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Frederik Broucke

Publications and source records attributed to Frederik Broucke.

17 recordsLinked to original sources
arXiv2026

On a question of Gowers related to Littlewood's conjecture

In a blogpost in 2009, Gowers raised a possible approach to Littlewood's conjecture in Diophantine approximation, leading to a question about the existence of sufficiently many points in the unit cube such that "hyperbolic distance" between any two of them is large. In this note we answer this question by an explicit construction. This shows that this approach to prove Littlewood's conjecture cannot work, unless some further refinements are added.

math.NT↗
arXiv2025

On the connection between zero-free regions and the error term in the Prime Number Theorem

We provide for a wide class of zero-free regions an upper bound for the error term in the Prime Number Theorem, refining works of Pintz (1980), Johnston (2024), and Révész (2024). Our method does not only apply to the Riemann zeta function, but to general Beurling zeta functions. Next we construct Beurling zeta functions having infinitely many zeros on a prescribed contour, and none to the right, for a wide class of such contours. We also deduce an oscillation result for the corresponding error term in the Prime Number Theorem, showing that our aforementioned refinement is close to being sharp.

math.NT↗
arXiv2024

On zero-density estimates for Beurling zeta functions

We show the zero-density estimate \[ N(ζ_{\mathcal{P}}; α, T) \ll T^{\frac{4(1-α)}{3-2α-θ}}(\log T)^{9} \] for Beurling zeta functions $ζ_{\mathcal{P}}$ attached to Beurling generalized number systems with integers distributed as $N_{\mathcal{P}}(x) = Ax + O(x^θ)$. We also show a similar zero-density estimate for a broader class of general Dirichlet series, consider improvements conditional on finer pointwise or $L^{2k}$-bounds of $ζ_{\mathcal{P}}$, and discuss some optimality questions.

math.NT↗
arXiv2024

A Mean Value Theorem for general Dirichlet Series

In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=ρx + O(x^β)$ for some $ρ>0$ and $β<1$. We prove that $\frac1T\int_0^T |f(σ+it)|^2\, dt \to \sum_{j=1}^\infty a_j^2n_j^{-2σ}$ for $σ>\frac{1+β}{2}$ and obtain an upper bound for this moment for $β<σ\le \frac{1+β}{2}$. We provide a number of examples indicating the sharpness of our results.

math.NT↗
arXiv2024

Some examples of well-behaved Beurling number systems

We investigate the existence of well-behaved Beurling number systems, which are systems of Beurling generalized primes and integers which admit a power saving in the error term of both their prime and integer-counting function. Concretely, we search for so-called $[α,β]$-systems, where $α$ and $β$ are connected to the optimal power saving in the prime and integer-counting functions. It is known that every $[α,β]$-system satisfies $\max\{α,β\}\ge1/2$. In this paper we show there are $[α,β]$-systems for each $α\in [0,1)$ and $β\in [1/2, 1)$. Assuming the Riemann hypothesis, we also construct certain families of $[α,β]$-systems with $β<1/2$.

math.NT↗
arXiv2024

A new generalized prime random approximation procedure and some of its applications

We present a new random approximation method that yields the existence of a discrete Beurling prime system $\mathcal{P}=\{p_{1}, p_{2}, \dotso\}$ which is very close in a certain precise sense to a given non-decreasing, right-continuous, nonnegative, and unbounded function $F$. This discretization procedure improves an earlier discrete random approximation method due to H. Diamond, H. Montgomery, and U. Vorhauer [Math. Ann. 334 (2006), 1-36], and refined by W.-B. Zhang [Math. Ann. 337 (2007), 671-704]. We obtain several applications. Our new method is applied to a question posed by M. Balazard concerning Dirichlet series with a unique zero in their half plane of convergence, to construct examples of very well-behaved generalized number systems that solve a recent open question raised by T. Hilberdink and A. Neamah in [Int. J. Number Theory 16 05 (2020), 1005-1011], and to improve the main result from [Adv. Math. 370 (2020), Article 107240], where a Beurling prime system with regular primes but extremely irregular integers was constructed.

math.NT↗
arXiv2023

The pointwise behavior of Riemann's function

We present a new and simple method for the determination of the pointwise Hölder exponent of Riemann's function $\sum_{n=1}^{\infty} n^{-2}\sin(πn^{2} x)$ at every point of the real line. In contrast to earlier approaches, where wavelet analysis and the theta modular group were needed for the analysis of irrational points, our method is direct and elementary, being only based on the following tools from number theory and complex analysis: the evaluation of quadratic Gauss sums, the Poisson summation formula, and Cauchy's theorem.

math.CA↗
arXiv2023

On the Lindelöf hypothesis for general sequences

In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences which coincides with the usual Lindelöf hypothesis for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the ''generic'' admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of ''generic'': we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems.

