arXiv subjects
Publications and source records attributed to Gregory Debruyne.
We provide a general quantified Ingham-Karamata Tauberian theorem with a flexible one-sided Tauberian condition under several types of boundary behavior for the Laplace transform. Our results in particular improve a theorem by Stahn, removing a vexing restriction on the growth of the Laplace transform. Improving existing optimality results, we also show that the obtained quantified rate is optimal in almost all cases.
In this expository article we provide an elegant proof of the one-sided Ingham-Karamata Tauberian theorem. As an application, we present a short deduction of the prime number theorem.
We study the range of validity of the density hypothesis for the zeros of $L$-functions associated with cusp Hecke eigenforms $f$ of even integral weight and prove that $N_{f}(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon}$ holds for $\sigma \geq 1407/1601$. This improves upon a result of Ivi\'{c}, who had previously shown the zero-density estimate in the narrower range $\sigma\geq 53/60$. Our result relies on an improvement of the large value estimates for Dirichlet polynomials based on mixed moment estimates for the Riemann zeta function. The main ingredients in our proof are the Hal\'{a}sz-Montgomery inequality, Ivi\'{c}'s mixed moment bounds for the zeta function, Huxley's subdivision argument, Bourgain's dichotomy approach, and Heath-Brown's bound for double zeta sums.
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 $[\alpha,\beta]$-systems, where $\alpha$ and $\beta$ are connected to the optimal power saving in the prime and integer-counting functions. It is known that every $[\alpha,\beta]$-system satisfies $\max\{\alpha,\beta\}\ge1/2$. In this paper we show there are $[\alpha,\beta]$-systems for each $\alpha \in [0,1)$ and $\beta \in [1/2, 1)$. Assuming the Riemann hypothesis, we also construct certain families of $[\alpha,\beta]$-systems with $\beta<1/2$.
We study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as $N(x) = Ax + O(x^{\theta})$. We obtain in particular \[ N(\alpha, T) \ll T^{\frac{c(1-\alpha)}{1-\theta}}\log^{9} T, \] 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.
We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given $\alpha\in (0,1]$ and $c>0$ (with $c\leq 1$ if $\alpha=1$), a generalized number system is constructed with Riemann prime counting function $ \Pi(x)= \operatorname*{Li}(x)+ O(x\exp (-c \log^{\alpha} x ) +\log_{2}x), $ and whose integer counting function satisfies the extremal oscillation estimate $N(x)=\rho x + \Omega_{\pm}(x\exp(- c'(\log x\log_{2} x)^{\frac{\alpha}{\alpha+1}})$ for any $c'>(c(\alpha+1))^{\frac{1}{\alpha+1}}$, where $\rho>0$ is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].
We study the family of Fourier-Laplace transforms $$ F_{\alpha,\beta}(z)= \operatorname*{F.p.} \int_{0}^{\infty} t^{\beta}\exp(\mathrm{i} t^{\alpha}-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{Im} z<0, $$ for $\alpha>1$ and $\beta\in\mathbb{C}$, where Hadamard finite part is used to regularize the integral when $\operatorname*{Re} \beta\leq -1$. We prove that each $F_{\alpha,\beta}$ 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.
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 $\pi(x)= \operatorname*{Li} (x)+ O(\sqrt{x})$, while its integer counting function satisfies the oscillation estimate $N(x) = \rho x + \Omega_{\pm}\bigl(x\exp(-c\sqrt{\log x\log\log x})\bigr)$ for some $c>0$, where $\rho>0$ 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.
We apply the saddle-point method to derive asymptotic estimates or asymptotic series for the number of partitions of a natural integer into parts chosen from a subset of the positive integers whose associated Dirichlet series satisfies certain analytic properties. This enables grouping in a single statement many cases studied in the literature, as well as a number of new ones.
We find asymptotics for $S_{K,c}(x)$, the number of positive integers below $x$ whose number of prime factors is $c \; \mathrm{mod}\; K$. We study this question in the context of Beurling integers.
We construct explicit counterexamples that show that it is impossible to get any remainder, other than the classical ones, in the Wiener-Ikehara theorem and the Ingham-Karamata theorem under just an additional analytic continuation hypothesis to a half-plane (or even to the whole complex plane).
We prove that a general version of the quantified Ingham-Karamata theorem for $C_0$-semigroups is sharp under mild conditions on the resolvent growth, thus generalising the results contained in a recent paper by the same authors. It follows in particular that the well-known Batty-Duyckaerts theorem is optimal even for bounded $C_0$-semigroups whose generator has subpolynomial resolvent growth. Our proof is based on an elegant application of the open mapping theorem, which we complement by a crucial technical lemma allowing us to strengthen our earlier results.
We provide a new and significantly shorter optimality proof of recent quantified Tauberian theorems, both in the setting of vector-valued functions and of $C_0$-semigroups, and in fact our results are also more general than those currently available in the literature. Our approach relies on a novel application of the open mapping theorem.
We show that Hal\'{a}sz's theorem holds for Beurling numbers under the following two mild hypotheses on the generalized number system: existence of a positive density for the generalized integers and a Chebyshev upper bound for the generalized primes.
We give a short proof of the $L^{1}$ criterion for Beurling generalized integers to have a positive asymptotic density. We actually prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate $m(x)=\sum_{n_{k}\leq x} μ(n_k)/n_k=o(1)$, with $μ$ the Beurling analog of the Moebius function.
We show that it is impossible to get a better remainder than the classical one in the Wiener-Ikehara theorem even if one assumes analytic continuation of the Mellin transform after subtraction of the pole to a half-plane. We also prove a similar result for the Ingham-Karamata theorem.
It is well known that there is an absolute constant $\mathfrak{C}>0$ such that if the Laplace transform $G(s)=\int_{0}^{\infty}ρ(x)e^{-s x}\:\mathrm{d}x$ of a bounded function $ρ$ has analytic continuation through every point of the segment $(-iλ,iλ)$ of the imaginary axis, then $$ \limsup_{x\to\infty} \left|\int_{0}^{x}ρ(u)\:\mathrm{d}u - G(0)\right|\leq \frac{ \mathfrak{C}}λ \: \limsup_{x\to\infty} |ρ(x)|. $$ The best known value of the constant $\mathfrak{C}$ was so far $\mathfrak{C}=2$. In this article we show that the inequality holds with $\mathfrak{C}=π/2$ and that this value is best possible. We also sharpen Tauberian constants in finite forms of other related complex Tauberian theorems for Laplace transforms.
We show that for Beurling generalized numbers the prime number theorem in remainder form $$π(x) = \operatorname*{Li}(x) + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n\in\mathbb{N}$$ is equivalent to (for some $a>0$) $$N(x) = ax + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n \in \mathbb{N},$$ where $N$ and $π$ are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299-307), but his article on the subject contains some mistakes. We also obtain an average version of this prime number theorem with remainders in the Cesàro sense.