Searcharxiv⌕ Search

arXiv subjects

Berend Ringeling

Publications and source records attributed to Berend Ringeling.

13 recordsLinked to original sources
arXiv2026

Machine learning the arithmetic of Boyd's Mahler measure conjectures

Boyd conjectured that the Mahler measure of $P_k(x,y)=x+y+\frac{1}{x}+\frac{1}{y}+k$ for $k$ an integer, is given by $r_kL'(E_k,0)$, where $E_k$ is the elliptic curve associated to the zero locus of $P_k$ and $r_k$ is a rational number. We study various arithmetic properties of $r_k$ using a dataset containing the first $250{,}000$ values of $k$, combining large-scale statistical analysis assisted by Claude with transformer-based experiments carried out using Axolver. We recover Boyd's observation that, apart from a few exceptions, $r_k$ is the reciprocal of an integer. The size of this integer is governed by the conductor of the elliptic curve. Moreover, its $p$-adic valuations display markedly different behavior according to the prime. For $p\geq 5$, the probability of $v_p(r_k)=-m$ for $m\geq 1$ appears to be $p^{-m}$. For the primes $2$ and $3$, however, we find additional arithmetic structure involving congruence conditions on $k$ and the primes of bad reduction of $E_k$. Although the neural networks do not predict $r_k$ exactly, they recover significant information about its magnitude and valuations. In particular, the experiments at the prime $2$ suggest arithmetic structure beyond the explicit predictor obtained from our statistical analysis.

math.NT↗
arXiv2026

Relating Mahler measures and Dirichlet $L$-values: new evidence for Chinburg's conjectures

Let $\chi_{-f}$ be the odd quadratic Dirichlet character of conductor $f$, and let $\mathrm{m}(P)$ denote the Mahler measure of a polynomial $P$. In 1984, Chinburg conjectured that for any such $\chi_{-f}$ there exist an integral bivariate rational function $P$ (and, in the strong form, an integral polynomial) such that $\mathrm{m}(P)$ is a rational multiple of $L'(\chi_{-f},-1)$. The strong form of the conjecture was previously known to hold for $18$ values of $f$. We double the number of numerical examples, giving $8$ new instances of the strong and $18$ new instances of the weak conjecture. Our examples arise from an explicit approach, which also captures almost all of the previously known results, and is based on work of Boyd and Rodriguez-Villegas. Moreover, we prove Chinburg's weak conjecture if we allow cyclotomic coefficients.

math.NT↗
arXiv2025

Geodesic clustering of zeros of Eisenstein series for congruence groups

We consider a set of generators for the space of Eisenstein series of even weight $k$ for any congruence group $\Gamma$ and study the set of all of their zeros taken for $\Gamma(1)$-conjugates of $\Gamma$ in the standard fundamental domain for $\Gamma(1)$. We describe (a) an upper bound $\kappa_\Gamma + O(1/k)$ for their imaginary part; (b) a finite configuration of geodesics segments to which all zeros converge in Hausdorff distance as $k \rightarrow \infty$; (c) a finite set containing all algebraic zeros for all weights. The bound in (a) depends on the (non-)vanishing of a new generalization of Ramanujan sums. The proof of (b) originates in a method used to study phase transitions in statistical physics. The proof of (c) relies on the theory of complex multiplication. The results can be made quantitative for specific groups. For $\Gamma=\Gamma(N)$ with $4 \nmid N$, $\kappa_\Gamma=1$ and the zeros tend to the unit circle, whereas if $4 \mid N$, $\kappa_\Gamma=2$ and the limit configuration includes parts of vertical geodesics and circles of radius $2$. In both cases, the only algebraic zeros are at $\mathrm{i}$ and $\exp(2\pi \mathrm{i}/3)$ for sufficiently large $k$. For $\Gamma(N)$ with $N$ odd, we use finer estimates to prove a trichotomy for the exact `convergence speed' of the zeros to the unit circle, as well as angular equidistribution of the zeros as $k \rightarrow \infty$.

math.NT↗
arXiv2025

The asymptotic Mahler measure of Gaussian periods

We construct a sequence of cyclotomic integers (Gaussian periods) of particularly small Mahler measure/height. We study the asymptotics of their Mahler measure as a function of their conductor, to find that the growth rate is the (multivariate) Mahler measure of a family of log Calabi-Yau varieties of increasing dimension. In turn, we study the asymptotics of some of these Mahler measures as the dimension increases, as well as properties of the associated algebraic dynamical system. We describe computational experiments that suggest that these cyclotomic integers realise the smallest non-zero logarithmic Mahler measure in the set of algebraic integers with cyclic Galois group of a given odd order. Finally, we discuss some precise conjectures that imply double logarithmic growth for those Mahler measures as a function of that order. The proofs use ideas from the theory of quantitative equidistribution, reflexive polytopes and toric varieties, the theory of random walks, Bessel functions, class field theory, and Linnik's constant.

