arXiv subjects

Dirk Zeindler

Publications and source records attributed to Dirk Zeindler.

At least 19 recordsLinked to original sources

arXiv2026

On the exponential sum over squarefree integers

Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on the minor arcs of the Hardy--Littlewood dissection throughout the range $Q\le N^{1/2}$. The estimates of Schlage-Puchta [SP] and of Tolev [T] have the same dependence on $q$ and $Q$ but carry a factor $N^{\varepsilon}$. The proof uses Heath-Brown's square sieve with sieving primes confined to an interval $(P,2P]$, where $P$ may be as small as a multiple of $\log N$; a finite Fej\'er majorant in place of a truncated Fourier series; and, after completion of the character sums, a count of representations that exploits the restriction on the primes in place of the divisor function.

math.NT↗

arXiv2026

Bilinear forms with Kloosterman fractions and applications

We establish improved bounds for bilinear forms with Kloosterman fractions of the form ${\sum\sum}_{m,n} \alpha_m \beta_n e(a\overline{m}/(bn))$ with $M<m\le 2M$, $N < n \le 2N$ and $(m,n)=1$. Our approach works directly with arbitrary coefficient sequences $(\alpha_m), (\beta_n) \in \mathbb{C}$, avoiding the temporary restriction to squarefree support used in prior work. While this requires handling additional arithmetic complexity, it yields strictly stronger bounds that improve upon the estimates of Duke, Friedlander, and Iwaniec \cite{DFI} and Bettin-Chandee \cite{BC}; in the balanced case $M \approx N$, the new saving over the trivial bound is $1/12$%, compared to $1/48$ in \cite{DFI} . As an application, we prove a generalized asymptotic formula for the twisted second moment of the Riemann zeta-function with Dirichlet polynomials of length $T^{1/2+\delta}$ for $\delta = 1/46$, extending beyond the previously limiting $\theta = 1/2$ barrier established by Bettin, Chandee, and Radziwi{\l}{\l} \cite{BCR}. We also establish bounds for related Hermitian sums involving Sali\'{e}-type exponential phases and develop techniques for more general bilinear forms with Kloosterman fractions.

math.NT↗

arXiv2024

Exponential sums twisted by general arithmetic functions

We examine exponential sums of the form $\sum_{n \le X} w(n) e^{2\pi i\alpha n^k}$, for $k=1,2$, where $\alpha$ satisfies a generalized Diophantine approximation and where $w$ are different arithmetic functions that might be multiplicative, additive, or neither. A strategy is shown on how to bound these sums for a wide class of functions $w$ belonging within the same ecosystem. Using this new technology we are able to improve current results on minor arcs that have recently appeared in the literature of the Hardy-Littlewood circle method. Lastly, we show how a bound on $\sum_{n \le X} |\mu(n)| e^{2\pi i\alpha n}$ can be used to study partitions asymptotics over squarefree parts and explain their connection to the zeros of the Riemann zeta-function.

math.NT↗

arXiv2024

Strange and pseudo-differentiable functions with applications to prime partitions

Let $\mathfrak{p}_{\mathbb{P}_r}(n)$ denote the number of partitions of $n$ into $r$-full primes. We use the Hardy-Littlewood circle method to find the asymptotic of $\mathfrak{p}_{\mathbb{P}_r}(n)$ as $n \to \infty$. This extends previous results in the literature of partitions into primes. We also show an analogue result involving convolutions of von Mangoldt functions and the zeros of the Riemann zeta-function. To handle the resulting non-principal major arcs we introduce the definition of strange functions and pseudo-differentiability.

