THE GEOMETRY OF THE DIRICHLET ETA FUNCTION

摘要

This paper presents a geometric interpretation of the Dirichlet eta function, and presents an analysis of the zeros of the eta function in that context. As the sum of a series of products, the eta function is recognized as two dot products involving three high-dimensional vectors. Two of these vectors, corresponding to the imaginary part of input value s, define a plane; the third vector corresponds to the real part of s. At a zero of the eta function, both of the dot products must be zero, meaning that the corresponding vectors are orthogonal. An analysis of the geometric properties of the curve traced by the vector corresponding to the real part of s as it varies from 0 to 1, and the geometric properties of the space perpendicular to the plane corresponding to the imaginary part of s leads to the conclusion that these elements can only intersect once for any given non-zero imaginary value of s. The implication is that there is only one zero of the eta function over that range of real values of s for a given non-zero imaginary value of s.