Abstract
We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were first systematically studied by Berndt and Zaharescu where their proofs involved complex contour integration. We show how to prove most of Berndt–Zaharescu's and some new identities by elementary methods of discrete Fourier analysis.
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