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1. Quadratic enrichment of the logarithmic derivative of the zeta function

   https://doi.org/10.1090/btran/201  

   Bilu, Margaret; Ho, Wei; Srinivasan, Padmavathi; Vogt, Isabel; Wickelgren, Kirsten (October 2024, Transactions of the American Mathematical Society, Series B)

   We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck–Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure. 

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2. Traces, high powers and one level density for families of curves over finite fields

   https://doi.org/10.1017/S030500411700041X  

   BUCUR, ALINA; COSTA, EDGAR; DAVID, CHANTAL; GUERREIRO, JOÃO; LOWRY–DUDA, DAVID (September 2018, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix Θ C . We develop and present a new technique to compute the expected value of tr(Θ C n ) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [ Rud10 ] and Chinis [ Chi16 ]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [ BDF + 16 ] and [ Zha ]. We extend [ BDF + 16 ] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L -functions L (1/2 + it , χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers. 

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3. Negative discrete moments of the derivative of the Riemann zeta‐function

   https://doi.org/10.1112/blms.13092  

   Bui, Hung_M; Florea, Alexandra; Milinovich, Micah_B (May 2024, Bulletin of the London Mathematical Society)

   Abstract We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For , our bounds for the ‐th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros. 

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4. Higher correlations and the alternative hypothesis

   https://doi.org/10.1093/qmathj/haz.043  

   Lagarias, Jeffrey C.; Rodgers, Brad (April 2020, The quarterly journal of mathematics)

   The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functionsof the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in 1/2 Z, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function. 

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5. From Curves to Words

   Chang, Hsien-Chih; Fasy, Brittany Terese; McCoy, Bradley; Millman, David L.; Wenk, Carola (June 2023, Young Researcher's Forum at CG Week)

   Blank, in his Ph.D. thesis on determining whether a planar closed curve $$\gamma$$ is self-overlapping, constructed a combinatorial word geometrically over the faces of $$\gamma$$ by drawing cuts from each face to a point at infinity and tracing their intersection points with $$\gamma$$. Independently, Nie, in an unpublished manuscript, gave an algorithm to determine the minimum area swept out by any homotopy from a closed curve $$\gamma$$ to a point. Nie constructed a combinatorial word algebraically over the faces of $$\gamma$$ inspired by ideas from geometric group theory, followed by dynamic programming over the subwords. In this paper, we examine the definitions of the two words and prove the equivalence between Blank's word and Nie's word under the right assumptions. 

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