Triple convolution sums of the generalised divisor functions and related sums over primes

We study the triple convolution sum of the generalised divisor functions $$\sum_{n\leq x} d_k(n+h)d_l(n)d_m(n-h),$$ where $h \le x^{1-\epsilon}$ for any $\epsilon>0$ and $d_k(n)$ denotes the generalised divisor function which counts the number of ways $n$ can be written as a product of $k$ many positive integers. The purpose of this paper is three-fold. Firstly, we note a predicted asymptotic estimate for the above sum, where the constant appearing in the estimate can be obtained from the theory of Dirichlet series of several complex variables and also using some probabilistic arguments. Then we show that a lower bound of the correct order can be derived using the several variable Tauberian theorems, where, more importantly, the constant in the predicted asymptotic can be recovered. Lastly, in the spirit of the Titchmarsh divisor problem, we consider this triple convolution sum over the prime numbers, which essentially leads to a shifted convolution sum. We use the Tauberian theory of multiple Dirichlet series along with the Bombieri-Vinogradov theorem to derive an explicit lower bound of this.