Applications Of Theta Function Identities In Two Problems In Number Theory

The representation of integers is one of the most important problem in number theory and combinatorics.The theory of theta function identities is an important tool to investigate the representations of integers.In this thesis,we mainly use theta function identities to study two topics:the theory of partitions and the representations of integers as sums of squares and sums of triangular numbers.In this thesis,we confirm four conjectures of partition congruences given by Lin and Wang and some conjectures due to Sun on the relations between sums of squares and sums of triangular number by utilizing theta function identities and the theory of modular forms.Congruence property of partition functions is one of the most important topic in the theory of partitions.Recently,Lin and Wang introduced two special partition functions RG1(n)and RG2(n),the generating functions of which are the reciprocals of two identities due to Ramanujan and Gordon.They established several congruences modulo 5 and 7 for RG1(n)and RG2(n)and posed four conjectures on congruences modulo 25 for RG1(n)and RG2(n)at the end of their paper.In Chapter 2,we confirm the four conjectures given by Lin and Wang by using theta function identities and Ramanujan's modular equation of fifth degree.Moreover,we also obtain new congruences modulo 25 for RG1(n)and RG2(n)based on Newman's identities.For a long time,the number of representations of a positive integer as sums of triangular numbers and sums of squares has been widely concerned.Lagrange,Gauss and others have made outstanding achievements in this field.Let T(a1,a2,...,ak;n)and N(a1,a2,...,ak;n)denote the number of representations of n as a1x1(x1+1)/2+a2x2(x2+1)/2+…+akxk(xk+1)/2 and a1y12+a2y22+…+akyk2,respectively,where a1,a2,...,ak are positive integers,n,x1,x2,...,xk are arbitrary negative integers,y1,y2,...,yk are integers.Recently,Sun proved a number of relations between T(a1,a2,...,ak;n)and N(a1,a2,...,ak;n)along with numerous conjectures on such relations.In Chapters 3 and 4,we confirm several conjectures of Sun by utilizing Ramanujan's theta function identities and a MAPLE packages thetaids proposed by Frye and Garvan based on the theory of modular forms respectively.