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## A Density Theorem for Dedekind Zeta Functions

Mathematica Pannonica
Author:
János Pintz

We apply a recent general zero density theorem of us (valid for a large class of complex functions) to improve earlier density theorems of Heath-Brown and Paul–Sankaranarayanan for Dedekind zeta functions attached to a number field 𝐾 of degree 𝑛 with 𝑛 > 2.

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## Kahane’s Upper Density and Syndetic Sets in LCA Groups

Mathematica Pannonica
Author:
Szilárd Gy. Révész

Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.

Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group 𝐺, a set 𝑆 ⊂ 𝐺 is syndetic, if there exists a compact subset 𝐶 ⋐ 𝐺 such that 𝑆𝐶 = 𝐺. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset 𝐴 ⊂ ℤ has positive a.u.u.d., then its difference set 𝐴 − 𝐴 is syndetic.

The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000’s several constructions were worked out to generalize it from the base cases of ℤ𝑑 and ℝ𝑑. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.

Here we work out various versions in a general locally compact Abelian group 𝐺 of the classical statement that if a set 𝑆 ⊂ 𝐺 has positive asymptotic uniform upper density, then the difference set 𝑆 − 𝑆 is syndetic.

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## Remark on a General Zero Density Theorem

Mathematica Pannonica
Author:
János Pintz

Recently [3] we proved a general zero density theorem for a large class of functions which included among others the Riemann zeta function, Dedekind zeta functions, Dirichlet 𝐿-functions. The goal of the present work is a (slight) improvement of this general theorem which might lead to stronger results in some applications.

Open access

## How to Approach Stability of Bi-Continuous Semigroups?

Mathematica Pannonica
Author:
Christian Budde

This paper serves as a kick-off to address the question of how to define and investigate the stability of bi-continuous semigroups. We will see that the mixed topology is the key concept in this framework.

Open access

## A Classification of 2-Dimensional Endo-Commutative Straight Algebras of Rank 1 over a non-Trivial Field

Mathematica Pannonica
Authors:
Sin-Ei Takahasi
,
Kiyoshi Shirayanagi
, and

An endo-commutative algebra is a nonassociative algebra in which the square mapping preserves multiplication. In this paper, we give a complete classification of 2-dimensional endo-commutative straight algebras of rank one over an arbitrary non-trivial field, where a straight algebra of dimension 2 satisfies the condition that there exists an element x such that x and x 2 are linearly independent. We list all multiplication tables of the algebras up to isomorphism.

Open access

## On the Simultaneous Sign Changes of Coefficients of Rankin–Selberg L-Functions over a Certain Integral Binary Quadratic Form

Mathematica Pannonica
Author:
Guodong Hua

In this paper, we consider the simultaneous sign changes of coefficients of Rankin–Selberg L-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant D. We provide a quantitative result for the number of sign changes of such sequence in the interval (x, 2x] for sufficiently large x.

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## Divisor Problem for the Greatest Common Divisor of Integers in Piatetski-Shapiro and Beatty Sequences

Mathematica Pannonica
Authors:
Sunanta Srisopha
,
Teerapat Srichan
, and
Pinthira Tangsupphathawat

In this paper, we derive several asymptotic formulas for the sum of d(gcd(m,n)), where d(n) is the divisor function and m,n are in Piatetski-Shapiro and Beatty sequences.

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## The Norming Sets of $L$ (2 d *(1, w)2)

Mathematica Pannonica
Author:
Sung Guen Kim

Let 𝑛 ∈ ℕ. An element (x 1, … , x 𝑛) ∈ En is called a norming point of T $L$ ( nE) if ‖x 1‖ = ⋯ = ‖xn ‖ = 1 and |T (x 1, … , xn )| = ‖T‖, where $L$ ( nE) denotes the space of all continuous n-linear forms on E. For T $L$ ( nE), we define

Norm(T) = {(x 1, … , x n) ∈ En ∶ (x 1, … , x n) is a norming point of T}.

Norm(T) is called the norming set of T. We classify Norm(T) for every T $L$ (2 𝑑 (1, w)2), where 𝑑 (1, w)2 = ℝ2 with the octagonal norm of weight 0 < w < 1 endowed with $x , y d * 1 , w = max x , y , x + y 1 + w$ .

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## On k-Strictly Quasi-Fredholm Linear Relations

Mathematica Pannonica
Authors:
Hafsa Bouaniza
,
Imen Issaoui
, and
Maher Mnif

In this paper, we introduce and study the class of k-strictly quasi-Fredholm linear relations on Banach spaces for nonnegative integer k. Then we investigate its robustness through perturbation by finite rank operators.

Open access

## A Continuum Dimensional Algebra of Nowhere Differentiable Functions

Mathematica Pannonica
Author:
Jan-Christoph Schlage-Puchta

We construct an algebra of dimension 2ℵ0 consisting only of functions which in no point possess a finite one-sided derivative. We further show that some well known nowhere differentiable functions generate algebras, which contain functions which are differentiable at some points, but where for all functions in the algebra the set of points of differentiability is quite small.

Open access