On primes and other interesting sequences in short intervals 

Kaisa Matomäki, University of Turku

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By the prime number theorem (known since 1896), the number of primes up to $x$ is asymptotically $x/\log x$. This suggests that whenever $H$ is reasonably large, the interval $[x, x+H]$ contains about $H/\log x$ primes. Huxley showed in 1972 that this holds when $H \geq x^{7/12+\epsilon}$. Since then this result has not been improved, except Heath-Brown managed to replace $\epsilon$ by $-o(1)$.

Write $\lambda(n)$ for the Liouville function which is $(-1)^{\Omega(n)}$, where $\Omega(n)$ is the number of prime factors of $n$. It is well-known that the prime number theorem is equivalent to the claim that $\sum_{n \leq x} \lambda(n) = o(x)$. Also the error term in the prime number theorem is closely connected to the amount of cancellation in this sum, and both questions are closely connected to the distribution of zeros of the Riemann zeta function. 

One can naturally ask when $\sum_{x < n < x + H} \lambda(n) = o(H)$. In 1970s Ramachandra and Motohashi independently showed a result corresponding to Huxley's prime number theorem, i.e. this holds for $H \geq x^{7/12+\varepsilon}$.

In addition to discussing these themes and other previous results, I will discuss my joint work with Joni Teräväinen where we showed that Ramachandra's and Motohashi's result can be extended to the range $H \geq x^{0.55+\varepsilon}$.