math.NT↗
arXiv2025

On twisted period functions and Moments of a weighted mean square of Dirichlet L-functions on the critical line

We extend to Dirichlet L-functions associated with arbitrary primitive characters a range of objects and properties -- including Eisenstein series and period functions -- that were originally introduced and studied by Lewis and Zagier (2001), and later by Bettin and Conrey (2013) in the case of the Riemann zeta function, and more recently by Lewis and Zagier (2019) for odd real characters. These tools yield closed-form expressions for the moments of a measure defined via a weighted mean square of the L-function. These moments not only provide a complete characterization of the modulus of the L-function on the critical line, but also imply an infinite number of non-trivial positivity conditions valid for all primitive characters, real or not. The methods also involve a general form of an asymptotic formula based on the shifted Euler--Maclaurin summation formula, which may be of independent interest.

math.NT↗
arXiv2025

Random walks through the areal Mahler measure: steps in the complex plane

We study the areal Mahler measure of the two-variable, $k$-parameter family $x+y+k$ and prove explicit formulas that demonstrate its relation to the standard Mahler measure of these polynomials. The proofs involve interpreting the areal Mahler measure as a random walk in the complex plane and utilizing the areal analogue of the Zeta Mahler function to arrive at the result. Using similar techniques, we also present formulas for a three-variable family $(x+1)(y+1)+kz$ in terms of the standard Mahler measure, along with terms that involve certain hypergeometric functions. For both families we show that its areal Mahler measure is, up to elementary functions, a linear combination of the normal Mahler measure and the volume of the Deninger cycle of the corresponding family.

math.NT↗
arXiv2022

On the modulo $p$ zeros of modular forms congruent to theta series

For a prime $p$ larger than $7$, the Eisenstein series of weight $p-1$ has some remarkable congruence properties modulo $p$. Those imply, for example, that the $j$-invariants of its zeros (which are known to be real algebraic numbers in the interval $[0,1728]$), are at most quadratic over the field with $p$ elements and are congruent modulo $p$ to the zeros of a certain truncated hypergeometric series. In this paper we introduce "theta modular forms" of weight $k \geq 4$ for the full modular group as the modular forms for which the first dim$(M_k)$ Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the $j$-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo $p$ all in the ground field with $p$ elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with $p$ elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.

math.NT↗
arXiv2022

Critical points of modular forms

We count the number of critical points of a modular form with real Fourier coefficients in a $\gamma$-translate of the standard fundamental domain $\mathcal{F}$ (with $\gamma\in \mathrm{SL}_2(\mathbb{Z})$). Whereas by the valence formula the (weighted) number of zeros of this modular form in $\gamma\mathcal{F}$ is a constant only depending on its weight, we give a closed formula for this number of critical points in terms of those zeros of the modular form lying on the boundary of $\mathcal{F},$ the value of $\gamma^{-1}(\infty)$ and the weight. More generally, we indicate what can be said about the number of zeros of a quasimodular form.

math.NT↗
arXiv2021

Exercising in complex Mahler measures: diamonds are not forever

Recently, Hang Liu and Hourong Qin came up with a numerical observation about the relation between the Mahler measures of one hyperelliptic and two elliptic families. The discoverers foresee a proof of the identities "by extending ideas in" two papers of Matilde Lal\'ın and Gang Wu, the ideas based on a theorem of Spencer Bloch and explicit diamond-operation calculations on the underlying curves. We prove the relation using the already available diamond-free methodology. While finding such relations for the Mahler measures remains an art, proving them afterwards is mere complex (analysis) exercising.

math.NT↗
arXiv2021

Special zeta Mahler functions

In 1969, I. Bernstein and S. Gelfand introduced an object, which is now called the zeta Mahler function (ZMF, also zeta Mahler measure) and related to the Mahler measure. Here we discuss a family of ZMFs attached to the Laurent polynomials $k + (x_1 + x_1^{-1}) \cdots (x_r + x_r^{-1})$, where $k$ is real. We give explicit formulae, present examples and establish properties for these ZMFs, such as an RH-type phenomenon. Further, we explore relations with the Mahler measure.

math.NT↗