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Abstract We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$ [ 1 , x ] . Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.
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This content will become publicly available on April 1, 2026
Poisson Approximation of Prime Divisors of Shifted Primes
Abstract We develop an analog for shifted primes of the Kubilius model of prime factors of integers. We prove a total variation distance estimate for the difference between the model and actual prime factors of shifted primes, and apply it to show that the prime factors of shifted primes in disjoint sets behave like independent Poisson variables. As a consequence, we establish a transference principle between the anatomy of random integers $$\leqslant x$$ and of random shifted primes p+a with $$p\leqslant x$$.
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- Award ID(s):
- 2301264
- PAR ID:
- 10611660
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 7
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 1-16
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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