arXiv subjects
Publications and source records attributed to Kathrin Bringmann.
In this note, we construct combinatorial statistics in terms of partitions, overpartitions, and two-color partitions and relate the corresponding generating functions to the universal mock theta functions. Our framework primarily builds on two $q$-series identities of Fine.
Motivated by conjectures of Schlosser and Zhou on sign patterns of Fourier coefficients of infinite products, we study two families of theta quotients and their positive integral powers. We determine asymptotic formulas for their Fourier coefficients and use them to obtain explicit eventual periodic sign patterns. Our results provide new instances of the sign phenomena observed by Schlosser and Zhou and illustrate how dominant cusps determine the resulting sign patterns. These results were formalized and verified in Lean by AxiomProver.
We develop an analytic framework to study hyperbolicity of Jensen polynomials and related inequalities for arithmetic sequences with general asymptotic growth. Our results provide partial converses to the Hermite--Jensen phenomenon of Griffin, Ono, Zagier, and the third author. As applications we prove several conjectures concerning log-concavity, higher-order Turan inequalities, Laguerre inequalities, infinite log-concavity, and Toeplitz determinants for partition-type functions.
A classical class number relation of Hurwitz expresses the Fourier coefficients of the product of a unary theta function and the class number generating function. Here, we establish an infinite family of analogous class number relations obtained by replacing the unary quadratic form $m^2$ by positive-definite binary quadratic forms. These identities involve Cohen's generalized class numbers and depend only on the genus of the underlying quadratic form. For this, we construct a genus-dependent level-lowering operator.
In this paper we study sign changes of an infinite class of $η$-quotients which are holomorphic modular forms. There is also a relation to Hurwitz class numbers.
In this paper, we prove a conjecture of Andrews and El Bachraoui concerning the parity of certain two-color partitions. Precisely, we show that if the Fourier coefficient $t_o(n)$ of the corresponding $q$-series is odd, then $8n+9$ is represented by the binary quadratic form $x^2+2y^2$.
The notion of depth two and higher mock modular forms have found important applications in mathematical physics and enumerative geometry since their inception through indefinite theta functions with general signature. These theta functions generalize Zwegers' work on Lorentzian signature lattices and the framework of mock modular forms that emanated from it. Mock modular forms can also be studied through Eisenstein and Poincaré series. The interaction of this second point of view with the indefinite theta function approach yields a wealth of tools to unearth the rich structure behind mock modular forms. For mock modular forms of higher depth, on the other hand, indefinite theta functions and their variants largely remained the only available approach. In this paper, we show that one can indeed get mock modular forms of depth two by "coupling" a pair of Eisenstein series that yield depth one mock modular forms, thereby providing a new and independent approach to higher depth mock modular forms. We exemplify this new perspective on a depth two object that appeared in the context of Vafa-Witten invariants.
In this paper, we give a direct conceptual proof of the main result of Mono, Rolen, and Stumpenhusen, using differential operators. More precisely, we realize their functions $ω_{k+1,D}$ as images of the quadratic form Poincaré series $f_{k,D}$ under the Maass raising operator. This perspective gives a natural explanation for the modularity and Laplace eigenvalue prop erties of $ω_{k+1,D}$. We further extend these results by investigating the images of more general local Maass forms under the Maass raising and lowering operators.
We study an extension of Ramanujan's identities for odd zeta values by Lim and introduce Jacobi analogues of classical Eichler integrals of Eisenstein series. In negative weight we construct explicit completions and embed these objects into a modular framework by showing that they are (singular) harmonic Maass--Jacobi forms. We further describe their non-holomorphic parts in terms of Eichler integrals, establish Ramanujan-type inversion formulas, and study their behavior under the Maass raising and lowering operators and at torsion points.
In this paper, we investigate congruences for meromorphic modular forms $F$ which have a pole at a single point $z$ in the fundamental domain of $\mathrm{SL}_2(\mathbb Z)$. For a prime $p$ with good supersingular reduction at the elliptic curve corresponding to $z$, we show that there exists a cusp form $f$ such that $F|U_p^m \equiv f|U_p^m \pmod{p^{κ_m}}$, where $κ_m=αm -β$ with $α$ only depending on the weight of $F$ and $β$ depending on $F$ and $p$ but is independent of $m$. In particular, if the space of cusp forms is trivial, then $F|U_p^m\equiv 0 \pmod{p^{κ_m}}$ vanishes $p$-adically to a high order. In order to prove these results, we use the fact that $p$ has supersingular reduction to realize $F$ as an overconvergent modular form and then utilize the theory of overconvergent forms to show the congruences.
In this paper, we prove a conjecture of Andrews and Bachraoui relating a generating function arising from two-color partitions (with odd smallest part and restrictions on the even parts) to a Hecke-type double sum. Our proof is based on Zwegers' theory of indefinite theta functions together with modular transformation properties of mock theta functions.
In recent years, there has been extensive work on inequalities among partition functions. In particular, Nicolas, and independently DeSalvo--Pak, proved that the partition function $p(n)$ is eventually log-concave. Inspired by this and other results, Chern--Fu--Tang first conjectured log-concavity of $k$-coloured partitions. Three of the authors and Tripp later proved this conjecture by introducing recursive sequences and a strict inequality for fractional partition functions, giving explicit errors. In this paper, we show that the log-concavity is, in fact, strict for $k\geq 2$. We shed further light on this phenomenon by utilizing Hardy--Littlewood--Pólya's notion of majorizing. We prove that for partitions $\bm{a},\bm{b}$ of $n\in\N$, if $\bm b$ majorizes $\bm a$, then $p_k(\bm{b})>p_k(\bm{a})$. Numerical calculations indicate that our result is sharp.
In this paper we study restricted overpartitions and concave compositions. In several cases the resulting generating functions involve simultaneously modular forms, mock theta functions, mock Maass theta functions, and false theta functions, illustrating the appearance of mixed modular structures in restricted partition problems. Moreover, we obtain their asymptotic main terms. We also study related rank statistics.
Andrews and the third author recently studied congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan-type congruences and a vanishing identity for the limiting sequence. In this paper, we settle these conjectures by relating the corresponding generating function to modular forms and mock theta functions.
In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations with non-negative parameters (hence counting the number of points in a regular $m$-gon) is asymptotically the same as $\frac{1}{16}$ times the number of such representations with arbitrary integer parameters (often called generalized polygonal numbers).
We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.
In this paper, we introduce a class of functions that behave like classical Eisenstein series in many ways, but with a key distinction: only their non-holomorphic completions transform like (quasi)modular forms. We show how the partition rank generating function can be expressed in terms of partition traces of these functions. A key feature of our construction is that the completions satisfy a holomorphic anomaly equation - a phenomenon typically seen in the context of quantum field theory and string theory. We also show that the Fourier coefficients of these Eisenstein-type series are integral.
Motivated by the fact that the classical Jacobi theta function $\vartheta$ is the exponential generating function of the Eisenstein series, we study the exponential Taylor coefficients (in the elliptic variable) of a related natural partial theta function, as well as a false theta function related to the Dedekind eta function. We prove that the space spanned by these objects is closed under differentiation, analogous to the space of quasimodular forms, and that it contains the quasimodular forms themselves. We further provide their Fourier expansions, establish quasimodular completions, and derive a recursive formula for the Taylor coefficients of the logarithm of the unimodal rank generating function, expressed as partition traces of the false and partial objects.