Dirichlet series and -log 2

In this paper, we show that any Dirichlet series \(\sum ^{\infty }_{n=1}\frac{f(n)}{n^s}\) ${\sum }_{n=1}^{\infty }\frac{f(n)}{n^s}$ , where \(s=\sigma +it\in C\) $s=\sigma +it\in C$ , \(|f(n)|=1\) $|f(n)|=1$ for all \(n\ge 1\) $n\ge 1$ , and \(f(1)=1\) $f(1)=1$ , has the property that \(\frac{a_n}{n}>-\text {log}\,2\) $\frac{a_n}{n}>-\text{log}2$ for \(n>1\) $n>1$ , where \(a_n =\inf \left\{ \Re s:S_{n}(s)=0\right\} \) $a_n=inf\left\{\Re ,s,:,S_n,(s),=,0\right\}$ and \(S_n(s):=\sum ^n_{k=1}\frac{f(k)}{k^s}\) $S_n(s):={\sum }_{k=1}^n\frac{f(k)}{k^s}$ . Furthermore, if f is completely multiplicative, then \(-\log 2 = \lim _{n\rightarrow \infty }\) $-log2={lim}_{n\to \infty }$   \(\frac{a_n}{n}\) $\frac{a_n}{n}$ .