Vieira de Vargas, Claudia Hopner Noguti, Fabiane Cristina

The following article is a piece of a Master’s research, in which investigates the teaching and learning of arithmetic progression in a learning environment through problem solving. The participants were students from the first year of high school at a private school in Santa Maria / RS. The data collect was developed in eight meetings and used the...

Löbrich, Steffen
Published in
Research in Number Theory

For a given generalized eta-quotient, we show that linear progressions whose residues fulfill certain quadratic equations do not give rise to a linear congruence modulo any prime. This recovers known results for classical eta-quotients, especially the partition function, but also yields linear incongruences for more general weakly holomorphic modul...

Chan, Vincent Łaba, Izabella Pramanik, Malabika
Published in
Journal d'Analyse Mathématique

Let E ⊆ ℝn be a closed set of Hausdorff dimension α. For m > n, let{B1, …, Bk} be n × (m − n) matrices. We prove that if the system of matrices Bj is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m dependi...

Gaze, Eric Gaze, Joseph
Published in
The Mathematical Intelligencer

Lester, Stephen Yesha, Nadav
Published in
Israel Journal of Mathematics

We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied by É. Fouvry, S. Ganguly, E. Kowalski and Ph. Michel. We prove a complementary result to their main theorem. We also show that in short intervals of certain lengths the divisor function has a Gaussian l...

Shteinikov, Yu. N.
Published in
Proceedings of the Steklov Institute of Mathematics

We estimate how many numbers in a given interval have the property that their residues modulo two different fixed numbers belong to two given sets. The estimates obtained are order sharp.

Conlon, David Fox, Jacob Zhao, Yufei
Published in
Geometric and Functional Analysis

The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple pro...

Tao, Terence Ziegler, Tamar
Published in
Israel Journal of Mathematics

We establish a version of the Furstenberg-Katznelson multi-dimensional Szemerédi theorem in the primes P:= {2, 3, 5, …}, which roughly speaking asserts that any dense subset of Pd contains finite constellations of any given rational shape. Our arguments are based on a weighted version of the Furstenberg correspondence principle, relative to a weigh...

Elliott, P.D.T.A.
Published in
Lithuanian Mathematical Journal

Fomenko, O. M.
Published in
Journal of Mathematical Sciences

Let P(x) and P3(x) be the error terms in the Gaussian circle problem and the sphere problem, respectively. We investigate the asymptotic behavior of the sums ∑k≤0k≡0modpPk,∑k≤xk≡0modpP3k.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepa...