论文标题
在猜想的soundararajan
On a conjecture of Soundararajan
论文作者
论文摘要
基于A. Harper（2012）的最新作品，并使用M. C. Chang（2014）和H. Iwaniec（1974）在零零的区域$ l $ l $ functions $ l（s，χ）$χ$的$ q $ quallus $ q $的$ q $ $ q $的$ q $ $ q $ qu. soundararajan（2008 $ qually $ quondie $ quondie $ quondie $ quondie norde norde norde norde norde norde norde quotude $ q $ q of Soundude $ q的平滑款项）。在我们的论点中，至关重要的成分是，对于这种$ Q $，最多有一个“问题字符”，其中$ l（s，χ）$具有较小的无零区域。同样，使用$ l $ functions的零零件的排斥性质的“ deuring-heilbronn”现象接近一个，我们还表明，soundararajan的猜想对一个拥有西格尔零的模量的家族而言。
Building on recent work of A. Harper (2012), and using various results of M. C. Chang (2014) and H. Iwaniec (1974) on the zero-free regions of $L$-functions $L(s,χ)$ for characters $χ$ with a smooth modulus $q$, we establish a conjecture of K. Soundararajan (2008) on the distribution of smooth numbers over reduced residue classes for such moduli $q$. A crucial ingredient in our argument is that, for such $q$, there is at most one "problem character" for which $L(s,χ)$ has a smaller zero-free region. Similarly, using the "Deuring-Heilbronn" phenomenon on the repelling nature of zeros of $L$-functions close to one, we also show that Soundararajan's conjecture holds for a family of moduli having Siegel zeros.
