Linear truncation for conditioned prime-factor fibres

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 $η_f(R)<sup>{r/(r+1)}$, 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.

• The paper replaces the earlier truncation error η_f(R)^{r/(r+1)} with the natural effective bound C_κ rη_f(R) under admissibility and Sathe–Selberg-type ratio estimates.
• The elementary counting argument applies to ω-fibres and, given corresponding ratio inputs, to Ω-, prime-set-restricted, and weighted fibres within the central range κ≤r≤1/κ.
• For f(p)=(log p)^{-ξ}, the improved truncation term reaches order (log x)^{-ξ} instead of (log x)^{-ξr/(r+1)}, while full large-prime ratio estimates remain open for several extensions.

This paper sharpens a technical step in the conditional effective Erdős–Wintner theorem on prime-factor fibres. In prior joint work with Tenenbaum (Tenenbaum et al., 2020), the truncation $f\mapsto f_R$ of a real strongly additive function $f$ on the fibre $\omega(n)=k$ produced a remainder of order $\eta_f(R)^{r/(r+1)}$, where $r=k/\log x$. The present work proves that, throughout the central window $\kappa\le r\le 1/\kappa$, this can be replaced by the natural linear scale $r\,\eta_f(R)$, provided a Sathe–Selberg-type ratio estimate holds for the fibre. The argument is elementary and effective; its scope extends to prime-set restrictions, $\Omega$-fibres, and weighted fibres whenever the corresponding ratio input is available.

The abstract truncation lemma

The setting is as follows. Let $f$ be real strongly additive satisfying the Erdős–Wintner hypotheses (convergence of $\sum_p \min(1,f(p)^2)/p$ and of $f$0) together with the continuity condition $f$1. A tail function $f$2 dominates both $f$3 and $f$4. Truncation at level $f$5 zeroes out $f$6 only for primes $f$7 with $f$8, so the exceptional set is governed by

$f$9

The paper isolates two structural properties of a generic fibre $\omega(n)=k$0, called admissibility: stability under removal of one prime factor ($\omega(n)=k$1), and the support bound $\omega(n)=k$2. The sole analytic input is Hypothesis 1.4: uniformly over the central window and over $\omega(n)=k$3,

$\omega(n)=k$4

Under these assumptions, the main theorem states that for all $\omega(n)=k$5,

$\omega(n)=k$6

The proof is a direct count: if truncation alters $\omega(n)=k$7 then $\omega(n)=k$8 has a divisor $\omega(n)=k$9, and the support bound restricts the sum to $\eta_f(R)^{r/(r+1)}$0, where the ratio hypothesis applies termwise; summing $\eta_f(R)^{r/(r+1)}$1 over $\eta_f(R)^{r/(r+1)}$2 and invoking the $\eta_f(R)^{r/(r+1)}$3 bound gives the result with $\eta_f(R)^{r/(r+1)}$4. All constants are effective, so any effective Tenenbaum–Verwee argument consuming the truncation step only through the exceptional-set count inherits an effective improvement.

The heuristic justification is transparent: on a fibre with $\eta_f(R)^{r/(r+1)}$5, a proportion $\eta_f(R)^{r/(r+1)}$6 of integers should contain a prime of $\eta_f(R)^{r/(r+1)}$7, so $\eta_f(R)^{r/(r+1)}$8 is the natural scale — the earlier Hölder-type exponent $\eta_f(R)^{r/(r+1)}$9 was an artifact of coarse counting.

Application to $r=k/\log x$0-fibres and the effective Erdős–Wintner theorem

Specializing to $r=k/\log x$1, admissibility holds and the required ratio estimate follows from the uniform asymptotics and local quotient bounds of Hildebrand–Tenenbaum (Duke Math. J. 1988). This yields Corollary 2.1, the linear bound on $r=k/\log x$2-fibres, and consequently Corollary 2.3: under the parameter regime of (Tenenbaum et al., 2020) (with $r=k/\log x$3, $r=k/\log x$4, $r=k/\log x$5 constrained by $r=k/\log x$6 etc.), the conditional distribution of $r=k/\log x$7 on the fibre satisfies

$r=k/\log x$8

where $r=k/\log x$9 comprises the smoothing term $\kappa\le r\le 1/\kappa$0, the mean-value term $\kappa\le r\le 1/\kappa$1, and the new truncation term $\kappa\le r\le 1/\kappa$2. Equivalently, in the earlier theorem's remainder $\kappa\le r\le 1/\kappa$3, the factor $\kappa\le r\le 1/\kappa$4 may be replaced by $\kappa\le r\le 1/\kappa$5 under the same admissibility conditions.

