Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$

Abstract We prove new large sieve inequalities for the Fourier coefficients $\rho _{j\mathfrak {a}}(n)$ 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.