# A possible refinement to an oscillation result

Let $\{a_n\}, \{b_m\}$ be two sequences of complex numbers satisfying $|a_n|, |b_m| \leq 1$ for all $m,n \geq 1$. For any positive numbers $N,M \geq 1$ and $\varepsilon > 0$, it is known that the estimate

$$\displaystyle \left \lvert \sum_{n \leq N} \sum_{m \leq M} a_n b_m \left(\frac{m}{n}\right) \right \rvert \ll_\varepsilon MN \left(N^{-1/2 + \varepsilon} + M^{-1/2 + \varepsilon} \right)$$

holds.

Let $\displaystyle \psi(m,n) = \frac{1}{2^{\omega(m)}} \sum_{k | m} \left(\frac{n}{k}\right)$, which can be interpreted as the indicator function for whether $n$ is a square mod $m$. What can one say about the sum

$\displaystyle \sum_{n \leq N} \sum_{m \leq M} a_n b_m \psi(m,n) \left(\frac{m}{n}\right)?$

• That's a Jacobi symbol, yes? – Gerry Myerson May 27 '18 at 23:37
• @GerryMyerson yes, $\left(\frac{m}{n}\right)$ is the Jacobi symbol – Stanley Yao Xiao May 28 '18 at 0:10