论文标题
关键线附近$ L $ functions的线性组合的零数
The number of zeros of linear combinations of $L$-functions near the critical line
论文作者
论文摘要
在本文中，我们调查了属于大型类的$ l $ functions的临界线性组合的关键线附近的零，这些零是由$ \ text {gl}（n）$上的自动形式表示产生的所有$ l $ functions。更准确地说，如果$ l_1，\ dots，l_j $是具有$ j \ ge 2 $的原始$ l $ functions，而$ b_j $是任何非零的实际数字，我们证明$ f（s）= \ sum_ = \ sum_ {j \ leq j} b_j l_j l_j l_j（s） 1/2+1/g（t）$和$ \ text {im}（s）\ in [t，2t] $渐近至$ k_0 t g（t）/\ sqrt {\ log g（t）} $在范围内均均匀地在$ \ log \ log \ log \ log \ log \ log t \ leq leq l log as a $ s a $ as a $ s $ s $ s $ s $ s $ s $ s $ s $ q $ q $ ther wer with取决于$ j $和$ l_j $。这确立了在这个范围内猜想的猜想的概括。此外，指数$ν$验证$ν\ asymp 1/j $是$ j $生长。
In this paper, we investigate the zeros near the critical line of linear combinations of $L$-functions belonging to a large class, which conjecturally contains all $L$-functions arising from automorphic representations on $\text{GL}(n)$. More precisely, if $L_1, \dots, L_J$ are distinct primitive $L$-functions with $J\ge 2$, and $b_j$ are any nonzero real numbers, we prove that the number of zeros of $F(s)=\sum_{j\leq J} b_j L_j(s)$ in the region $\text{Re}(s)\geq 1/2+1/G(T)$ and $\text{Im}(s)\in [T, 2T]$ is asymptotic to $K_0 T G(T)/\sqrt{\log G(T)}$ uniformly in the range $ \log \log T \leq G(T)\leq (\log T)^ν$, where $K_0$ is a certain positive constant that depends on $J$ and the $L_j$'s. This establishes a generalization of a conjecture of Hejhal in this range. Moreover, the exponent $ν$ verifies $ν\asymp 1/J$ as $J$ grows.
