A triple convolution sum of the divisor function

We study the triple convolution sum of the divisor function given by $$\sum_{n\leq x} d(n)d(n-h)d(n+h)$$ for $h\neq 0$ and $d(n)$ denotes the number of positive divisors of $n$. Based on algebraic and geometric considerations, Browning conjectured that the above sum is asymptotic to $c_hx(\log x)^3$, for a suitable constant $c_h\neq 0$, as $x\to \infty$. This conjecture is still unproved. Using sieve-theoretic results of Wolke and Nair (respectively), it is possible to derive the exact order of the sum. The lower bound of the correct order of magnitude can also be derived by very elementary arguments. In this paper, using the Tauberian theory for multiple Dirichlet series, we prove an explicit lower bound and provide a new theoretical framework to predict Browning's conjectured constant $c_h$.