Building on recent work of Harper, and using various results of Chang and Iwaniec on the zero-free regions of $L$𝐿$L$-functions $L(s,\chi )$$L(s,\chi )$ for characters $\chi$$\chi$ with a smooth modulus $q$$q$, we establish a conjecture of Soundararajan on the distribution of smooth numbers over reduced residue classes for such moduli $q$$q$. A crucial ingredient in our argument is that, for such $q$$q$, there is at most one ‘problem character’ for which $L(s,\chi )$$L(s,\chi )$ has a smaller zero-free region. Similarly, using the ‘Deuring–Heilbronn’ phenomenon on the repelling nature of zeros of $L$$L$-functions close to one, we also show that Soundararajan's conjecture holds for a family of moduli having Siegel zeros.

中文翻译：

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关于 Soundararajan 的猜想

以 Harper 最近的工作为基础，并使用 Chang 和 Iwaniec 在零自由区域的各种结果$\mathrm{大号}$𝐿$L$-职能$\mathrm{大号}(s,\chi )$$L(s,\chi )$对于字符$\chi$$\chi$具有平滑模量$q$$q$，我们建立了 Soundararajan 的猜想，关于平滑数在此类模的减少残差类别上的分布$q$$q$. 我们论证中的一个关键因素是，对于这样的$q$$q$，至多有一个“问题人物”$\mathrm{大号}(s,\chi )$$L(s,\chi )$具有较小的零自由区域。类似地，使用“Deuring-Heilbronn”现象对零点的排斥性质$\mathrm{大号}$$L$-函数接近一，我们还证明 Soundararajan 猜想适用于具有 Siegel 零的模族。