arXiv subjects

Johann Stumpenhusen

Publications and source records attributed to Johann Stumpenhusen.

8 recordsLinked to original sources

arXiv2026

Local Weak Maass Forms and Traces of Cycle Integrals

Using relations established by Bringmann and Kane in a recent preprint based on the function $ω_{k+1,D}$ studied by Mono, Rolen, and the second author, we introduce two further functions naturally associated with a construction by Bringmann and Mono. We exploit results by Löbrich--Schwagenscheidt, Mono, and the second author on representations of these functions as traces of cycles integrals in order to provide more identities of the same kind.

math.NT↗

arXiv2026

A Positive Proportion of the Reduced D'Arcais Polynomials is not Hurwitz

Heretofore, the second and third author conjectured that the D'Arcais polynomials, related to the coefficients of the powers of the Dedekind $η$-function, are Hurwitz polynomials except for a root at the origin. We show that this does in fact not hold for a positive proportion of all natural numbers.

math.NT↗

arXiv2026

Modular Forms Related to Real Quadratic Fields as Traces of Cycle Integrals

A decade ago, locally harmonic Maaß forms were first established by Bringmann, Kane, and Kohnen in negative weights and independently by Hövel in weight $0$. Since then, they have seen important applications related to twisted central $L$-values of newforms. However, there are just a few examples of these forms known so far, and in particular there is no unifying theory behind these examples. Towards this direction, we show that the representations of Zagier's $f_{k,D}$ and Mono's $g_{k+1,D}$ functions as traces of cycle integrals are not unique.

math.NT↗

arXiv2026

On the Log-Concavity of the D'Arcais Polynomials for Normalised Functions

A sequence $(a_n)_{n \in \mathbb{N}}$ of non-negative real numbers is called log-concave at $n$ if $a_n^2 \geq a_{n+1}a_{n-1}$. This property has been generalised in various ways to families of polynomials. We introduce a new variant and show that certain types of D'Arcais polynomials have the respective properties at certain points.

math.NT↗

arXiv2026

The difference of the sums of odd and even parts of restricted partitions

For a subset $D \subset \mathbb{N}$, denote by $S\left(D,n\right)$ the total sum of all odd parts minus the sum of all even parts of all partitions of $n$ in which parts from $D$ do not repeat. In this paper, we derive the generating function for $S\left(2\mathbb{N},n\right)$ and use it to give some congruences modulo 4. We also derive the generating function for $S\left(2\mathbb{N}-1,n\right)$ and give two congruences for these functions, one of which was previously proved by Garvan and Sarma.

math.NT↗

arXiv2026

On the Smallest Counterexample to the Log-Concavity of the D'Arcais Polynomials

Recently, Starr used asymptotic methods to disprove a conjecture by Heim--Neuhauser and Abdesselam about the log-concavity of the D'Arcais polynomials, without giving an explicit counterexample. We refine the asymptotics, to give the necessary estimates on convolutions of $σ_{-1}$, and identify the first counterexample at $λ= 65\,214\,507\,758\,400$. We also consider the asymptotic density of such counterexamples.

math.NT↗

arXiv2025

On the Detection of Non-Roots of D'Arcais Polynomials

The Lehmer conjecture states that the non-constant Fourier coefficients of the 24th power of the Dedekind eta function are non-zero. In a recent preprint, Neuhauser and the first author exploited an easily accessible tool from algebraic number theory, namely the Dedekind--Kummer Theorem, to prove the non-vanishing of the Fourier coefficients of certain powers of the Dedekind eta function at roots of unity. We extend the application of this method to enlarge the scope of non-roots of the related D'Arcais polynomials.

math.NT↗

arXiv2025

On a Divisor Modular Form and a Theta Lift

In 1975, Zagier introduced the highly influential hyperbolic Poincaré series $f_{k,D}$. We connect the divisor modular form of $f_{k,D}$ to a new weak Maass form $ω_{k+1,D}$. Furthermore, we show that the generating function of $ω_{k+1,D}$ has the same modularity properties as Kohnen and Zagier's fruitful theta kernel generating the $f_{k,D}$'s. This yields a new theta lift.

math.NT↗