In this paper, we apply the saddle-point method in conjunction with the theory of the Nörlund–Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function $F$ in the Selberg class $\mathcal{S}$ and under the Generalized Riemann Hypothesis, we have

$${{\lambda }_{F}}(n)\,=\,\frac{{{d}_{F}}}{2}n\,\log \,n\,+\,{{c}_{F}}n\,+\,O(\sqrt{n}\,\log \,n),$$

with

$${{c}_{F}}\,=\,\frac{{{d}_{F}}}{2}(\gamma \,-\,1)\,+\,\frac{1}{2}\log (\lambda \text{Q}_{F}^{2}),\,\,\lambda \,=\,\prod\limits_{j=1}^{r}{\lambda _{j}^{2{{\lambda }_{j}}}},$$

where $\gamma $ is the Euler's constant and the notation is as below.

We prove that if the Fourier transform of a compactly supported measure is in ${{L}^{2}}$ of a half-space, then the measure is absolutely continuous to Lebesgue measure. We then show how this result can be used to translate information about the dimensionality of a measure and the decay of its Fourier transform into geometric information about its support.

Writing $s\,=\,\sigma \,+\,it$ for a complex variable, it is proved that the modulus of the gamma function, $\left| \Gamma (s) \right|$, is strictly monotone increasing with respect to $\sigma $ whenever $\left| t \right|\,>\,5/4$. It is also shown that this result is false for $\left| t \right|\,\le \,1$.

Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number $\alpha $ by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure.

We obtain new results about the number of trinomials ${{t}^{n}}\,+\,at\,+\,b$ with integer coefficients in a box $(a,\,b)\,\in \,[C,\,C\,+\,A]\,\times \,[D,\,D\,+\,B]$ that are irreducible modulo a prime $p$. As a by-product we show that for any $p$ there are irreducible polynomials of height at most ${{p}^{1/2+o(1)}}$, improving on the previous estimate of ${{p}^{2/3+o(1)}}$ obtained by the author in 1989.

An important conjecture in the theory of Borel sets in non-separable metric spaces is whether any point-countable Borel-additive family in a complete metric space has a $\sigma $-discrete refinement. We confirm the conjecture for point-countable $\Pi _{3}^{0}$-additive families, thus generalizing results of R. W. Hansell and the first author. We apply this result to the existence of Borel measurable selectors for multivalued mappings of low Borel complexity, thus answering in the affirmative a particular version of a question of J. Kaniewski and R. Pol.

Soient ${{p}_{1}},\,{{p}_{2}},\,{{p}_{3}}$ et $q$ des nombres premiers distincts tels que ${{p}_{1}}\,\equiv \,{{p}_{2}}\,\equiv \,{{p}_{3}}\,\equiv \,-q\,\equiv \,1\,(\bmod \,4)$, $k=\mathbf{Q}(\sqrt{{{p}_{1}}},\sqrt{{{p}_{2}}},\sqrt{{{p}_{3}}},\sqrt{q})$ et $\text{C}{{\text{l}}_{2}}(k)$ le 2-groupe de classes de $k$. A. Fröhlich a démontré que $\text{C}{{\text{l}}_{2}}(k)$ n’est jamais trivial. Dans cet article, nous donnons une extension de ce résultat, en démontrant que le rang de $\text{C}{{\text{l}}_{2}}(k)$ est toujours supérieur ou égal à 2. Nous démontrons aussi, que la valeur 2 est optimale pour une famille infinie de corps $k$.

In this paper we characterize the positive definite measures with discrete Fourier transform. As an application we provide a characterization of pure point diffraction in locally compact Abelian groups.

We study Szegö's theorem for a uniform algebra. In particular, we do it for the disc algebra or the bidisc algebra.

Let $A({{n}_{1}},\,{{n}_{2}},\ldots ,\,{{n}_{m}}-1)$ be the normalized Fourier coefficients of a Maass cusp form on $\text{GM(}m\text{)}$. In this paper, we study the cancellation of $A({{n}_{1}},\,{{n}_{2}},\ldots ,\,{{n}_{m}}-1)$ over Beatty sequences.

We show that there is an infinite family of hyperbolic knots such that each knot admits a cyclic surgery $m$ whose adjacent surgeries $m\,-\,1$ and $m\,+\,1$ are toroidal. This gives an affirmative answer to a question asked by Boyer and Zhang.

In this note we give a brief review of the construction of a toric variety $\mathcal{V}$ coming from a genus $g\,\ge \,2$ Riemann surface ${{\sum }^{g}}$ equipped with a trinion, or pair of pants, decomposition. This was outlined by J. Hurtubise and L. C. Jeffrey. A. Tyurin used this construction on a certain collection of trinion decomposed surfaces to produce a variety $D{{M}_{g,}}$ the so-called Delzant model of moduli space, for each genus $g$. We conclude this note with some basic facts about the moment polytopes of the varieties $\mathcal{V}$. In particular, we show that the varieties $D{{M}_{g}}$ constructed by Tyurin, and claimed to be smooth, are in fact singular for $g\,\ge \,3$.

We shall present examples of Schauder bases in the preduals to the hyperfinite factors of types $\text{I}{{\text{I}}_{1}},\,\text{I}{{\text{I}}_{\infty }},\,\text{II}{{\text{I}}_{\lambda }},\,0\,<\,\lambda \,\le \,1$. In the semifinite (respectively, purely infinite) setting, these systems form Schauder bases in any associated separable symmetric space of measurable operators (respectively, in any non-commutative ${{L}^{p}}$-space).

We consider approximation of multivariate functions in Sobolev spaces by high order Parzen windows in a non-uniform sampling setting. Sampling points are neither i.i.d. nor regular, but are noised from regular grids by non-uniform shifts of a probability density function. Sample function values at sampling points are drawn according to probability measures with expected values being values of the approximated function. The approximation orders are estimated by means of regularity of the approximated function, the density function, and the order of the Parzen windows, under suitable choices of the scaling parameter.

Two theorems regarding the asymptotic behavior of evolution families are established in terms of the solutions of a certain Lyapunov operator equation.

This note considers the problem of approximating continuous maps from sets in complex spaces into complex manifolds by holomorphic maps.

Klep and Velušček generalized the Krull–Baer theorem for higher level preorderings to the non-commutative setting. A $n$-real valuation $v$ on a skew field $D$ induces a group homomorphism $\overline{v}$. A section of $\overline{v}$ is a crucial ingredient of the construction of a complete preordering on the base field $D$ such that its projection on the residue skew field ${{k}_{v}}$ equals the given level 1 ordering on ${{k}_{v}}$. In the article we give a proof of the existence of the section of $\overline{v}$, which was left as an open problem by Klep and Velušček, and thus complete the generalization of the Krull–Baer theorem for preorderings.