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On the counts of p-rough numbers

Fred B. Holt

Abstract

The p-rough numbers are those numbers all of whose prime factors are greater than p. These are exactly those numbers left after Eratosthenes sieve has been advanced from 2 through the prime p. Here we show that for fixed p there is a line of symmetry for the function $Φ(x,p)$, and we introduce the function $ΔΦ(x,p)$ which is the difference between $Φ(x,p)$ and the line of symmetry. $ΔΦ(x,p)$ is periodic and bounded and has a rotational symmetry.

On the counts of p-rough numbers

Abstract

The p-rough numbers are those numbers all of whose prime factors are greater than p. These are exactly those numbers left after Eratosthenes sieve has been advanced from 2 through the prime p. Here we show that for fixed p there is a line of symmetry for the function , and we introduce the function which is the difference between and the line of symmetry. is periodic and bounded and has a rotational symmetry.
Paper Structure (5 sections, 4 theorems, 43 equations, 10 figures, 2 tables)

This paper contains 5 sections, 4 theorems, 43 equations, 10 figures, 2 tables.

Table of Contents

  1. Setting
  2. The function $\Delta \Phi(x,p) = \Phi(x,p) - \frac{\phi({p}^{\#})}{{p}^{\#}} \cdot x$.
  3. Bounds on $\Delta \Phi(x,p)$
  4. Asymptotics in $p$.
  5. Conclusion

Key Result

Lemma 2.1

Waypoints at ends of cycles: There are waypoints at the end of each cycle ${x_k = k \cdot {p}^{\#}}$, for which

Figures (10)

  • Figure 1: ${y=\Phi(x,5)}$ and ${y = \frac{\phi({5}^{\#})}{{5}^{\#}} x }$ are shown over two periods. The cycle ${\mathcal{G}}({5}^{\#})$ has length $8$ and span $30$. We mark the waypoints at the ends of the periods and at the midpoints, and the periods of span $30$ are delimited, to highlight the translational and rotational symmetries of $\Phi(x,5)$.
  • Figure 2: The equation ${\Delta \Phi(x,p) = \Phi(x,p) - \frac{1}{\mu} x }$ for ${p=3}$. The cycle ${\mathcal{G}}({3}^{\#})$ has length $2$ and span $6$. The waypoints are highlighted and the periods of span $6$ are delimited, to highlight the symmetries. Note the translational symmetry across periods of the cycle, and the rotational symmetry around the waypoints $x_k$ and $\tilde{x}_k$.
  • Figure 3: The equation ${\Delta \Phi(x,5) = \Phi(x,5) - \frac{1}{\mu} x }$ is shown over three periods. The cycle ${\mathcal{G}}({5}^{\#})$ has length $8$ and span $30$. The waypoints are highlighted and the periods of span $30$ are delimited, to highlight the symmetries.
  • Figure 4: The equation ${\Delta \Phi(x,5) = \Phi(x,5) - \frac{\phi({5}^{\#})}{{5}^{\#}} x }$ is shown over one period. We mark the downward segments with the corresponding gap $g$ in the cycle ${\mathcal{G}}({5}^{\#}) = 6 4 2 4 2 4 6 2$.
  • Figure 5: The equation ${\Delta \Phi(x,7) = \Phi(x,7) - \frac{\phi({7}^{\#})}{{7}^{\#}} x }$ is shown over two periods. The cycle ${\mathcal{G}}({7}^{\#})$ has length $48$ and span $210$. The waypoints are highlighted, and the cycles of span $210$ are delimited, to highlight the symmetries.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 3.1
  • proof