• Updates on my research and expository papers, discussion of open problems, and other maths-related topics. • By Terence Tao Home About Career advice On writing Books Mastodon+ Applets Subscribe to feed Quantitative correlations and some problems on prime factors of consecutive integers 1 December, 2025 inmath.NT,paper| Tags:correlation,Joni Teravainen,prime divisors| byTerence Tao Joni Teravainenand I have uploaded to the arXiv our paper “Quantitative correlations and some problems on prime factors of consecutive integers”. • This paper applies modern analytic number theory tools - most notably, the Maynard sieve and therecent correlation estimatesfor bounded multiplicative functions of Pilatte - to resolve (either partially or fully) some old problems of Erdős, Strauss, Pomerance, Sárközy, and Hildebrand, mostly regarding the prime counting function Our first result,answering a question of Erdős, shows that there are infinitely manyfor which one has the bound However, with the advent of the Maynard sieve (also sometimes referred to as the Maynard-Tao sieve), it turns out to be possible to sieve for the conditionsfor allsimultaneously (roughly speaking, by sieving out anyfor whichis divisible by a primefor a large), and then performing a moment calculation analogous to the standard proof (due to Turán) of the Hardy-Ramanujan law, but weighted by the Maynard sieve. • (In order to get good enough convergence, one needs to control fourth moments as well as second moments, but these are standard, if somewhat tedious, calculations). • Our second result, which answers a separatequestion of Erdős, establishes that the quantity Our final result concerns the asymptotic behavior of the density Share this: Print (Opens in new window)Print Email a link to a friend (Opens in new window)Email More Share on X (Opens in new window)X Share on Facebook (Opens in new window)Facebook Share on Reddit (Opens in new window)Reddit Share on Pinterest (Opens in new window)Pinterest Recent Comments
Article Summaries:
- Researchers Joni Teravainen and the author have posted a new arXiv paper that applies the Maynard sieve and recent correlation estimates for bounded multiplicative functions to tackle longstanding questions about prime factors of consecutive integers. The authors prove that there are infinitely many integers n for which the number of prime factors of n, n+1, … n+k satisfies a uniform upper bound for all shifts up to k, a result that was previously only expected for fixed k. They also establish that a certain limiting constant involving the average number of prime factors is irrational, removing the need for the prime‑tuple conjecture. These advances bring analytic number‑theory tools closer to resolving the joint distribution of ω(n) for consecutive n.
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