Prahlad Sharma: Bilinear sums with (GL(2)) coefficients and the exponent of distribution of (d_3)

Abstract:

We obtain the exponent of distribution \(1/2+1/30\) for the ternary divisor function \(d_3\) to square-free and prime power moduli, improving the previous results of Fouvry–Kowalski–Michel, Heath-Brown, and Friedlander–Iwaniec. The key input is certain estimates on bilinear sums with \(GL(2)\) coefficients obtained using the delta symbol approach.