analytic number theory - Continuation up to zero of a Dirichlet series with bounded coefficients

The following paper seems to (among other things) give a detailed construction roughly along the lines of my comment above:

---

Bhowmik, Gautami, Schlage-Puchta, Jan-Christoph
Natural boundaries of Dirichlet series. (English summary)
Funct. Approx. Comment. Math. 37 (2007), part 1, 17--29.

In this paper, the authors prove some conditions for the existence of natural boundaries of Dirichlet series, and give applications to the determination of asymptotic results.

Let $n_nu$ be rational integers, assume the series $sumfrac{n_nu}{2^{epsilonnu}}$ converges absolutely for every $epsilon>0$, and let $mathcal{P}$ be the set of prime numbers $p$ such that $n_p>0$. Assume that the Riemann $zeta$-function has infinitely many zeros on the line $frac{1}{2}+it$, and suppose that $f$ is a function of the form $$ f(s)=prod_{nugeq1}zetaleft(muleft(s-frac{1}{2}right)+frac{1}{2}right)^{n_nu}. $$ Then $f$ is holomorphic in the half-plane $Re s>1$ and has a meromorphic continuation to the half-plane $Re s>frac{1}{2}$. If, for all $epsilon>0$, $ mathcal{P}((1+epsilon)x)-mathcal{P}(x)gg x^{frac{sqrt{5}-1}{2}}log^2x, $ then the line $Im s =frac{1}{2}$ is the natural boundary of $f$; more precisely, every point of this line is an accumulation point of zeros of $f$.

As an example on the existence of a natural boundary, $Omega$-results for Dirichlet series associated to counting functions are obtained. It is proved that if $D(s)=sumfrac{a(n)}{{n^s}}$ has a natural boundary at $Re s=sigma$, then there does not exist an explicit formula of the form $ A(s) := sum_{nleq x}a_n=sum_{rho}c_rho x^rho+O(x^sigma), $ where $rho$ is a zero of the Riemann zeta-function, and hence it is possible to obtain a term $Omega(x^{sigma-epsilon})$ in the asymptotic expression for $A(x)$.

Reviewed by Roma Kačinskaitė

---

In the above review, where $Im(s) = frac{1}{2}$ appears, I'm sure $Re(s) = frac{1}{2}$ is intended. Also the "assume" is a bit strange, since it is an old, famous theorem of G.H. Hardy that $zeta(s)$ has infinitely many zeros on the critical line.