TITLE: Who proved the upper bound for the autocorrelation of higher order divisor functions?
QUESTION [7 upvotes]: 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.

REPLY [9 votes]: 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, Acta Math. 180, No.1, 119-144 (1998). ZBL0917.11048.
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, Math. Proc. Camb. Philos. Soc. 152, No. 3, 405-424 (2012); erratum ibid. 157, No. 2, 375-377 (2014). ZBL1255.11048.
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