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• Title: Twists of $L$-functions and applications 

• Abstract: The talk is devoted to twists of $L$-functions in the Selberg class and their applications to converse theorems and moment problems. Twisting techniques provide a powerful tool for uncovering the structural properties of $L$-functions. We survey both older and recent results illustrating the effectiveness of this approach, with particular emphasis on the classification of low-degree $L$-functions. We also discuss recent developments concerning self-dual twists and explain how their conjectured analytic properties may lead to sharp asymptotic formulas for moments on the critical line. The talk is based on a series of joint papers with Alberto Perelli. 

• Title: On the Bogomolny-Schmit Conjecture for Maass Forms 

• Abstract: We introduce the Bogomolny-Schmit conjecture. 

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• Title: Sign changes of Fourier coefficients of modular forms and their applications 

• Abstract: We survey recent developments on sign changes in the Fourier coefficients of modular forms. Such sign changes reflect the oscillatory nature of these coefficients and are connected with arithmetic properties, such as the Hecke relations and the Sato-Tate distribution of Hecke eigenvalues, as well as with the analytic properties of L-functions. We discuss selected results on sign changes for holomorphic cusp forms and, where appropriate, for Maass forms. We also highlight the main methods involved and present a recent application in number theory. 

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• Title: The hyperbolic lattice point problem 

• Abstract: The Gauss circle problem is a classical problem in number theory and harmonic analysis that asks to find the number of integer lattice points within a disc of radius $R$ for large $R$. In this talk I will describe this problem and discuss an analogue of the Gauss circle problem in hyperbolic space. Given two points $z, w$ that lie in the hyperbolic upper half‑plane, the hyperbolic lattice point problem is to determine the number of $SL_2(\mathbb Z)$ translates of $w$ that lie in the hyperbolic disk centred at $z$ with radius $arcosh(R/2)$ for large $R$. Selberg proved that the error term in this problem is $O(R^{2/3})$. I will describe some joint work in which we improve the error term to $o(R^{2/3})$ as $R$ tends to infinity, for $z,w$ that are CM-points of different, square-free discriminants. This is joint work with Dimitrios Chatzakos, Giacomo Cherubini, and Morten Risager.

• Title: Biases in the gaps between zeros of Dirichlet L-functions 

• Abstract: I will describe an infinite family of Dirichlet L-functions whose members provably have an unusual value distribution and, experimentally, exhibit a substantial and previously undetected bias in the distribution of gaps between their zeros. This bias has an arithmetic explanation, corresponding to the nonvanishing of a certain Gauss-type sum. We classify the characters for which these Gauss-type sums are nonzero, and count the number of such characters. It turns out that this Gauss-type sum vanishes for 100% of primitive Dirichlet characters, so our newly discovered family is thin, corresponding to a zero-density subset of the primitive characters. I will also describe some experimental results concerning a Chebyshev-type bias in the gaps between the zeros of the Riemann zeta-function, as well as related phenomena for a different thin family of Dirichlet L-functions. This is joint work with Jonathan Bober (Bristol) and Zhenchao Ge (Waterloo). 

• Title: Connection between the Riemann zeta-function and random matrices via hyperfunctions 

• Abstract: In this talk, we extend Bohr's theory on the value-distribution of the Riemann zeta-function to the space of hyperfunctions. We introduce certain random hyperfunctions associated with the value-distribution of the Riemann zeta-function on the critical line. Furthermore, we consider the random matrix analogue using the circular unitary ensemble. We then derive a relationship between these random hyperfunctions which is consistent with the Keating-Snaith conjecture on the moments of the Riemann zeta-function. 

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• Title: Applications of the screw function to the Weil quadratic form 

• Abstract: The Weil quadratic form is defined using the Guinand-Weil explicit formula, a generalization of the classical explicit formulas in analytic number theory. Motivated by the analogy with the proof of the Riemann Hypothesis for zeta functions of one-dimensional function fields via Castelnuovo's inequality, Weil formulated his positivity criterion in 1952, asserting that the Riemann Hypothesis is equivalent to the positivity of the Weil quadratic form. Although little progress was made for a long time, Yoshida (1992) obtained several pioneering results by localizing the Weil quadratic form to finite intervals [-a,a]. This localized approach was subsequently adopted and extended to spectral-theoretic studies in the major contributions by Bombieri (2001), Connes and Consani (2023), and Connes, Consani, and Moscovici (2025+).

The purpose of this talk is to explain how these works can be understood in a unified way from a variational viewpoint by using the theory of screw functions, which was introduced into number theory in 2023. Based on this framework, we formulate a conjecture stating that a self-adjoint operator whose eigenvalues are the imaginary parts of the non-trivial zeros of the Riemann zeta function can be obtained as the limit, as $a \to \infty$, of self-adjoint operators arising from nonlocal realizations of the first-order differential operator on the finite interval [-a,a]. This conjecture should be compared with the limit formula for the Riemann zeta function expressed in terms of zeta-regularized products proposed by Connes, Consani, and Moscovici, and it sheds new light on the spectral-theoretic interpretation of the non-trivial zeros of the Riemann zeta function.