This paper gives a representation-theoretic interpretation of the Lerch zeta function and related Lerch @math -functions twisted by Dirichlet characters. These functions are associated to a four-dimensional solvable real Lie group @math , called here the sub-Jacobi group, which is a semi-direct product of @math with the Heisenberg group @math . The Heisenberg group action on L^2-functions on the Heisenberg nilmanifold @math decomposes as @math , where each space @math consists of @math copies of an irreducible representation of @math with central character @math . The paper shows that show one can further decompose @math into irreducible @math -modules @math indexed by Dirichlet characters @math for @math , each of which carries an irreducible @math -action. On each @math there is an action of certain two-variable Hecke operators @math ; these Hecke operators have a natural global definition on all of @math , including the space of one-dimensional representations @math . For @math with @math suitable Lerch @math -functions on the critical line @math form a complete family of generalized eigenfunctions (pure continuous spectrum) for a certain linear partial differential operator @math . These Lerch @math -functions are also simultaneous eigenfunctions for all two-variable Hecke operators @math and their adjoints @math , provided @math . Lerch @math -functions are characterized by this Hecke eigenfunction property.
We write the complex variable @math where @math are real. We reserve @math for a real variable, viewed in the multiplicative group @math . The additive Fourier transform of a function @math is @math , following Tate @cite_12 . A Hilbert space inner product is denoted @math , with @math and @math . We use the convention that divisibility relations @math have @math but @math may have a sign, and we interpret it as @math divides @math .
@cite_12
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This is the first volume of a two-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years. It provides an introduction to analytic number theory suitable for undergraduates with some background in advanced calculus, but with no previous knowledge of number theory. Actually, a great deal of the book requires no calculus at all and could profitably be studied by sophisticated high school students. Number theory is such a vast and rich field that a one-year course cannot do justice to all its parts. The choice of topics included here is intended to provide some variety and some depth. Problems which have fascinated generations of professional and amateur mathematicians are discussed together with some of the techniques for solving them. One of the goals of this course has been to nurture the intrinsic interest that many young mathematics students seem to have in number theory and to open some doors for them to the current periodical literature. It has been gratifying to note that many of the students who have taken this course during the past 25 years have become professional mathematicians, and some have made notable contributions of their own to number theory. To all of them this book is dedicated.