Abstract In a previous paper, we proved Carlson‐type density theorem for zeroes in the critical strip Beurling zeta functions satisfying Axiom A of Knopfmacher. There needed to invoke two additional conditions: integrality norm (Condition B) and an “average Ramanujan condition” arithmetical function counting number different integers same G). Here, implement new approach Pintz using classic zero‐detecting sums coupled with Halász' method, but otherwise arguing elementary way avoiding,...

If $\Delta$ stands for the region enclosed by triangle in ${\mathsf R}^2$ of vertices $(0,0)$, $(0,1)$ and $(1,0)$ (or simplex short), we consider space ${\mathcal P}(^2\Delta)$ 2-homogeneous polynomials on endowed with norm given $\|ax^2+bxy+cy^2\|_\Delta:=\sup\{|ax^2+bxy+cy^2|:(x,y)\in\Delta\}$ every $a,b,c\in{\mathsf R}$. We investigate some geometrical properties this norm. provide an explicit formula $\|\cdot\|_\Delta$, a full description extreme points corresponding unit ball...

Abstract For a fixed positive integer n consider continuous functions $$K_1,\dots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> </mml:mrow> </mml:math> , K_n:[-1,1]\rightarrow {{\mathbb {R}}}\cup \{-\infty \}$$ <mml:mi>n</mml:mi> <mml:mo>:</mml:mo> <mml:mo>[</mml:mo> <mml:mo>-</mml:mo> <mml:mo>]</mml:mo> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> <mml:mo>∪</mml:mo>...

We study the following question posed by Turán. Suppose <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a convex body in Euclidean space alttext="double-struck R Superscript d"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD">...

We study the following question: given an open set Ω, symmetric about 0, and a continuous, integrable, positive definite function f, supported in Ω with f(0) = 1, how large can ∫ f be? This problem has been studied so far mostly for convex domains Euclidean space. In this paper we question arbitrary locally compact abelian groups more general domains. Our emphasis is on finite as well spaces ℤd. exhibit upper bounds assuming geometric properties of two types: (a) packing (b) spectral Ω....

We prove three results on the density resp. local and clustering of zeros Beurling zeta function $\zeta(s)$ close to one-line $\sigma:=\Re s=1$. The analysis here brings about some news, sometimes even for classical case Riemann function. Theorem 4 provides a zero estimate, which is complement known Selberg class. Note that class rely use functional equation $\zeta$, we do not assume in context. In 5 deduce variant well-known theorem Tur\'an, extending its range validity rectangles height...

We prove two results, generalizing long existing knowledge regarding the classical case of Riemann zeta function and some its generalizations. These are concerned with question Ingham, who asked for optimal explicit order estimates error term Δ(x):=ψ(x)−x, given any zero-free region D(η):={s=σ+it∈C:σ:=ℜs≥1−η(t)}. In essentially sharp results due to 40 years old work Pintz. Here we consider a system Beurling primes P, generated arithmetical semigroup G, corresponding integer counting N(x),...

We investigate the existence of well-behaved Beurling number systems, which are systems generalized primes and integers admit a power saving in error term both their prime integer-counting function. Concretely, we search for so-called <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket alpha comma beta right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo>...

The classical Bernstein pointwise estimate of the (first) derivative a univariate algebraic polynomial on an interval has natural extensions to multivariate setting. However, in several variables domain boundedness, even if convex, considerable geometric variety. In 1990, Y. Sarantopoulos satisfactorily settled case centrally symmetric convex body by method we may call "the inscribed ellipses." On other hand, for general nonsymmetric bodies are only within constant factor exact inequality....

We extend some equilibrium-type results first conjectured by Ambrus, Ball and Erdélyi, then proved recenly Hardin, Kendall Saff. work on the torus T ≃ [ 0 , 2 π ) but motivation comes from an analogous setup unit interval, investigated earlier Fenton. The problem is to minimize — with respect arbitrary translates y = j ∈ 1 ⋯ n maximum of sum function F : K + ∑ ( · − where functions are certain fixed 'kernel functions'. In our setting, has singularities at while in between these nodes it...

We consider the classical Wiener–Ikehara Tauberian theorem, with a generalized condition of slow decrease and some additional poles on boundary convergence Laplace transform. In this generality, we prove otherwise known asymptotic evaluation transformed function, when usual conditions theorem hold. However, our version also provides an effective error term, not thus far in generality. The crux proof is proper, variation lemmas Ganelius Tenenbaum, constructed for sake theorem.

We say that Wiener's property holds for the exponent p>0 whenever a positive definite function f, which belongs to Lp(−ε,ε) some ε>0, necessarily Lp(T), too. This true p∈2N by classical result of Wiener. Recently various concentration results were proved idempotents and functions on measurable sets torus. They enable us prove sharp version failure p∉2N, strengthening Wainger Shapiro. To cite this article: A. Bonami, S.Gy. Révész, C. R. Acad. Sci. Paris, Ser. I 346 (2008). On dit que...

We prove that for all p&gt;1/2 there exists a constant $γ_p&gt;0$ such that, any symmetric measurable set of positive measure $E\subset \TT$ and $γ γ\int_{\TT} |f|^p$. This disproves conjecture Anderson, Ash, Jones, Rider Saffari, who proved the existence p&gt;1 conjectured it does not p=1. Furthermore, we one can take $γ_p=1$ when is an even integer, polynomials f be chosen with arbitrarily large gaps $p\neq 2$. shows striking differences case p=2, which best strictly smaller than 1/2, as...