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Linear truncation for conditioned prime-factor fibres

Johann Verwee

Abstract

In previous joint work with Tenenbaum, the truncation step $f \mapsto f_R$ in the conditional effective Erdos-Wintner theorem on the fibre $ω(n)=k$ yields, in the continuous case for real strongly additive $f$, a remainder of size $η_f(R)^{r/(r+1)}$, where $R$ is the truncation level and $r=k/\log\log x$. We prove an effective linear truncation lemma showing that, in the central window $κ\le r \le 1/κ$, this bound improves to the natural linear scale $rη_f(R)$ under an effective Sathe-Selberg-type ratio estimate for the fibre. This yields a direct effective sharpening of the truncation step in the previous joint work. The same truncation upgrade also applies to prime-set restrictions, $Ω$-fibres, and weighted fibres whenever the corresponding ratio estimate is available.

Linear truncation for conditioned prime-factor fibres

Abstract

In previous joint work with Tenenbaum, the truncation step in the conditional effective Erdos-Wintner theorem on the fibre yields, in the continuous case for real strongly additive , a remainder of size , where is the truncation level and . We prove an effective linear truncation lemma showing that, in the central window , this bound improves to the natural linear scale under an effective Sathe-Selberg-type ratio estimate for the fibre. This yields a direct effective sharpening of the truncation step in the previous joint work. The same truncation upgrade also applies to prime-set restrictions, -fibres, and weighted fibres whenever the corresponding ratio estimate is available.
Paper Structure (9 sections, 6 theorems, 76 equations)

This paper contains 9 sections, 6 theorems, 76 equations.

Table of Contents

  1. A linear truncation lemma
  2. Truncation tails
  3. A ratio hypothesis on a fibre
  4. Linear truncation
  5. Application to the function $\omega$
  6. Further conditioned fibres
  7. Prime factors in a prime set
  8. Conditioning on Omega(n)
  9. Weighted fibres (abstract form)

Key Result

Theorem 1.1

Assume that $(\mathcal{F}(x;k))$ is an admissible fibre and hyp:ratio holds. Then there exists a constant $C_\kappa>0$ such that, uniformly for $x\geqslant 3$, $k\geqslant 1$ with eq:window, and all $R\in[3,x]$,

Theorems & Definitions (14)

  • Definition 1
  • Theorem 1.1
  • proof
  • Corollary 1
  • Corollary 2
  • Example 1
  • Proposition 1
  • proof
  • Corollary 3
  • proof
  • ...and 4 more