On zero-density estimates and the PNT in short intervals for Beurling generalized numbers
arxiv(2022)
摘要
We study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as $N(x) = Ax + O(x^{\theta})$. We obtain in particular \[ N(\alpha, T) \ll T^{\frac{c(1-\alpha)}{1-\theta}}\log^{9} T, \] for a constant $c$ arbitrarily close to $4$, improving significantly the current state of the art. We also investigate the consequences of the obtained zero-density estimates on the PNT in short intervals. Our proofs crucially rely on an extension of the classical mean-value theorem for Dirichlet polynomials to generalized Dirichlet polynomials.
更多查看译文
关键词
short intervals,pnt,beurling,zero-density
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要