math.NT↗

arXiv2023

Partitions into semiprimes

Let $\mathbb{P}$ denote the set of primes and $\mathcal{N}\subset \mathbb{N}$ be a set with arbitrary weights attached to its elements. Set $\mathfrak{p}_{\mathcal{N}}(n)$ to be the restricted partition function which counts partitions of $n$ with all its parts lying in $\mathcal{N}$. By employing a suitable variation of the Hardy-Littlewood circle method we provide the asymptotic formula of $\mathfrak{p}_{\mathcal{N}}(n)$ for the set of semiprimes $\mathcal{N} = \{p_1 p_2 : p_1, p_2 \in \mathbb{P}\}$ in different set-ups (counting factors, repeating the count of factors, and different factors). In order to deal with the minor arc, we investigate a double Weyl sum over prime products and find its corresponding bound thereby extending some of the results of Vinogradov on partitions. We also describe a methodology to find the asymptotic partition $\mathfrak{p}_{\mathcal{N}}(n)$ for general weighted sets $\mathcal{N}$ by assigning different strategies for the major, non-principal major, and minor arcs. Our result is contextualized alongside other recent results in partition asymptotics.

math.NT↗

arXiv2020

Precise asymptotics of longest cycles in random permutations without macroscopic cycles

We consider Ewens random permutations of length $n$ conditioned to have no cycle longer than $n^β$ with $0<β<1$ and to study the asymptotic behaviour as $n\to\infty$. We obtain very precise information on the joint distribution of the lengths of the longest cycles; in particular we prove a functional limit theorem where the cumulative number of long cycles converges to a Poisson process in the suitable scaling. Furthermore, we prove convergence of the total variation distance between joint cycle counts and suitable independent Poisson random variables up to a significantly larger maximal cycle length than previously known. Finally, we remove a superfluous assumption from a central limit theorem for the total number of cycles proved in an earlier paper.

math.PR↗

arXiv2020

Long cycle of random permutations with polynomially growing cycle weights

We study the asymptotic behavior of the long cycles of a random permutation of $n$ objects with respect to multiplicative measures with polynomial growing cycle weights. We show that the longest cycle and the length differences between the longest cycles converge, after suitable normalisation, in distribution to iid random variables $(Z_j)_{j\in\mathbb{N}}$ such that $\exp(-Z_j)$ is exponentially distributed. Our method is based on generating functions and the saddle point method.

math.PR↗

arXiv2018

Random permutations without macroscopic cycles

We consider uniform random permutations of length $n$ conditioned to have no cycle longer than $n^β$ with $0<β<1$, in the limit of large $n$. Since in unconstrained uniform random permutations most of the indices are in cycles of macroscopic length, this is a singular conditioning in the limit. Nevertheless, we obtain a fairly complete picture about the cycle number distribution at various lengths. Depending on the scale at which cycle numbers are studied, our results include Poisson convergence, a central limit theorem, a shape theorem and two different functional central limit theorems.

math.PR↗

arXiv2018

Random permutations with logarithmic cycle weights

We consider random permutations on $\Sn$ with logarithmic growing cycles weights and study asymptotic behavior as the length $n$ tends to infinity. We show that the cycle count process converges to a vector of independent Poisson variables and also compute the total variation distance between both processes. Next, we prove a central limit theorem for the total number of cycles. Furthermore we establish a shape theorem and a functional central limit theorem for the Young diagrams associated to random permutations under this measure. We prove these results using tools from complex analysis and combinatorics. In particular we have to apply the method of singularity analysis to generating functions of the form $\exp\left( (-\log(1-z))^{k+1} \right)$ with $k\geq 1$, which have not yet been studied in the literature.

math.PR↗

arXiv2018

More than five-twelfths of the zeros of $\zeta$ are on the critical line

The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $(\mu \star \Lambda_1^{\star k_1} \star \Lambda_2^{\star k_2} \star \cdots \star \Lambda_d^{\star k_d})$ is computed unconditionally by means of the autocorrelation of ratios of $\zeta$ techniques from Conrey, Farmer, Keating, Rubinstein and Snaith (2005), Conrey, Farmer and Zirnbauer (2008) as well as Conrey and Snaith (2007). This in turn allows us to describe the combinatorial process behind the mollification of \[ \zeta(s) + \lambda_1 \frac{\zeta'(s)}{\log T} + \lambda_2 \frac{\zeta''(s)}{\log^2 T} + \cdots + \lambda_d \frac{\zeta^{(d)}(s)}{\log^d T}, \] where $\zeta^{(k)}$ stands for the $k$th derivative of the Riemann zeta-function and $\{\lambda_k\}_{k=1}^d$ are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (2017), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.

