The motivation for this enquiry is to understand something about the impact of multiplicativity for $f:\mathbb{N}\rightarrow\mathbb{C}$ on the conditional convergence of Dirichlet series 


$$F(s)=\sum\_{1}^{\infty}\frac{f(n)}{n^{s}}.$$


The existence of an abscissa of absolute convergence $\sigma\_a$ naturally implies the existence of an abscissa of conditional convergence $$\sigma\_c\leq\sigma\_a$$ yet, in many cases of interest to number theorists, improvements on this upper bound for $\sigma\_c$ are presently unavailable. 


I want to know about the impact of multiplicativity on lower bounds for $\sigma\_c$ in the same "functional" setting, i.e. in relation to $\sigma\_a$. For example, although one can easily find non-multiplicative $f$ for which $\sigma\_c/\sigma\_a=0$, one is tempted to suggest that multiplicativity for $f$ implies that
 $$ l\leq \sigma\_c/\sigma\_a $$ 


for some $0<l\leq 1,$ 
which is equivalent to saying that
 $$\limsup\_{x\rightarrow\infty}\sum\_{n\leq x}|f(n)|\leq\limsup\_{x\rightarrow\infty} \left|\sum\_{n\leq x}f(n)\right|^{1/l} $$
for all (or perhaps some class of) multiplicative arithmetic functions $f$. 


Specifically, I would like to ask if the assumption of multiplicativity (perhaps with additional hypotheses) leads to such (or similar) functional inequalities?



Lower bounds for partial sums of multiplicative functions
If $f(n)$ is a multiplicative function then so is $f(n)n^{-w}$ for any fixed complex number $w$. In particular, find a multiplicative $f(n)$ for which $\sigma\_c$ is strictly smaller than $\sigma\_a$, and take $w=\sigma\_c-\epsilon$; the modified function $\tilde f(n) = f(n)n^{-w}$ will have its $\tilde\sigma\_c = \epsilon$ and its $\tilde\sigma\_a$ bounded below by the positive number $\sigma\_a-\sigma\_c$. This example shows that $\tilde\sigma\_c/\tilde\sigma\_a$ can be arbitrarily close to 0.


Although it doesn't directly address your question: perhaps you already know that the inequalities $\sigma\_c \le \sigma\_a \le \sigma\_c+1$ are true and optimal, in the class of Dirichlet series as a whole (see for example Chapter 1 of Montgomery/Vaughan). Because of the ability to translate Dirichlet series as in the above construction, it seems to be that the difference between the two abscissae is more important than the quotient.


