Seminare und Vorträge im WS 2025/2026

am Montag, 27. Oktober :

Oberseminar Zahlentheorie

Debmalya Basak
Title: Distributions Associated to Small Quadratic Non-Residues and their Partitions
Abstract: Assuming the Generalized Riemann Hypothesis, it is known that the least quadratic nonresidue modulo a prime \(p\) is less than or equal to \( (\mathrm{log} \, p)^2\). In this talk, we will discuss unconditional results on the distribution of quadratic nonresidues and their associated partitions in significantly smaller intervals of length \( (\mathrm{log} \, p)^A \) with 0 < A < 2. Along the way, we shall introduce an analogue of the Dedekind Eta function for Hecke Groups \(H(\sqrt{D})\), leading to some interesting conjectures. This is based on joint projects with Bruce Berndt, Dorian Goldfeld, Kunjakanan Nath, Nicolas Robles and Alexandru Zaharescu.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 17. November :

Oberseminar Zahlentheorie

Ozlem Imamoglu
Title: A Lyapunov exponent attached to modular functions
Abstract: Recently, in joint work with Paloma Bengoechea and Sebastian Herrero we defined a Lyapunov exponent attached to modular functions and proved its properties. Our work was motivated by the work of Spalding and Veselov as well as conjectures of Kaneko on the values of modular functions at real quadratic irrationalities. In this talk I will first explain the results of Spalding and Veselov and conjectures of Kaneko. I will then talk about the new results.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 01. Dezember :

Oberseminar Zahlentheorie

Likun Xie
Title: On a Conjecture of Erdős over Function Fields
Abstract: I will discuss a function-field analogue of a classical conjecture of Erdős. Given a squarefree polynomial f over F_q of degree n, one can ask whether every residue class modulo f can be written as a product of two irreducible polynomials of degree at most n. I will explain how Katz’s equidistribution framework leads to an affirmative answer when q is large in terms of n. Sawin previously proved this result with stronger square-root cancellation using a higher-dimensional geometric method. I will instead present a one-dimensional approach that produces a natural q^{-1/2} saving and is effective in the large-q regime.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 15. Dezember :

Oberseminar Zahlentheorie

Eleanor McSpirit
Title: Quantum Modular forms and Resurgence
Abstract: In 2010, Zagier described a new phenomenon which he called quantum modularity. This connected various examples coming from disparate fields that exhibit near modular behavior. Since then, Zagier’s quantum modular forms have been connected to Ramanujan’s mock theta-functions and have informed new developments in areas such as quantum topology and physics. More recently, the concept of holomorphic quantum modularity has emerged, pointing to a clearer structure for Zagier’s original examples. These new developments suggest connections to the theory of resurgence. In joint work with Rolen, we unify the examples of quantum modular forms in Zagier’s original paper under the umbrella of resurgence. In doing so, we strengthen known quantum modularity results for holomorphic Eichler integrals of half-integer weight modular forms.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 12. Januar :

Oberseminar Zahlentheorie

Harald Andrés Helfgott
Title: Optimal bounds for sums of arithmetic functions
Abstract: (Joint with Andrés Chirre) Let \(A(s) = \sum_n a_n n^{-s}\) be a Dirichlet series with meromorphic continuation. Say we are given information on the poles of \(A(s)\) with \(|\Im s| \leq T\) for some large constant \(T\). What is the best way to use such finite spectral data to give explicit estimates for sums \(\sum_{n\leq x} a_n\)? The problem of giving explicit bounds on the Mertens function \(M(x) = \sum_{n\leq x} \mu(n)\) illustrates how open this basic question was. Bounding \(M(x)\) might seem equivalent to estimating \(\psi(x) = \sum_{n\leq x} \Lambda(n)\) or the number of primes \(\leq x\). However, we have long had fairly good explicit bounds on prime counts, while bounding \(M(x)\) remained a notoriously stubborn problem. We prove a sharp, general result on sums \(\sum_{n\leq x} a_n n^{-\sigma}\) for \(a_n\) bounded, giving a optimal way to use information on the poles of \(A(s)\) with \(|\Im s|\leq T\) and no data on the poles above. Our bounds on \(M(x)\) are stronger than previous ones by many orders of magnitude. We also give a sharp result on such sums for a_n non-negative and not necessarily bounded, and apply it to obtain optimal bounds on psi(x)-x given finite verifications of RH. Our proofs mixes a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type as in (Graham--Vaaler, 1981) and (Carneiro--Littmann, 2013); for \(\sigma=1\), the approximants in (Vaaler, 1985) and in Beurling and Selberg's work reappear.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 19. Januar :

Oberseminar Zahlentheorie

Mads Christensen
Title: Linking numbers and non-holomorphic Siegel modular forms
Abstract: A compelling selling point for the theory of classical modular forms is the frequent connections between their Fourier coefficients and other areas of mathematics. I will give some examples of this, first classical theta functions, then indefinite theta series in the style of Zwegers, Funke, and Kudla, and ending with new results that relate certain non-holomorphic Siegel modular forms of genus 2 to arithmetic hyperbolic 3-manifolds.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 26. Januar :

Oberseminar Zahlentheorie

Oliver Schlotterer
Title: Single-valued elliptic polylogarithms from zeta generators
Abstract: Multiple polylogarithms at genus zero are by themselves multi-valued functions but can be completed to single-valued polylogarithms by adding suitable combinations with their complex conjugates and multiple zeta values. As a generalization to genus one, this talk presents an explicit construction of single-valued elliptic polylogarithms which depend on a marked point on the torus and its modular parameter. Our single-valued elliptic polylogarithms do not have any monodromies from the homology cycles or loops around singular points of the integrand and transform as non-holomorphic modular forms under SL(2,Z). The construction is carried out at the level of Lie-algebra valued generating series where the combinations of elliptic polylogarithms, their complex conjugates and (elliptic) multiple zeta values are controlled by certain derivations on the generators. The series in single-valued polylogarithms at genus zero and genus one exhibit striking parallels since the appearance of multiple zeta values is governed by so-called zeta generators on the Lie-algebra setup of both cases.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 02. Februar :

Oberseminar Zahlentheorie

Aritram Dhar
Title: Extension of Bressoud's generalization of Borwein's conjecture, some exact results, and applications
Abstract: In this talk, we conjecture an extension of Bressoud's 1996 generalization of Borwein's famous 1990 conjecture. We then state a few infinite hierarchies of non- negative q-series identities which are interesting examples of our proposed conjecture and Bressoud's generalized conjecture. Using certain positivity-preserving transformations for q-binomial coefficients due to Berkovich and Warnaar, we prove the non-negativity of the infinite families. We also present some applications of cubic positivity-preserving transformations where we establish new identities analogous to Andrews' representations of the Borwein polynomials. This talk is based on recent joint works with Alexander Berkovich.

14:00 Seminarraum 3 des Mathematischen Instituts

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