例外Maass形式的大型筛不等式及n²+1的最大大素因子
Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$
AI总结：
本文通过证明例外Maass形式的新型大型筛不等式，改进了n²+1的最大大素因子的下限，并应用于素数和光滑数在算术级数中的分布指数。
摘要：
我们为给定级别的例外Maass形式的傅里叶系数ρ_{jℌ}(n)证明了新的大型筛不等式，这些不等式以具有稀疏傅里叶变换的序列(a_n)加权，包括在散射方法中出现的两种关键类型序列。这些结果在临界情况下实现了对例外谱的首次改进，即当序列长度与级别相同时，从而改进了各种Kloosterman和的多重线性形式的界限。作为应用，我们证明了n²+1的最大大素因子无限次大于n^{1.3}，改进了Merikoski之前的阈值n^{1.279}。我们还宣布了关于算术级数中素数和光滑数分布指数的应用。
英文摘要：
We prove new large sieve inequalities for the Fourier coefficients $ρ_{j\mathfrak{a}}(n)$ of exceptional Maass forms of a given level, weighted by sequences $(a_n)$ with sparse Fourier transforms - including two key types of sequences that arise in the dispersion method. These give the first savings in the exceptional spectrum for the critical case of sequences as long as the level, and lead to improved bounds for various multilinear forms of Kloosterman sums. As an application, we show that the greatest prime factor of $n^2+1$ is infinitely often greater than $n^{1.3}$, improving Merikoski's previous threshold of $n^{1.279}$. We also announce applications to the exponents of distribution of primes and smooth numbers in arithmetic progressions.
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