0

Is there an example of an ordinary Dirichlet series such that

(a) the Dirichlet series diverges to infinity at the real point (R > 0) on the line of convergence, and

(b) R is not a pole of the function represented by the Dirichlet series.

nickkatz2018
  • 27

1 Answers1

0

I think the following example answers my question: $$\sum_{n=1}^{\infty}[(-1)^{n+1}n^{-s+\frac{1}{2}}][\log \ n].$$

nickkatz2018
  • 27
  • I have reservations about saying that diverges to infinity at $s = \frac{1}{2}$, since the even-indexed partial sums converge (or diverge) to $-\infty$ and the odd-indexed partial sums con/diverge to $+\infty$. Pretty famous examples are $\log \zeta(s)$, the prime zeta function ($P(s) = \sum \frac{1}{p^s}$ with the sum running over all primes $p$) and the zeta integral $$\sum_{n = 2}^{\infty} \frac{1}{n^s\log n},.$$ These all have non-negative coefficients, the sum diverges to $+\infty$ for $s = 1$, and that's not a pole but a logarithmic branch point. – Daniel Fischer Jun 12 '18 at 17:51
  • Thank you for your answer. Actually, I was thinking in terms of a Dirichlet series that is bounded in a neighborhood of its abscissa of convergence, but divergent (in the sense that the partial sums diverge to plus or minus infinity) at the abscissa of convergence. The alternating zeta function would fit the bill, except that it diverges only by finite oscillation at the abscissa of convergence. – nickkatz2018 Jun 12 '18 at 19:27
  • You can't have that. If the series diverges to $+\infty$ for $s = s_0$, then you have $$\lim_{\varepsilon \downarrow 0} \sum_{n = 1}^{\infty} \frac{a_n}{n^{s_0 + \varepsilon}} = +\infty,.$$ If the function given by the Dirichlet series is bounded on its half-plane of convergence, the series cannot diverge to $+\infty$ or $-\infty$ on any point of the line of convergence. Oscillation with unbounded amplitude is then the most you can have. – Daniel Fischer Jun 12 '18 at 19:46
  • Thank you again. I should have been more precise in distinguishing between 'unbounded oscillation' and 'divergence to infinity'. – nickkatz2018 Jun 12 '18 at 21:44