math.NT↗

arXiv2017

On mean values of mollifiers and L-functions associated to primitive cusp forms

We study the second moment of the L-function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers. This family is a natural extension of the mollifers considered by Conrey and by Bui, Conrey and Young. As an application, we improve the current lower bound on critical zeros of holomorphic primitive cusp forms.

math.NT↗

arXiv2016

On a mollifier of the perturbed Riemann zeta-function

The mollification $ζ(s) + ζ'(s)$ put forward by Feng is computed by analytic methods coming from the techniques of the ratios conjectures of $L$-functions. The current situation regarding the percentage of non-trivial zeros of the Riemann zeta-function on the critical line is then clarified.

math.NT↗

arXiv2015

The order of large random permutations with cycle weights

The order $O_n(σ)$ of a permutation $σ$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $σ$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdös and Turán who proved in 1965 that $\log O_n$ satisfies a central limit theorem. We extend this result to the so-called \textit{generalized Ewens measure} in a previous paper. In this paper, we establish a local limit theorem as well as, under some extra moment condition, a precise large deviations estimate. These properties are new even for the uniform measure. Furthermore, we provide precise large deviations estimates for random permutations with polynomial cycle weights.

math.PR↗

arXiv2014

Total variation distance and the Erdős-Turán law for random permutations with polynomially growing cycle weights

We study the model of random permutations of $n$ objects with polynomially growing cycle weights, which was recently considered by Ercolani and Ueltschi, among others. Using saddle-point analysis, we prove that the total variation distance between the process which counts the cycles of size $1, 2, ..., b$ and a process $(Z_1, Z_2, ..., Z_b)$ of independent Poisson random variables converges to $0$ if and only if $b=o(\ell)$ where $\ell$ denotes the length of a typical cycle in this model. By means of this result, we prove a central limit theorem for the order of a permutation and thus extend the Erdős-Turán Law to this measure. Furthermore, we prove a Brownian motion limit theorem for the small cycles.

math.PR↗

arXiv2014

Asymptotic statistics of cycles in surrogate-spatial permutations

We propose an extension of the Ewens measure on permutations by choosing the cycle weights to be asymptotically proportional to the degree of the symmetric group. This model is primarily motivated by a natural approximation to the so-called spatial random permutations recently studied by V. Betz and D. Ueltschi (hence the name "surrogate-spatial"), but it is of substantial interest in its own right. We show that under the suitable (thermodynamic) limit both measures have the similar critical behaviour of the cycle statistics characterized by the emergence of infinitely long cycles. Moreover, using a greater analytic tractability of the surrogate-spatial model, we obtain a number of new results about the asymptotic distribution of the cycle lengths (both small and large) in the full range of subcritical, critical and supercritical domains. In particular, in the supercritical regime there is a parametric "phase transition" from the Poisson-Dirichlet limiting distribution of ordered cycles to the occurrence of a single giant cycle. Our techniques are based on the asymptotic analysis of the corresponding generating functions using Polya's Enumeration Theorem and complex variable methods.

math.PR↗

arXiv2014

The limit shape of random permutations with polynomially growing cycle weights

In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set $\{1,\dots,n\}$ under a particular class of multiplicative measures. Our method is based on generating functions and complex analysis (saddle point method). We show that fluctuations near a point behave like a normal random variable and that the joint fluctuations at different points of the limiting shape have an unexpected dependence structure. We will also compare our approach with the so-called randomization of the cycle counts of permutations and we will study the convergence of the limit shape to a continuous stochastic process.

math.PR↗

arXiv2013

The Characteristic Polynomial of a Random Permutation Matrix at Different Points

We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show that the behavior at different points is independent in the limit and are asymptotically normal. Our methods enables us to study more general matrices, closely related to permutation matrices, and multiplicative class functions.

math.PR↗

arXiv2013

Random permutation matrices under the generalized Ewens measure

We consider a generalization of the Ewens measure for the symmetric group, calculating moments of the characteristic polynomial and similar multiplicative statistics. In addition, we study the asymptotic behavior of linear statistics (such as the trace of a permutation matrix or of a wreath product) under this new measure.

math.PR↗