A Bombieri-Vinogradov Theorem For Higher-rank Groups

One of the central problems in analytic number theory is the distribution of primes in arithmetic progressions.The Siegel-Walfisz theorem claims that there are infinitely many primes in arithmetic progressions when the modulus is relatively small,and gives an asymptotic formula.However,the larger the modulus is,the more difficult this question is.It is known that the Prime Number Theorem in arithmetic progressions with good error terms is equivalent to the Grand Riemann Hypothesis.The celebrated Bombieri-Vinogradov theorem gives an average form.It in some sense shows that the Grand Riemann Hypothesis holds on average and it can be served as a substitute for the Grand Riemann Hypothesis in many applications.This theorem is closely related to many other problems,such as the Goldbach conjecture,the twin prime conjecture and so on.In this paper,we discuss some higher-rank analogues of the classical Bombieri-Vinogradov theorem.Namely,we mainly consider the distribution of the Dirichlet coefficients of automorphic L-functions on higher-rank groups in arithmetic progressions.Moreover,our results hold for all number fields.To be precise,let π be an irreducible cuspidal automorphic representation of GLn over a number field F.We denote by λπ(n)the Dirichlet coefficients of the associated automorphic L-function L(s,π)If we assume that π is self-dual and π(?)π(?)χ for any primitive Hecke characterχ,then we unconditionally establish a result of Bombieri-Vinogradov type for λπ(p).Moreover,we improve the level of distribution.Compared to the classical Bombieri-Vinogradov theorem,the higher-rank case is more difficult.On the one hand,we don’t know if the generalized Ramanujan conjecture holds for general π.On the other hand,we don’t know any result about the exceptional zeros when the rank is bigger than 3.One of the novelties in our work is that our result does not rely on any unproven progress.To overcome these difficulties,we give an extension of Siegel’s theorem of twisted automorphic L-functions in the character aspect by constructing a nice Dirichlet series.Moreover,we use the dual Pieri rule and the Rankin-Selberg theory to get rid of the usage of the generalized Ramanujan conjecture.As applications,we prove a GL4 analogue of the Titchmarsh divisor problem and a nontrivial bound for a certain GL4×GL2 shifted convolution sum for the first time.