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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 $$p\leqslant x$$. 

We show that for any polynomial $f:\mathbb{Z}\to \mathbb{Z}$with positive leading coefficient and irreducible over $\mathbb{Q}$, if $x$is large enough then there is a string of $(\log x)(\log \log x{)}^{1/835}$consecutive integers $n\in [1,x]$for which $f\left(n\right)$is composite. This improves the result by Kevin Ford, Sergei Konyagin, James Maynard, Carl Pomerance, and Terence Tao [J. Eur. Math. Soc. (JEMS) 23 (2023), pp. 667–700], which has the exponent of $\log \log x$being a constant depending on $f$which can be exponentially small in the degree of $f$. 

Abstract We give an improved lower bound for the average of the Erdős–Hooley function , namely for all and any fixed , where is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of of Hall and Tenenbaum, and can be compared to the recent upper bound of of the second and third authors. 

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. 

Abstract We study the extent to which divisors of a typical integer n are concentrated. In particular, defining $$\Delta (n) := \max _t \# \{d | n, \log d \in [t,t+1]\}$$ Δ ( n ) : = max t # { d | n , log d ∈ [ t , t + 1 ] } , we show that $$\Delta (n) \geqslant (\log \log n)^{0.35332277\ldots }$$ Δ ( n ) ⩾ ( log log n ) 0.35332277 … for almost all n , a bound we believe to be sharp. This disproves a conjecture of Maier and Tenenbaum. We also prove analogs for the concentration of divisors of a random permutation and of a random polynomial over a finite field. Most of the paper is devoted to a study of the following much more combinatorial problem of independent interest. Pick a random set $${\textbf{A}} \subset {\mathbb {N}}$$ A ⊂ N by selecting i to lie in $${\textbf{A}}$$ A with probability 1/ i . What is the supremum of all exponents $$\beta _k$$ β k such that, almost surely as $$D \rightarrow \infty $$ D → ∞ , some integer is the sum of elements of $${\textbf{A}} \cap [D^{\beta _k}, D]$$ A ∩ [ D β k , D ] in k different ways? We characterise $$\beta _k$$ β k as the solution to a certain optimisation problem over measures on the discrete cube $$\{0,1\}^k$$ { 0 , 1 } k , and obtain lower bounds for $$\beta _k$$ β k which we believe to be asymptotically sharp. 

Abstarct Given disjoint subsets T 1 , …, T m of “not too large” primes up to x , we establish that for a random integer n drawn from [1, x ], the m -dimensional vector enumerating the number of prime factors of n from T 1 , …, T m converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T 1 , …, T m are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors. 

Abstract We determine, up to multiplicative constants, the number of integers $$n\leq x$$ that have a divisor in $(y,2y]$ and no prime factor $$\leq w$$ . Our estimate is uniform in $x,y,w$ . We apply this to determine the order of the number of distinct integers in the $$N\times N$$ multiplication table, which are free of prime factors $$\leq w$$ , and the number of distinct fractions of the form $$(a_{1}a_{2})/(b_{1}b_{2})$$ with $$1\leq a_{1}\leq b_{1}\leq N$$ and $$1\leq a_{2}\leq b_{2}\leq N$$ .