# Almost-Primes in Short Intervals

Let $$S$$ be the set of integers which are a product of $$k$$ distinct primes, $$k$$ a fixed positive integer (the condition that the primes are distinct is not crucial). Landau used the Prime Number Theorem to prove that $$\frac{1}{x}\sum_{n \in S:\, n \le x} = \frac{\log^{k-1} \log x}{(k-1)!\log x} (1+o(1)),$$ and he obtains some information on the size of the error term.

I am trying to find what is known, conditionally (on RH) and unconditionally, on the asymptotics of $$\sum_{n\in S: x\le n \le x+x^{c}} 1$$ as $$x$$ goes to infinity. Landau's result answers this for $$c=1$$.

For $$k=1$$, I am just asking about actual primes in short intervals. RH gives the asymptotics when $$c>1/2$$, and the work of Huxley gives the asymptotics unconditionally when $$c>7/12$$. For $$k>1$$ all I can find are 'almost everywhere' results. What results are known for all intervals when $$k>1$$?

• Why post a question and then answer it yourself one hour later? Sep 27, 2018 at 17:47
• @GregMartin I did not find anything relevant for quite a while before posting the question here -- I really thought that perhaps an expert is required here. In the meanwhile I continued sifting through the literature and actually found the right paper -- not sure if by luck or `skill'. Sep 27, 2018 at 17:53

## 1 Answer

Kátai, building on an important paper of Ramachandra, was able to prove that $$\sum_{n \in S: x \le n \le x+x^c} 1 \sim x^c \frac{\sum_{n \in S: n \le x}}{x}$$ as $$x \to \infty$$, for $$c=\frac{1}{2}+\varepsilon$$ conditionally and $$c=\frac{7}{12}+\varepsilon$$ unconditionally. Moreover, he achieves some uniformity in the parameter $$k$$.

• Nice. Could you please specify Landau's result in either the question or the answer to make the answer more explicit. Sep 27, 2018 at 9:07
• @kodlu Thanks for the suggestion, I've edited the question accordingly. Sep 27, 2018 at 9:46