The Series $\sum\sum\frac{e^{2\pi imnx}}{mn}$ and a Problem of Chowla

The double trigonometric series $U(x):=\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{e^{2\pi imnx}}{\pi mn}$ and $U(\chi,x):=\sum_{m=1}^\infty\sum_{n=1}^\infty\chi_{m,n}\frac{e^{2\pi imnx}}{\pi mn}$ with the hyperbolic phase and coordinate-wise slow multipliers $\chi_{m,n}$ are studied. Complete descriptions of the $\mathcal K$-convergence (summability) sets of the sine series $\Im U(x)$ and the cosine series $\Re U(x)$ are given. The $\mathcal K$-sum of a double series is defined as the common value of the limits of partial sums over expanding families of kites in $\mathbb N^2$. The latter include convex domains in the usual sense, such as rectangles, as well as nonconvex domains, for example, hyperbolic crosses $\{(m,n):1\le mn\le N\}$.