# Dirichlet series, abscissa of absolute convergence $\neq$ abscissa of uniform convergence

It is well known that Dirichlet series, series of the form $$\sum_{n=1}^{\infty}\dfrac{a_n}{n^s},$$ where $\{a_n\}$ is a complex sequence and s is a complex variable, converge in half planes. The example $$\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n^s}$$ shows that in general Dirichlet series, in contrast to power series, doesn't always converge absolutely when converging. It is immediate that there is a half plane of absolute convergence. In addition, the example shows that Dirichlet series, in general, doesn't converge uniformly in every half-plane inside the half-plane of convergence. It is immediate that a Dirichlet series converge uniformly in every half plane inside the half-plane of absolute convergence.

So if we denote with $\sigma_a$ the abscissa of absolute convergence, the smallest number such that the Dirichlet series is absolutely convergent in the half-plane defined by that number.

And denote with $\sigma_u$ the abscissa of uniform convergence, the smallest number such that the Dirichlet series converge uniformly in every half plane beyond the half plane defined by that number.

The distance between this to abscissas can be 0, by the above example. It is known that the distance can be anything between 0 and 1/2, including 1/2. But I don't know any example that the two abscissas are different.

Can you provide an example of a Dirichlet series that the abscissas are different? Or any reason to believe that they can be different?