ON THE LEVEL CURVES OF THE XI FUNCTION JON BRESLAW Abstract. The properties of the saddle point on the critical line is examined to establish the behavior of the zero level curves of the Xi function. These properties are then used to analyze the intersection of the real and imaginary zero level curves. It is shown that a pair of zeros off the critical line is not consistent with these properties, thus validating the Riemann hypothesis. 1. Introduction Riemann’s ξ function is defined [4] for complex z = x + iy as: ξ (z )= z (z − 1)Γ(z/2)ζ (z ) 2π z/2 (1) where ζ (z ) is the Riemann zeta function and Γ(z ) is the gamma function. It follows that the zeros of the ξ function are identical to those of the ζ function. The idea behind this paper is simple. A zero of the ξ function can only occur at the intersection of the two level curves ℜξ (z ) = 0 and ℑξ (z ) = 0. For the ξ function, there are only four possible scenarios, and hence an analysis of each scenario should indicate whether such an intersection is feasible. The plan of the paper is as follows. In Section 2, the properties of the ξ function are described, and the consequence of thes properties are discussed. In Section 3, the four possible level curve scenarios are presented, and the feasibility of each is ascertained. Date : January 1, 2015. 2000 Mathematics Subject Classification. Primary 11M06, 11M26; Secondary 30D05. Key words and phrases. Riemann hypothesis, Xi function, functional equation, level curves. Address: Department of Economics, Concordia University, Montreal, QC, Canada H3G 1M8. Email: jon.breslaw@concordia.ca. 1 arXiv:1002.0352v8 [math.GM] 9 Jan 2015