We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$ and study the distribution of zeros of Dirichlet polynomials $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ corresponding to these functions. We prove that the known non-trivial zero-free half plane for Dirichlet polynomials associated to this class of multiplicative functions is optimal. We also introduce a characterization of elements in this class based on a new parameter depending on the Dirichlet series $F(s) = \sum_{n=1}^\infty f(n) n^{-s}$. In this context, we obtain non-trivial regions in which the associated Dirichlet polynomials do have zeros.

中文翻译：

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Dirichlet 多项式的零点

我们考虑某一类乘法函数 $f：\mathbb N \rightarrow \mathbb C$ 并研究 Dirichlet 多项式的零点分布 $F_N(s)= \sum_{n\le N} f(n)n^{ -s}$ 对应这些函数。我们证明了与此类乘法函数相关的狄利克雷多项式的已知非平凡无零半平面是最优的。我们还基于狄利克雷级数 $F(s) = \sum_{n=1}^\infty f(n) n^{-s}$ 的新参数，介绍了此类中元素的表征。在这种情况下，我们获得了非平凡区域，其中相关的狄利克雷多项式确实具有零。