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Power-Kernel Fluctuations, the Lambert W Subpolynomial Source, and a Local CRT Conditional Law

Primarily human-written textHuman understanding: all partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Zeng Chunping ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

For a positive integer nn we define the power kernel dn=n/rad⁡(n)d_n = n/\operatorname{rad}(n) and the fluctuation Xn=log⁡dnX_n = \log d_n, where rad⁡(n)=∏p∣np\operatorname{rad}(n) = \prod_{p \mid n} p. This paper develops three mutually supporting lines around this ``relative fluctuation''. First, the microscopic theory. From the kernel decomposition n=vdsn = v_d s (ss squarefree, (s,d)=1(s,d)=1, vd=drad⁡(d)v_d = d \operatorname{rad}(d)) we obtain the exact discrete law P(Xn=log⁡d)=6π2w(d)drad⁡(d)\mathbb{P}(X_n = \log d) = \frac{6}{\pi^2} \frac{w(d)}{d \operatorname{rad}(d)}, w(d):=∏p∣dpp+1w(d) := \prod_{p \mid d} \frac{p}{p+1}, together with a three-branch parity-conditioned refinement. Parity is a first-order effect: the measured gap E[Xn∣n even]−E[Xn∣n odd]=0.69008\mathbb{E}[X_n \mid n \text{ even}] - \mathbb{E}[X_n \mid n \text{ odd}] = 0.69008 matches log⁡2=0.69315\log 2 = 0.69315. Second, the conditional law. For fixed cc, the distribution of Xc−aX_{c-a} is completely determined by mask(a)\mathrm{mask}(a) through a three-case local CRT convolution: the local contribution vanishes identically when p∣cp \mid c or p∣ap \mid a, and otherwise follows a corrected law rather than the usual one, differing by a factor pp−1\frac{p}{p-1} (a factor 22 at p=2p=2). The corrected law agrees with 220220 masks to 0.9930.993, while the usual law underestimates the tail by 47%47\%--78%78\%. This replaces an earlier erroneous assertion of equidistribution. Third, the subpolynomial source. For radical-type Euler products ∏p(1+η(p)pσ−1)\prod_p \bigl(1 + \frac{\eta(p)}{p^\sigma - 1}\bigr) with η(p)→a\eta(p) \to a, one has as σ→0+\sigma \to 0^+ the asymptotic Φ(σ)=aσW(a/σ)[1+κ1W+κ2W2+⋯ ]\Phi(\sigma) = \frac{a}{\sigma W(a/\sigma)}\bigl[1 + \frac{\kappa_1}{W} + \frac{\kappa_2}{W^2} + \cdots\bigr], where WW is the Lambert function. The main term comes not from small primes but from the prime range [ev∗,e1/σ][e^{v^*}, e^{1/\sigma}], which explains why no finite prime table can reproduce it. The coefficients satisfy κn=(n−1)!+∑jcn,jζ(2j)\kappa_n = (n-1)! + \sum_j c_{n,j} \zeta(2j), a sum over even zetas only, because the two-sided integral coefficients satisfy ι(r)=(−1)r+1ρ(r)\iota(r) = (-1)^{r+1} \rho(r), forcing odd-zeta terms to cancel exactly at every order. This unifies T(x)=∑n≤x1/rad⁡(n)T(x) = \sum_{n \le x} 1/\operatorname{rad}(n), Σ(D)\Sigma(D) and Σ⊥(D;Q)\Sigma_\perp(D;Q), and explains at one stroke why all previously fitted ``constants'' are transient. Applications include: a quantitative formula for Mazur parameters in the open range 1<s<21 < s < 2 (keff=1.91±0.17k_{\mathrm{eff}} = 1.91 \pm 0.17, excluding k=1,3k=1,3); the ceiling theorem q≤log⁡clog⁡2+log⁡rad⁡(c)q \le \frac{\log c}{\log 2 + \log \operatorname{rad}(c)} (equality iff {a,b}={1,2k}\{a,b\} = \{1,2^k\}); the rad--P+P^+ duality (principal versus negative Lambert branch); an exact SS-unit reformulation of ρ(c)=min⁡arad⁡(a)rad⁡(c−a)\rho(c) = \min_a \operatorname{rad}(a)\operatorname{rad}(c-a); and a mechanistic decomposition of the exponent of N0(N)\mathcal{N}_0(N). We report honestly: the abc conjecture itself is not attackable within this framework. The three structural walls are: (i) the conditional law is provable for D≤c1/2−δD \le c^{1/2-\delta} while abc solutions live at D≳c0.78D \gtrsim c^{0.78}; (ii) relaxing any one of the three radicals inflates the count by a factor 387387; (iii) the Fourier circle method is controllable iff α+β>2\alpha + \beta > 2, while Mazur parameters always have α,β<1\alpha, \beta < 1. Our weak abc bound is weaker than de Bruijn's 1962 trivial bound; this is recorded as well.

Tools used

Magma
Magma

References

  1. J. Oesterlé, Nouvelles approches du ``théorème'' de Fermat , Astérisque 161--162 (1988), 165--186.
  2. D. W. Masser, Note on a conjecture of Szpiro , Astérisque 183 (1990), 19--23.
  3. N. G. de Bruijn, On the number of positive integers ≤x\le x and free of prime factors >y>y , Nederl.\ Akad.\ Wetensch.\ Proc.\ Ser.\ A 54 (1951), 50--60.
  4. N. G. de Bruijn, On the number of integers whose prime factors divide nn , Illinois J.\ Math.\ 6 (1962), 137--141.
  5. T. D. Browning, J. Lichtman, J. Teräväinen, On the radical of sums of two integers , preprint, 2024.
  6. J. Li, On the abc conjecture and the radical of a+b=ca+b=c , preprint, 2025.
  7. D. M. Kane, The number of solutions to a+b=ca+b=c with $\rad(abc)$ small , preprint.
  8. N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular Variation , Cambridge University Press, 1987.
  9. R. R. Bahadur, R. R. Rao, On deviations of the sample mean , Ann.\ Math.\ Statist.\ 31 (1960), 1015--1027.
  10. J.-H. Evertse, On equations in SS-units and the Thue--Mahler equation , Invent.\ Math.\ 75 (1984), 561--584.
Show all 16 referencesShow fewer references
  1. G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory , 3rd ed., AMS, 2015.
  2. J. Knopfmacher, Abstract Analytic Number Theory , 2nd ed., Dover, 1990.
  3. S. W. Golomb, Random permutations , Bull.\ Amer.\ Math.\ Soc.\ 67 (1961), 619.
  4. K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude , Ark.\ Mat.\ Astr.\ Fys.\ 22A (1930), 1--14.
  5. G. Halász, Remarks on the remainder in the prime number theorem , Period.\ Math.\ Hungar.\ 15 (1984), 299--314.
  6. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, On the Lambert WW function , Adv.\ Comput.\ Math.\ 5 (1996), 329--359.

Version history

  1. v1Submitted by Zeng ChunpingInitial depositCurrentOct 29, 2026