  A result of Helson on general Dirichlet series $\sum a_{n} e^{-\lambda_{n}s}$
states that, whenever $(a_{n})$ is $2$-summable and $\lambda=(\lambda_{n})$
satisfies a certain condition introduced by Bohr, then for almost all
homomorphism $\omega \colon (\mathbb{R},+) \to \mathbb{T}$ the Dirichlet series
$\sum a_{n} \omega(\lambda_{n})e^{-\lambda_{n}s}$ converges on the open right
half plane $[Re>0]$. For ordinary Dirichlet series $\sum a_n n^{-s}$ Hedenmalm
and Saksman related this result with the famous Carleson-Hunt theorem on
pointwise convergence of Fourier series, and Bayart extended it within his
theory of Hardy spaces $\mathcal{H}_p$ of such series. The aim here is to prove
variants of Helson's theorem within our recent theory of Hardy spaces
$\mathcal{H}_{p}(\lambda),\,1\le p \le \infty,$ of general Dirichlet series. To
be more precise, in the reflexive case $1 < p < \infty$ we extend Helson's
result to Dirichlet series in $\mathcal{H}_{p}(\lambda)$ without any further
condition on the frequency $\lambda$, and in the non-reflexive case $p=1$ to
the wider class of frequencies satisfying the so-called Landau condition (more
general than Bohr's condition). In both cases we add relevant maximal
inequalities. Finally, we give several applications to the structure theory of
Hardy spaces of general Dirichlet series.
