Zeros of Families of Automorphic L-functions Close to 1
نویسنده
چکیده
For many L-functions of arithmetic interest, the values on or close to the edge of the region of absolute convergence are of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to non-vanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet L-functions (e.g. because of the Dirichlet class-number formula) and the symmetric square L-functions of classical automorphic forms. For analytic purposes, in the absence of the Generalized Riemann Hypothesis, it is very useful to have an upper-bound, on average, for the number of zeros of the L-functions which are very close to 1. We prove a very general statement of this type for forms on GL(n)/Q for any n > 1, comparable to the log-free density theorems for Dirichlet characters first proved by Linnik.
منابع مشابه
Zeros of Families of Automorphic L-functions
For many L-functions of arithmetic interest, the values on or close to the edge of the region of absolute convergence are of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to non-vanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet L-functions (e.g., because of the Dirichlet class-number formula) and the symmetric square L-functions of...
متن کاملNotes on Talks at the Simons Symposium on Families of Automorphic Forms and the Trace Formula
The following are notes on the talks at the Simons Symposium on Families of Automorphic Forms and the Trace Formula, held from January 26 to February 1 in Puerto Rico. All errors should be attributed solely to the typist, Steven J. Miller; these notes were TeX-ed in real time (with no effort made to go back and correct mistakes!).. See https://web.math.princeton.edu/~templier/families/ for more...
متن کاملOn tensor product $L$-functions and Langlands functoriality
In the spirit of the Langlands proposal on Beyond Endoscopy we discuss the explicit relation between the Langlands functorial transfers and automorphic $L$-functions. It is well-known that the poles of the $L$-functions have deep impact to the Langlands functoriality. Our discussion also includes the meaning of the central value of the tensor product $L$-functions in terms of the Langl...
متن کاملA Relation between the Zeros of Different Two L-functions Which Have the Euler Product and Functional Equation
As automorphic L-functions or Artin L-functions, several classes of Lfunctions have Euler products and functional equations. In this paper we study the zeros of L-functions which have the Euler products and functional equations. We show that there exists some relation between the zeros of the Riemann zetafunction and the zeros of such L-functions. As a special case of our results, we find the r...
متن کاملSuperposition of zeros of distinct L-functions
In this paper we ®rst prove a weighted prime number theorem of an ``o ̈-diagonal'' type for Rankin-Selberg L-functions of automorphic representations of GLm and GLm 0 over Q. Then for m 1, or under the Selberg orthonormality conjecture for mV 2, we prove that nontrivial zeros of distinct primitive automorphic L-functions for GLm over Q are uncorrelated, for certain test functions whose Fourier...
متن کامل