Lethbridge Number Theory and Combinatorics Seminar: Farzad Aryan

Let $d(n)=\sum_{d|n} 1$. This is known as the divisor function. It counts the number of divisors of an integer. Consider the following shifted convolution sum  \begin{equation*}
 \sum_{an-m=h}d(n) \, d(m) \, f(an, m),  \end{equation*} where $f$ is a smooth function which is supported on $[x, 2x]\times[x, 2x]$ and oscillates mildly. In 1993, Duke, Friedlander, and Iwaniec proved that $$ \sum_{an-m=h}d(n) \, d(m) \, f(an, m) = \textbf{Main term}(x)+ \mathbf{O}(x^{0.75}).$$ Here, we improve (unconditionally) the error term in the above formula to $\mathbf{O}(x^{0.61})$, and conditionally, under the assumption of the Ramanujan-Petersson conjecture, to $\mathbf{O}(x^{0.5})$. We will also give some new results on shifted convolution sums of functions coming from Fourier coefficients of modular forms.