21. The theta function of a lattice is then a holomorphic function on the upper half-plane. 22. One of the major achievements of Bernhard Riemann was his theory of complex tori and theta functions . 23. This was also a beginning of a substantial development by others, connecting representation theory and theta functions . 24. These formulas can, therefore, be used to define the Theta functions over other-adic numbers. 25. In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions. 26. This is applied in the theory of theta functions , and is a possible method in geometry of numbers. 27. The next simplest cases of a disc or square involve, respectively, Bessel functions and Jacobi theta functions . 28. The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's mock theta functions . 29. The theta function of a unimodular positive definite lattice is a modular form whose weight is one half the rank. 30. He worked with ideas related to those of Weierstrass, Riemann and Heinrich Weber on theta functions and Gaussian sums.
