On the Exponent of Distribution of the Ternary Divisor Function

For any positive integer k ě 1, we denote by dk the k–fold divisor function: for n a positive integer, dkpnq is the number of solutions of the equation n “ n1 . . . nk, where the ni are positive integers. The purpose of this paper is to investigate the exponent of distribution of the ternary divisor function d3 in arithmetic progressions. More generally, we will say that a real number Θ ą 0 is an exponent of distribution for dk restricted to a set Q of moduli if, for any ε ą 0, for any q P Q with q ď xΘ ́ε and any residue class a mod q with pa, qq “ 1, we have a uniform asymptotic formula