Examples of divergent series summed by means of the analytic continuation of the corresponding For my Bachelor's thesis, I am investigating divergent series. This is (yet another) question on this topic. Apparently, a divergent series $$ S = \sum_{n=1}^{\infty} a_{n} $$ can be summed by means of analytic continuation of the corresponding dirichlet series $$ f(s) = \sum_{n=1}^{\infty} \frac{ a_{n} }{n^{s}} $$ at $s=0$. Howevever, I can not find any examples of divergent series being summed by means of this method (for example, I can't find anything in Hardy's _Divergent Series_ ). Can you please provide me with some examples and/or theory which show how this method can be used to sum divergent series? References are also very much appreciated.

For example, for any Dirichlet character $\chi$, the sums $\sum_{n=1}^\infty \chi(n)$ can be summed by analytic continuation of $\sum_n \chi(n)/n^s$, which has a meromorphic continuation.

Similarly, for many other arithmetical functions (such as coefficients of modular forms) the analogous sum _has_ a meromorphic continuation, so is summable in this sense.

However, not every reasonable sequence of coefficients, even if admitting reasonable-looking arithmetic descriptions, gives a Dirichlet series with a meromorphic continuation. Some have natural boundaries, as was discovered by Estermann c. 1928. An account of some relatively elementary examples is in my course notes, linked-to from the HTML with title "Estermann phenomenon", from my course-notes page at <

In fairly immediate situations, and in general, the question of whether a Dirichlet series has a meromorphic continuation (beyond a fairly obvious half-plane) is a difficult open question.