Quasialgebraicity of Picard--Vessiot fields

We prove that under certain spectral assumptions on the monodromy group, solutions of Fuchsian systems of linear equations on the Riemann sphere admit explicit global bounds on the number of their isolated zeros.Comment: 39 pages (AmSLaTeX/amsart). Second revision: typos corrected, one figure adde

On limit cycles appearing by polynomial perturbation of Darbouxian integrable systems

We prove an existential finiteness Varchenko-Khovanskii type result for integrals of rational 1-forms over the level curves of Darbouxian integrals.Comment: 21 pages, 2 figures, LaTe

Zeros of systems of exponential sums and trigonometric polynomials

Gelfond and Khovanskii found a formula for the sum of the values of a Laurent polynomial at the zeros of a system of n Laurent polynomials in the complex n-torus whose Newton polyhedra have generic mutual positions. An exponential change of variables gives a similar formula for exponential sums with rational frequencies. We conjecture that this formula holds for exponential sums with real frequencies. We give an integral formula which proves the existence-part of the conjectured formula not only in the complex situation but also in a very general real setting. We also prove the conjectured formula when it gives answer zero, which happens in most cases.Comment: 16 pages, 2 figure

Quantitative theory of ordinary differential equations and tangential Hilbert 16th problem

These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this study was its potential application to the tangential Hilbert 16th problem on zeros of complete Abelian integrals. The exposition consists mostly of examples illustrating various phenomena related to this problem. Sometimes these examples give an insight concerning the proofs, though the complete exposition of the latter is mostly relegated to separate expositions. For related and quoted articles, check the author's homepage http://www.wisdom.weizmann.ac.il/~yakov .Comment: Expanded lecture notes of the course delivered on the Workshop "Asymptotic series, differential algebra and finiteness theorems" (Montreal, June-July, 2000), 78 pages. Typos corrected, TeX style and references update

Determinacy for measures

We study the general moment problem for measures on the real line, with polynomials replaced by more general spaces of entire functions. As a particular case, we describe measures that are uniquely determined by a restriction of their Fourier transform to a finite interval. We apply our results to prove an extension of a theorem by Eremenko and Novikov on the frequency of oscillations of measures with a spectral gap (high-pass signals) near infinity

On the convergence of type I Hermite-Pad\'e approximants

Pad\'e approximation has two natural extensions to vector rational approximation through the so called type I and type II Hermite-Pad\'e approximants. The convergence properties of type II Hermite-Pad\'e approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite-Pad\'e approximants for Nikishin systems of functions.Comment: 20 page

Two inverse spectral problems for a class of singular Krein strings

We solve the inverse problem from the spectral measure and the inverse three-spectra problem for the class of singular Krein strings on a finite interval with trace class resolvents. In particular, this includes a complete description of all possible spectral measures and three (Dirichlet) spectra associated with this class of Krein strings. The solutions of these inverse problems are obtained by approximation with Stieltjes strings.Comment: 16 page

A New Proof of Newman's Conjecture and a Generalization

Newman's conjecture (proved by Rodgers and Tao in 2018) concerns a certain family of deformations $\{\xi_t(s)\}_{t \in \mathbb{R}}$ of the Riemann xi function for which there exists an associated constant $\Lambda \in \mathbb{R}$ (called the de Bruijn-Newman constant) such that all the zeros of $\xi_t$ lie on the critical line if and only if $t \geq \Lambda$. The Riemann hypothesis is equivalent to the statement that $\Lambda \leq 0$, and Newman's conjecture states that $\Lambda \geq 0$. In this paper we give a new proof of Newman's conjecture which avoids many of the complications in the proof of Rodgers and Tao. Unlike the previous best methods for bounding $\Lambda$, our approach does not require any information about the zeros of the zeta function, and it can be readily be applied to a wide variety of $L$-functions. In particular, we establish that any $L$-function in the extended Selberg class has an associated de Bruijn-Newman constant and that all of these constants are nonnegative. Stated in the Riemann xi function case, our argument proceeds by showing that for every $t < 0$ the function $\xi_t$ can be approximated in terms of a Dirichlet series $\zeta_t(s)=\sum_{n=1}^{\infty}\exp(\frac{t}{4} \log^2 n)n^{-s}$ whose zeros then provide infinitely many zeros of $\xi_t$ off the critical line.Comment: 29 pages, 4 figure

On the argument of LL-functions

For $L(\cdot,\pi)$ in a large class of $L$-functions, assuming the generalized Riemann hypothesis, we show an explicit bound for the function $S_1(t,\pi)=\frac{1}{\pi}\int_{1/2}^\infty\log|L(\sigma+it,\pi)|\,d\sigma$, expressed in terms of its analytic conductor. This enables us to give an alternative proof of the most recent (conditional) bound for $S(t,\pi)=\frac{1}{\pi} \,arg\,L(\tfrac12+it,\pi)$, which is the derivative of $S_1(\cdot,\pi)$ at $t$.Comment: 12 page

A method for proving the completeness of a list of zeros of certain L-functions

When it comes to partial numerical verification of the Riemann Hypothesis, one crucial part is to verify the completeness of a list of pre-computed zeros. Turing developed such a method, based on an explicit version of a theorem of Littlewood on the average of the argument of the Riemann zeta function. In a previous paper we suggested an alternative method based on the Weil-Barner explicit formula. This method asymptotically sacrifices fewer zeros in order to prove the completeness of a list of zeros with imaginary part in a given interval. In this paper, we prove a general version of this method for an extension of the Selberg class including Hecke and Artin L-series, L-functions of modular forms, and, at least in the unramified case, automorphic L-functions. As an example, we further specify this method for Hecke L-series and L-functions of elliptic curves over the rational numbers.Comment: final version, 19 pages. To appear in Mathematics of Computatio