Irrationality of the Angles of Kloosterman Sums over Finite Field

We prove that the angles of Kloosterman sums over arbitrary finite field are incommensurable with the constant π. In particular, this implies that the Weil bound for Kloosterman sums over finite fields cannot be reached.

Ihara Zeta function, coefficients of Maclaurin series and Ramanujan graphs

Let [Formula: see text] denote a connected [Formula: see text]-regular undirected graph of finite order [Formula: see text]. The graph [Formula: see text] is called Ramanujan whenever [Formula: see text] for all nontrivial eigenvalues [Formula: see text] of [Formula: see text]. We consider the variant [Formula: see text] of the Ihara Zeta function [Formula: see text] of [Formula: see text] defined by [Formula: see text] The function [Formula: see text] satisfies the functional equation [Formula: see text]. Let [Formula: see text] denote the number sequence given by [Formula: see text] In this paper, we establish the equivalence of the following statements: (i) [Formula: see text] is Ramanujan; (ii) [Formula: see text] for all [Formula: see text]; (iii) [Formula: see text] for infinitely many even [Formula: see text]. Furthermore, we derive the Hasse–Weil bound for the Ramanujan graphs.

On the curve Yn = Xℓ(Xm + 1) over finite fields

Let 𝓧 be the nonsingular model of a plane curve of type yn = f(x) over the finite field F of order q2, where f(x) is a separable polynomial of degree coprime to n. If the number of F-rational points of 𝓧 attains the Hasse–Weil bound, then the condition that n divides q+1 is equivalent to the solubility of f(x) in F; see [20]. In this paper, we investigate this condition for f(x) = xℓ(xm+1).