A worked example quantifies the gain. For $\kappa\le r\le 1/\kappa$6 with $\kappa\le r\le 1/\kappa$7, choosing $\kappa\le r\le 1/\kappa$8, $\kappa\le r\le 1/\kappa$9, and $r\,\eta_f(R)$0 gives

$r\,\eta_f(R)$1

and for $r\,\eta_f(R)$2 simply $r\,\eta_f(R)$3. The truncation contribution drops from $r\,\eta_f(R)$4 to $r\,\eta_f(R)$5 — a gain of a factor $r\,\eta_f(R)$6, up to the bounded factor $r\,\eta_f(R)$7. The verification uses $r\,\eta_f(R)$8 and the Fourier decay $r\,\eta_f(R)$9 (with $\Omega$0) to obtain $\Omega$1 via Esseen-type smoothing.

Extensions to other conditioned fibres

Prime-set restrictions. For a prime set $\Omega$2 with harmonic density $\Omega$3, $\Omega$4, Proposition 3.3 establishes the two-sided local quotient bounds

$\Omega$5

uniformly for primes $\Omega$6, where $\Omega$7. The proof combines the quantitative Sathe–Selberg estimates of Tenenbaum (2017) with the observation that $\Omega$8 in this range, so the exponential and power corrections are both $\Omega$9. However, the paper is explicit that this covers only the range $f$0; a full linear truncation bound for $f$1 would require separate treatment of the complementary range $f$2 inside the exceptional-set count, which is not carried out here.

$f$3-fibres. For $f$4 counted with multiplicity, the needed ratio estimate (Hypothesis 3.5) is stated but treated as an external input rather than proved; given it, Corollary 3.6 delivers the same linear bound, since the support bound $f$5 again confines the sum to the range where the hypothesis applies.

Weighted fibres. For $f$6 with bounded nonnegative weights, Hypothesis 3.8 posits local ratio bounds against the neighbouring fibres at $f$7 and $f$8 relative to a scale $f$9 with $\sum_p \min(1,f(p)^2)/p$0. Corollary 3.11 then yields the linear bound $\sum_p \min(1,f(p)^2)/p$1; the proof splits each $\sum_p \min(1,f(p)^2)/p$2 contribution according to whether $\sum_p \min(1,f(p)^2)/p$3 after writing $\sum_p \min(1,f(p)^2)/p$4, using that $\sum_p \min(1,f(p)^2)/p$5 depends only on the set of prime divisors. Two examples probe the hypothesis: constant weights $\sum_p \min(1,f(p)^2)/p$6 on a density-$\sum_p \min(1,f(p)^2)/p$7 prime set reduce to the $\sum_p \min(1,f(p)^2)/p$8 situation and recover the ratio pattern locally (but not the full hypothesis, again because large primes are untreated); and the finite perturbation $\sum_p \min(1,f(p)^2)/p$9, $f$00 otherwise reduces to standard $f$01-fibres up to the prime 2, strongly suggesting verifiability from Hildebrand–Tenenbaum bounds after separate treatment of $f$02 — again left open.

Limitations and open questions

The paper is candid about what remains unproved. The central-window restriction $f$03 is essential to the method, since the ratio hypothesis is formulated only there. For prime-set-restricted and weighted fibres, the ratio estimates are established or conjectured only for $f$04 (respectively, expected from Selberg–Delange and saddle-point analysis of a two-parameter Dirichlet series); verifying Hypotheses 3.5 and 3.8 in full, including the complementary large-prime ranges, is the concrete open problem left by the paper. Whether the linear scale persists outside the central window is likewise not addressed.

Conclusion

The paper replaces a Hölder-exponent truncation bound $f$05 by the natural linear bound $f$06 on conditioned prime-factor fibres, via an elementary, fully effective counting argument requiring only admissibility and a Sathe–Selberg-type ratio estimate. Inserted into the Tenenbaum–Verwee conditional effective Erdős–Wintner theorem, it yields a direct sharpening of the truncation contribution, illustrated concretely by a gain of $f$07 for $f$08. The framework applies uniformly to $f$09-, $f$10-, prime-set-restricted, and weighted fibres, contingent on the corresponding ratio estimates, whose complete verification for the latter classes remains open.