The influence of the first term of an arithmetic progression

TL;DR

This paper investigates how the initial term of an arithmetic progression affects the distribution of a class of sequences within arithmetic progressions, revealing significant influence even after excluding the first term.

Contribution

It demonstrates that the first term of an arithmetic progression has a notable impact on the distribution of certain sequences, a previously underexplored aspect.

Findings

The first term significantly influences distribution in arithmetic progressions.

The effect persists even after removing the initial term.

Results apply to sequences of arithmetic interest.

Abstract

The goal of this article is to study the discrepancy of the distribution of arithmetic sequences in arithmetic progressions. We will fix a sequence $\A=\{\a(n)\}_{n\geq 1}$ of non-negative real numbers in a certain class of arithmetic sequences. For a fixed integer $a\neq 0$, we will be interested in the behaviour of $\A$ over the arithmetic progressions $a \bmod q$, on average over $q$. Our main result is that for certain sequences of arithmetic interest, the value of $a$ has a significant influence on this distribution, even after removing the first term of the progressions.