A page in Ramanujan´s lost notebook contains two identities for trigonometric sums in terms of doubly infinite series of Bessel functions. One is related to the famous “circle problem” and the other to the equally famous “divisor problem”. We first discuss these classical unsolved problems. Each identity can be interpreted in three distinct ways. We discuss various methods that have been devised to prove the identities under these different interpretations. Weighted divisor sums naturally arise, and new methods for estimating trigonometric sums need to be developed. Trigonometric analogues and extensions of Ramanujan´s identities are discussed. The research to be described is joint work with Sun Kim and Alexandru Zaharescu. (The lecture will be entirely expository, except for two short proofs due to Gauss and Dirichlet.)

Bruce Berndt attended college at Albion College, graduating in 1961, where he also ran track. He received his master´s and doctoral degrees from the University of Wisconsin–Madison. He lectured for a year at the University of Glasgow and then, in 1967, was appointed an assistant professor at the University of Illinois at Urbana-Champaign, where he has remained since. In 1973–74 he was a visiting scholar at the Institute for Advanced Study in Princeton. He is currently (as of 2006) Michio Suzuki Distinguished Research Professor of Mathematics at the University of Illinois. Berndt is an analytic number theorist who is probably best known for his work explicating the discoveries of Srinivasa Ramanujan. He is a coordinating editor of The Ramanujan Journal and, in 1996, received an expository Steele Prize from the American Mathematical Society for his work editing Ramanujan´s Notebooks. In 2012 he became a fellow of the American Mathematical Society. In December 2012 he received an honorary doctorate from SASTRA University in Kumbakonam, India