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1. Orthogonal Dirichlet Polynomials

   https://doi.org/10.1007/978-3-030-84122-5_30  

   Lubinsky, D. (January 2022, Approximation and Computation in Science and Engineering)

   Daras, N.; Rassias, T. (Ed.)

   Abstract. Let fj g1 j=1 be a sequence of distinct positive numbers. Let w be a nonnegative function, integrable on the real line. One can form orthogonal Dirichlet polynomials fng from linear combinations of n 

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2. Nonvanishing of Dirichlet L-functions, II

   https://doi.org/10.1007/s00209-021-02821-8  

   Khan, Rizwanur; Milićević, Djordje; Ngo, Hieu T. (February 2022, Mathematische Zeitschrift)
3. Dirichlet, Sierpiński, and Benford

   https://doi.org/10.1016/j.jnt.2021.12.010  

   Pollack, Paul; Singha Roy, Akash (October 2022, Journal of Number Theory)
4. 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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5. Local Bayesian Dirichlet mixing of imperfect models

   https://doi.org/10.1038/s41598-023-46568-0  

   Kejzlar, Vojtech; Neufcourt, Léo; Nazarewicz, Witold (December 2023, Scientific Reports)

   Abstract To improve the predictability of complex computational models in the experimentally-unknown domains, we propose a Bayesian statistical machine learning framework utilizing the Dirichlet distribution that combines results of several imperfect models. This framework can be viewed as an extension of Bayesian stacking. To illustrate the method, we study the ability of Bayesian model averaging and mixing techniques to mine nuclear masses. We show that the global and local mixtures of models reach excellent performance on both prediction accuracy and uncertainty quantification and are preferable to classical Bayesian model averaging. Additionally, our statistical analysis indicates that improving model predictions through mixing rather than mixing of corrected models leads to more robust extrapolations. 

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