An Elementary Dyadic Riemann Hypothesis

Abstract: The connection zeta function of a finite abstract simplicial complex G is defined as zeta_L(s)=sum_x 1/lambda_x^s, where lambda_x are the eigenvalues of the connection Laplacian L defined by L(x,y)=1 if x and y intersect and 0 else. (I) As a consequence of the spectral formula chi(G)=sum_x (-1)^dim(x) = p(G)-n(G), where p(G) is the number of positive eigenvalues and n(G) is the number of negative eigenvalues of L, both the Euler characteristic chi(G)=zeta(0)-2 i zeta'(0)/pi as well as determinant det(L)=e^zeta'(0)/pi can be written in terms of zeta. (II) As a consequence of the generalized Cauchy-Binet formula for the coefficients of the characteristic polynomials of a product of matrices we show that for every one-dimensional simplicial complex G, the functional equation zeta(s)=zeta(-s) holds, where zeta(s) is the Zeta function of the positive definite squared connection operator L^2 of G. Equivalently, the spectrum sigma of the integer matrix L^2 for a 1-dimensional complex always satisfies the symmetry sigma = 1/sigma and the characteristic polynomial of L^2 is palindromic. The functional equation extends to products of one-dimensional complexes. (III) Explicit expressions for the spectrum of circular connection Laplacian lead to an explicit entire zeta function in the Barycentric limit. The situation is simpler than in the Hodge Laplacian H=D^2 case where no functional equation was available. In the connection Laplacian case, the limiting zeta function is a generalized hypergeometric function which for an integer s is given by an elliptic integral over the real elliptic curve w^2=(1+z)(1-z)(z^2-4z-1), which has the analytic involutive symmetry (z,w) to (1/z,w/z^2).