Primes In Fractional Sequences

TL;DR

This paper proves that certain fractional sequences, including those related to primes and primes in arithmetic progressions, contain infinitely many primes, with improved error estimates and extensions to polynomial-generated sequences.

Contribution

The paper establishes the presence of infinitely many primes in fractional sequences and improves error bounds, extending results to polynomial-generated sequences.

Findings

Sequences contain infinitely many primes as x approaches infinity.

Primes in arithmetic progressions are included in these sequences.

Error terms for the distribution of primes in these sequences are improved.

Abstract

The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these sequences of fractional numbers contain the set of primes, and the set primes in arithmetic progressions as $x \to \infty$ respectively. Furthermore, the corresponding error terms for these sequences are improved. Other results considered are the fractional sequences of integers such as the sequence $\left \{[x/n]^2+1:n \leq x\right \}$ generated by the quadratic polynomial $n^2+1$, and the sequence $\left \{[x/n]^3+2:n \leq x\right \}$ generated by the cubic polynomial $n^3+2$. It is shown that each of these sequences of fractional numbers contains infinitely many primes as $x \to \infty$.