Correlations of von Mangoldt and higher order divisor functions

I will discuss joint work with M. Radziwill and T. Tao on asymptotics for the sums $\sum_{n \leq x} \Lambda(n) \Lambda(n+h)$ and $\sum_{n \leq x} d_k(n) d_l(n+h)$ where $\Lambda$ is the von Mangoldt function and $d_k$ is the kth divisor function. For the first sum we show that the expected asymptotics hold for almost all $|h| \leq X^{8/33}$ and for the second sum we show that the expected asymptotics hold for almost all $|h| \leq (\log X)^{O_{k, l} ( 1) }$.

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