the Dirichlet series of a product of an arbitrary (finite) number of shifted $$L$$-functions from the family. Assuming the Generalized Lindelöf Hypothesis for Dirichlet $$L$$-functions, we prove an asymptotic formula for averages of these Dirichlet polynomials. Our result agrees with the prediction of the recipe of Conrey, Farmer, Keating, Rubinstein, and Snaith for these averages. One may view our result as evidence for the “one-swap” terms in the recipe prediction for the general $2k$th moment of the family of Dirichlet $$L$$-functions.
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Free, publicly-accessible full text available February 1, 2026
On the Montgomery–Odlyzko method regarding gaps between zeros of the zeta-function

https://doi.org/10.1016/j.jmaa.2023.127548

Goldston, Daniel A.; Trudgian, Timothy S.; Turnage-Butterbaugh, Caroline L. (November 2023, Journal of Mathematical Analysis and Applications)

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Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function

Simonič, Aleksander (May 2022, Transactions of the American Mathematical Society)

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On a conjecture for $$\ell$$-torsion in class groups of number fields: from the perspective of moments

https://doi.org/10.4310/MRL.2021.v28.n2.a9

Pierce, Lillian B.; Turnage-Butterbaugh, Caroline L.; Matchett Wood, Melanie (January 2021, Mathematical Research Letters)
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An effective Chebotarev density theorem for families of number fields, with an application to $$\ell$$-torsion in class groups

https://doi.org/10.1007/s00222-019-00915-z

Pierce, Lillian B.; Turnage-Butterbaugh, Caroline L.; Wood, Melanie Matchett (February 2020, Inventiones mathematicae)

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