Some Applications of Modular Forms [Hardcover]

An account of unexpected applications of number theory to practical questions that arise in mathematics and computer science.The theory of modular forms and especially the so-called 'Ramanujan Conjectures' have been applied to resolve problems in combinatorics, computer science, analysis and number theory. This tract, based on the Wittemore Lectures given at Yale University, is concerned with describing some of these applications.The theory of modular forms and especially the so-called 'Ramanujan Conjectures' have been applied to resolve problems in combinatorics, computer science, analysis and number theory. This tract, based on the Wittemore Lectures given at Yale University, is concerned with describing some of these applications.The theory of modular forms and especially the so-called Ramanujan Conjectures have recently been applied to resolve problems in combinatorics, computer science, analysis, and number theory. Professor Sarnak begins by developing the necessary background material in modular forms. He then considers in detail the solution of three problems: the Rusiewisz problem concerning finitely additive rotationally invariant measures on the sphere; the explicit construction of highly connected but sparse graphs, e.g. expander graphs and Ramanujan graphs; and the Linnik problem concerning the distribution of integers that represent a given large integer as a sum of three squares.Introduction; 1. Modular forms; 2. Invariant means on L(Sn); 3. Ramanujan graphs; 4. Bounds for Fourier coefficients of 1/2-integral weight; Bibliogrpahy; Index. ...treats in detail a remarkable range of ideas and beautiful mathematics. It is highly recommended to everyone interested in modular forms. Solomon Friedberg, Mathematical Reviews