math.NT↗
arXiv2023

A note on Bohr's theorem for Beurling integer systems

Given a sequence of frequencies $\{λ_n\}_{n\geq1}$, a corresponding generalized Dirichlet series is of the form $f(s)=\sum_{n\geq 1}a_ne^{-λ_ns}$. We are interested in multiplicatively generated systems, where each number $e^{λ_n}$ arises as a finite product of some given numbers $\{q_n\}_{n\geq 1}$, $1 < q_n \to \infty$, referred to as Beurling primes. In the classical case, where $λ_n = \log n$, Bohr's theorem holds: if $f$ converges somewhere and has an analytic extension which is bounded in a half-plane $\{\Re s> θ\}$, then it actually converges uniformly in every half-plane $\{\Re s> θ+\varepsilon\}$, $\varepsilon>0$. We prove, under very mild conditions, that given a sequence of Beurling primes, a small perturbation yields another sequence of primes such that the corresponding Beurling integers satisfy Bohr's condition, and therefore the theorem. Applying our technique in conjunction with a probabilistic method, we find a system of Beurling primes for which both Bohr's theorem and the Riemann hypothesis are valid. This provides a counterexample to a conjecture of H. Helson concerning outer functions in Hardy spaces of generalized Dirichlet series.

math.NT↗
arXiv2022

On zero-density estimates and the PNT in short intervals for Beurling generalized numbers

We study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as $N(x) = Ax + O(x^θ)$. We obtain in particular \[ N(α, T) \ll T^{\frac{c(1-α)}{1-θ}}\log^{9} T, \] for a constant $c$ arbitrarily close to $4$, improving significantly the current state of the art. We also investigate the consequences of the obtained zero-density estimates on the PNT in short intervals. Our proofs crucially rely on an extension of the classical mean-value theorem for Dirichlet polynomials to generalized Dirichlet polynomials.

math.NT↗
arXiv2022

The optimal Malliavin-type remainder for Beurling generalized integers

We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given $α\in (0,1]$ and $c>0$ (with $c\leq 1$ if $α=1$), a generalized number system is constructed with Riemann prime counting function $ Π(x)= \operatorname*{Li}(x)+ O(x\exp (-c \log^α x ) +\log_{2}x), $ and whose integer counting function satisfies the extremal oscillation estimate $N(x)=ρx + Ω_{\pm}(x\exp(- c'(\log x\log_{2} x)^{\fracα{α+1}})$ for any $c'>(c(α+1))^{\frac{1}{α+1}}$, where $ρ>0$ is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].

math.NT↗
arXiv2022

Distributed-order time-fractional wave equations

Distributed-order time-fractional wave equations appear in the modeling of wave propagation in viscoelastic media. The material characteristics of the medium are modeled through constitutive functions or distributions in the distributed-order constitutive law. In this work we propose to take positive Radon measures for the constitutive "functions". First, we derive a thermodynamical restriction on the constitutive measures which is easy to check, and therefore suitable for applications. Then we prove that the setting with measures in combination with the derived thermodynamical restriction guarantee existence and uniqueness of solutions for the distributed-order fractional wave equation. We further discuss the support and regularity of the fundamental solution, and conclude with a discussion on wave velocities.

math.AP↗
arXiv2020

Note on a conjecture of Bateman and Diamond concerning the abstract PNT with Malliavin-type remainder

Given $β\in(0,1)$, we show the existence of a Beurling generalized number system whose integer counting satisfies $N(x) = ax + O\bigl(x\exp(-c\log^β x)\bigr)$ for some $a>0$ and $c>0$, and whose prime counting function satisfies $π(x) = \mathrm{Li}(x) + Ω\bigl(x\exp(-c'(\log x)^{\fracβ{β+1}})\bigr)$ for some $c'>0$. This is done by generalizing a construction of Diamond, Montgomery, and Vorhauer. This Beurling system serves as additional motivation for a conjecture of Bateman and Diamond from 1969, concerning the PNT with Malliavin-type remainder.

math.NT↗
arXiv2020

An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions

We study the family of Fourier-Laplace transforms $$ F_{α,β}(z)= \operatorname*{F.p.} \int_{0}^{\infty} t^β\exp(\mathrm{i} t^α-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{Im} z<0, $$ for $α>1$ and $β\in\mathbb{C}$, where Hadamard finite part is used to regularize the integral when $\operatorname*{Re} β\leq -1$. We prove that each $F_{α,β}$ has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin. We also apply our ideas to show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a quantified version of the Ingham-Karamata theorem, supplying new simple and constructive proofs of optimality results for these complex Tauberian theorems.

math.CV↗
arXiv2020

Beurling integers with RH and large oscillation

We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies $π(x)= \operatorname*{Li} (x)+ O(\sqrt{x})$, while its integer counting function satisfies the oscillation estimate $N(x) = ρx + Ω_{\pm}\bigl(x\exp(-c\sqrt{\log x\log\log x})\bigr)$ for some $c>0$, where $ρ>0$ is its asymptotic density. The construction is inspired by a classical example of H. Bohr for optimality of the convexity bound for Dirichlet series, and combines saddle-point analysis with the Diamond-Montgomery-Vorhauer probabilistic method via random prime number system approximations.

math.NT↗