Gap Between Abscissae of Conditional Convergence and Holomorphicity for Dirichlet Series

For a Dirichlet series, $D = \sum_n a_n n^{-s}$ we may define the abscissae, in (non-strictly) increasing order

• $\sigma_c(D) = \inf\{\sigma : D \text{ converges in } \mathrm{Re} s > \sigma \}$, the abscissa of conditional convergence,
• $\sigma_b(D) = \inf\{\sigma : D \text{ defines a bounded, holomorphic (in$s$) function in } \mathrm{Re} s > \sigma \}$, the abscissa of boundedness,
• $\sigma_u(D) = \inf\{\sigma : D \text{ converges uniformly (in$s$) in } \mathrm{Re} s > \sigma \}$, the abscissa of uniform convergence, and
• $\sigma_a(D) = \inf\{\sigma : D \text{ converges absolutely in } \mathrm{Re} s > \sigma \}$, the abscissa of absolute convergence.

H. Bohr [B1913] studies these, finding $\sigma_b(D) = \sigma_u(D)$. He then turns to the gap $\sigma_a(D) - \sigma_u(D)$. D. Carando and P. Sevilla-Peris [CS2014] review Bohr's methods, the result of H.F. Bohnenblust and E. Hille [BH1931] that the supremum of this gap over all $D$ is $\frac{1}{2}$, and then generalize the results. (This gap has previously appeared on MathOverflow.)

Is anything known about the "other" gap, $\sigma_u(D) - \sigma_c(D)$, and its relation with Bohr's gap (beyond the elementary: the sum of the two gaps is at most $1$)?

References

[B1913] H. Bohr. Über die gleichmäßige Konvergenz Dirichletscher Reihen. J. Reine Angew. Math., 143:203–211, 1913.

[BH1931] H. F. Bohnenblust and E. Hille. On the absolute convergence of Dirichlet series. Ann. of Math. (2), 32(3):600–622, 1931. MR 1503020.

[CS2014] D. Carando and P. Sevilla-Peris. On the convergence of some classes of Dirichlet series. Actas del XII Congreso Dr. Antonio Monteiro (2013), :57-66, 2014. URL

• @July : I don't object to the abscissae being $\pm \infty$, but it's a fairly common exercise to show (using summation by parts) that if either is finite, $\sigma_a - \sigma_c \leq 1$. Hardy observes this (The General Theory of Dirichlet's Series, Cambridge Tracts in Mathematical Physics, no. 18, p. 9 (PDF page 17)): "If [in a generalized Dirichlet series], $\lambda_n = \log n$, the maximum possible distance between the lines of convergence is $1$." – Eric Towers Sep 21 '17 at 19:10
• Though it may not definitely answer your question, you may be interested in the following article by Kaczorowski and Perelli : arxiv.org/abs/1506.07630 – Sylvain JULIEN Sep 21 '17 at 19:26
• You may find much more literature on Laplace transforms. Also using that $D(s) \Gamma(s)$ is fast decreasing then $D(s)$ is holomorphic for $\Re(s) > -K-\epsilon$ iff $\sum_{n=1}^\infty a_n e^{-nx} - \sum_{k=0}^K b_k x^k e^{-x} = \mathcal{O}(x^{K+\epsilon})$ where $b_k = n!(-1)^n\sum_{m=0}^k D(m)\frac{(-1)^m}{(k-m)! }$ – reuns Sep 22 '17 at 8:19