On correlation of the 3-fold divisor function with itself

Let $\zeta^k(s) = \sum_{n=1}^\infty \tau_k(n) n^{-s}, \Re s > 1$. We present three conditional results on the ternary additive correlation sum $$\sum_{n\le X} \tau_3(n) \tau_3(n+h),\quad (h\ge 1),$$ and give numerical verifications of our method. The first is a conditional proof for the full main term of the above correlation sum for any composite shift $1 \le h \le X^{2/3}$, on assuming an averaged level of distribution for the three-fold divisor function $\tau_3(n)$ in arithmetic progressions to level two-thirds. The second is a conditional derivation for the leading order main term asymptotics of this correlation sum, also valid for any composite shift $1 \le h \le X^{2/3}$. The third result gives a complete expansion of the polynomial for the full main term for the special case $h=1$ from both our method and from the delta-method, showing that our answers match. Our method is essentially elementary, especially for the $h=1$ case, uses congruences, and, as alluded to earlier, gives the same answer as in prior prediction of Conrey and Gonek [Duke Math. J. 107 (3) 2002], previously computed by Ng and Thom [Funct. Approx. Comment. Math. 60(1) 2019], and unpublished heuristic probabilistic arguments of Tao. Our procedure is general and works to give the full main term with a power-saving error term for any correlations of the form $\sum_{n\le X} \tau_k(n) f(n+h)$, to any composite shift $h$, and for a wide class of arithmetic function $f(n)$.