In Stein’s Fourier Analysis: an introduction, Dirichlet’s hyperbola method is introduced, which serves as a key step to prove Dirichlet’s theorem on arithmetic progressions. For a real, nontrivial character, define a function $F(m,n)=\frac{\chi(n)}{(mn)^\frac{1}{2}}$, and the sum $S_N=\sum_{mn\leq N}F(m,n)$. Applying Dirichlet’s hyperbola method, we obtain the following results:
i) $S_N \geq c \log{N}$ for some $c>0$;
ii)$S_N=2N^{1/2}L(\chi,1)+O(1)$, and the non-vanishing of L-function follows.
I wonder the motivation of defining $F(m,n)$ for the summation along hyperbolas. Does it involve some deeper results?