Mean square of quadratic Hecke character sum

On the square root of quadratic matrices

Here we present a new approach to calculating the square root of a quadratic matrix. Actually, the purpose of this article is to show how the Cayley-Hamilton theorem may be used to determine an explicit formula for all the square roots of $2times 2$ matrices.

On a Sum Analogous to Dedekind Sum and Its Mean Square Value Formula

Character Rings and Hecke Algebras

In this project, we are going to study some results concerning the character rings and Heche Algebras in Sk and GL(k, F ). In next section, we will introduce some notation system in Sk and GL(k, F ) first, and with that, we will deduce some interesting and important results in this field, such as the results about the length function on W and the double B − B cosets, where W is the Weyl group o...

The Spectral Mean Square of Hecke L-functions on the Critical Line

The Hecke L-function Hj(s) attached to the jth Maass form for the full modular group is estimated in the mean square over a spectral interval for s = 1 2 + it. As a corollary, we obtain the estimate Hj( 1 2 + it) ¿ t1/3+ε for t À κ j , where 1/4 + κj is the respective jth eigenvalue of the hyperbolic Laplacian. This extends a result due to T. Meurman.

On the Quadratic Character of Quadratic Units

Let p ≡ 1 (mod 4) be a prime. Let a, b ∈ Z with p a(a2 + b2). In the paper we mainly determine ( b+ √ a2+b2 2 ) p−1 2 (mod p) by assuming p = c2 + d2 or p = Ax2 + 2Bxy + Cy2 with AC − B2 = a2 + b2. As an application we obtain simple criteria for εD to be a quadratic residue (mod p), where D > 1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the qu...