A Hilbert space of Dirichlet series is obtained by considering the Dirichlet series $f(s)=\sum_{n=1}^\infty a_n n^{-s}$ that satisfy $\sum_{n=0}^\infty|a_n|^2<+\infty$. These series converge in the half plane $\re s >\frac12$ and define a functions that are locally $L^2$ on the boundary $\re s=\frac12$. An analog of Carleson's celebrated convergence theorem is obtained: Each such Dirichlet series converges almost everywhere on the critical line $\re s=\frac12$. To each Dirichlet series of the above type corresponds a ``trigonometric'' series $\sum_{n=1}^\infty a_n \chi(n)$, where $\chi$ is a multiplicative character from the positive integers to the unit circle. The space of characters is naturally identified with the infinite-dimensional torus ${\mathbb T}^\infty$, where each dimension comes from a a prime number. The second analog of Carleson's theorem reads: The above ``trigonometric'' series converges for almost all characters $\chi$.