Topics In Theta Functions And Integer Partitions

In this thesis,we investigate some problems on analytic and combinatorial number theory involving integer partitions,theta functions and modular forms.The main results as follows:Using the basic asymptotic analysis and some basic techniques from analytic number theory,we establish some uniform asymptotic formulas for the Fourier coefficients of the inverse of Jacobi theta functions,and the crank,rank and k-rank statistics of integer partitions introduced by Dyson,Andrews and Garvan.These quantities plays an important role and significance in the integer partitions,algebraic geometry and theoretical physics.The main results improve the recent works of Bringmann,Manschot and Dousse and many others on this topic.Some of the results also improve and generalize the recent works of Bringmann,Dousse and Mertens and many others on the Dyson's asymptotic problem of the crank statistics of integer partitions.Using the Hardy–Ramanujan asymptotic formula and some basic techniques from analytic number theory,we establish some uniform asymptotic formulas for the rank statistic of a strongly concave composition introduced by Andrews,Rhoades and Zwegers,and the number of partitions of the bipartite number into steadily decreasing parts introduced by Carlitz.Using the theory of theta functions,we establish a new trigonometric identity of Ramanujan type,and prove and generalize a conjecture on the identity of modular functions proposed by Farkas and Kra in 2001.