Who first published a proof that 


$$\sum\_{n\leq x}d\_{k}(n)d\_k(n+h)=O(x(\log x)^{2k-2})$$


for fixed $k$ and $h$ please? I am struggling to find a reference. Thank you. 



Who proved the upper bound for the autocorrelation of higher order divisor functions?
I'm not sure if it was the first, but the bounds would follow from the main theorem in


*Nair, Mohan; Tenenbaum, G\'erald*, [**Short sums of certain arithmetic functions**](http://dx.doi.org/10.1007/BF02392880), Acta Math. 180, No.1, 119-144 (1998). [ZBL0917.11048](https://zbmath.org/?q=an:0917.11048).


See also the more uniform bounds (in your case, this would give uniformity in $h$, after taking the singular series into account) obtained in


*Henriot, Kevin*, [**Nair-Tenenbaum bounds uniform with respect to the discriminant**](http://dx.doi.org/10.1017/S0305004111000752), Math. Proc. Camb. Philos. Soc. 152, No. 3, 405-424 (2012); erratum ibid. 157, No. 2, 375-377 (2014). [ZBL1255.11048](https://zbmath.org/?q=an:1255.11048).


The result also follows from the properties of the pseudorandom majorants for multiplicative functions such as $d\_k$ that were constructed by Matthiesen, see e.g. <https://arxiv.org/abs/1606.04482>


