Rational functions, cotangent sums and Eichler integrals

摘要

With the help of so called pre-weak functions, we formulate a very general transformation law for some holomorphic functions on the upper half plane and motivate the term of a generalized Eisenstein series with real-exponent Fourier expansions. Using the transformation law in the case of negative integers k , we verify a close connection between finite cotangent sums of a specific type and generalized L -functions at integer arguments. Finally, we expand this idea to Eichler integrals and period polynomials for some types of modular forms.