Quantitative correlations and some problems on prime factors of consecutive integers

TL;DR

The paper resolves several longstanding questions about prime factors in consecutive integers by blending probabilistic methods with advanced sieve techniques and quantitative correlation estimates for multiplicative functions. It proves an Erdős–Straus-type bound via a Maynard sieve construction, establishes the unconditional irrationality of the series ∑ ω(n)/2^n, and derives asymptotics for the event ω(n)=ω(n+1) (as well as Ω and τ) for almost all x, supported by a robust local-limit framework. A central technical advance is a quantitative correlation estimate (building on Pilatte’s decoupling) that governs high-dimensional sieve arguments and smooth-number correlations, enabling precise control of near/far-shift behaviors and two-point statistics. The results illuminate the stochastic structure of prime divisors across consecutive integers and provide a versatile toolkit for further explorations of multiplicative-function correlations and their arithmetic consequences.

Abstract

We consider several old problems involving the number of prime divisors function $ω(n)$, as well as the related functions $Ω(n)$ and $τ(n)$. Firstly, we show that there are infinitely many positive integers $n$ such that $ω(n+k) \leq Ω(n+k) \ll k$ for all positive integers $k$, establishing a conjecture of Erdős and Straus. Secondly, we show that the series $\sum_{n=1}^{\infty} ω(n)/2^n$ is irrational, settling a conjecture of Erdős. Thirdly, we prove an asymptotic formula conjectured by Erdős, Pomerance and Sárközy for the number of $n\leq x$ satisfying $ω(n)=ω(n+1)$, for almost all $x$, with similar results for $Ω$ and $τ$. Common to the resolution of all these problems is the use of the probabilistic method. For the first problem, this is combined with computations involving a high-dimensional sieve of Maynard-type. For the second and third problems, we instead make use of a general quantitative estimate for two-point correlations of multiplicative functions with a small power of logarithm saving that may be of independent interest. This correlation estimate is derived by using recent work of Pilatte.

Figures (6)

• Figure 1: A schematic diagram (not to scale) of the various ranges of primes for a given shift. We do not perform any subdivision of primes when $k$ is a very far shift. We perform an exact sieve at tiny primes and a Selberg type sieve at medium primes, in order to ensure $\omega(\mathbf{n}+k) \ll_{C_0} k$ with high probability for all shifts $k$.
• Figure 2: The probability of the event that $\tau(\mathbf{n}+1)/\tau(\mathbf{n})$ is a power of $2$ numerically converges (fairly rapidly) to about $0.4888$. For comparison we also include the larger events $\nu_p(\tau(\mathbf{n}+1)/\tau(\mathbf{n}))=0$ for $p=3$, $p=3,5$, and $p=3,5,7$, which are easier to calculate asymptotically (they correspond to return probabilities of certain one-dimensional, two-dimensional, and three-dimensional random walks respectively), as well as the two smaller events that $\tau(\mathbf{n}), \tau(\mathbf{n}+1)$ are both powers of two, or either both powers of two or both three times powers of two. The parameter $x$ ranges between $10$ and $10^7$ (and restricted to multiples of $10$ or $10^2$ for $x \geq 10^3$ and $x \geq 10^4$ respectively) and is plotted logarithmically.
• Figure 3: The probabilities $\mathbf{P}(f(\mathbf{n}) = f(\mathbf{n}+1))$ for $f=\omega,\Omega,\tau$ and $10 \leq n \leq 10^7$, normalized by multiplying against $2\sqrt{\pi(\log_2 x + B_5)}$, $2\sqrt{\pi(\log_2 x + B_6)}$, and $2\sqrt{\pi \log_2 x}/c_\tau$ respectively, with the $x$-axis plotted using $\log_2 x$. (Due to the negativity of $B_5$, the normalized $f=\omega$ probability vanishes for small $x$.) \ref{['consecutive-omega-thm']} asserts that these densities converge to $1$ asymptotically, although the convergence is very slow numerically, suggesting the presence of additional correction terms that decay as some negative power of $\log_2 x$, as well as some negative correction $B_7$ in the $f=\tau$ case.
• Figure 4: Empirical values of $B_5, B_6, B_7$ imputed from treating the heuristic approximations \ref{['b5']}, \ref{['b6']}, \ref{['b7']} as exact, compared against the predicted values of $B_5$ and $B_6$. These numerics tentatively suggest that $B_7 \approx -1$ to one significant figure.
• Figure 5: Histograms of $\omega(n+1)-\omega(n)$ and $\Omega(n+1)-\Omega(n)$ for $n \leq 10^7$, compared to the Gaussian with the empirical mean $\hat{\mu}$ and variance $\hat{\sigma}^2$ for these data sets, as well as the predicted Gaussian behavior using \ref{['b5']}, \ref{['b6']}. At this scale the effects of the lower order terms $B_5, B_6$ are significant.

Theorems & Definitions (17)

• Theorem 1.1: Erdős #248
• Remark 1.2
• Theorem 1.3: Erdős #69
• Definition 1.4: Definition of $c_\tau$
• Remark 1.5
• Theorem 1.6: Problems of Erdős--Pomerance--Sárközy and Hildebrand
• Theorem 1.7: Consecutive smooth numbers
• Remark 2.1
• Theorem 3.1: A quantitative correlation estimate
• Theorem 3.3: Pilatte's decoupling inequality