论文标题
在零密度估计上和PNT的短时间间隔间隔为beurling generalive数字
On zero-density estimates and the PNT in short intervals for Beurling generalized numbers
论文作者
论文摘要
我们研究了与Beurling通用质数系统相关的ZETA函数的零的分布，其整数被分布为$ n（x）= ax + o（x^θ）$。我们特别获得\ [ N（α，t）\ ll t^{\ frac {c（1-α）} {1-θ}}} \ log^{9} t，\]对于常数$ c $任意接近$ 4 $，从而显着改善了艺术的现状。我们还研究了所获得的零密度估计值对PNT的后果。我们的证据至关重要地依赖于对迪基莱特多项式多项式多项式多项式多项式的经典均值定理的扩展。
We study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as $N(x) = Ax + O(x^θ)$. We obtain in particular \[ N(α, T) \ll T^{\frac{c(1-α)}{1-θ}}\log^{9} T, \] 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.
