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