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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 $$1\leq a_{2}\leq b_{2}\leq N$$ .