The number of positive integers less than or equal to a positive real number x is given by [x], the integral part of x. The number of positive integers less than or equal to a positive real number x, in a given arithmetic progression modulo q, is given by x/q up to an error which is bounded above by 1. In the analogous setup of rings of integers of a number field,
a result due to Weber gives an asymptotic with an error term for the number of integral
ideals with absolute norm less than or equal to x. The implied constant in the O error term in Weber's theorem is effective but inexplicit. In this talk we will first talk about improving the constant given by Weber's method. We further extend these ideas towards counting ideals in ray classes (bounding the implied constant in the error term in a theorem of Tatuzawa in terms of the modulus and number field invariants).The work on these results is motivated by a myriad of applications. One important application is towards proving a number field analogue of a theorem of Ramare, Srivastava and Serra on Linnik's problem of finding the least prime in an arithmetic progression. In addition, these ideas can be used to prove new variants of the Brun-Titchmarsh theorem and finding upper bounds for the average value of Ihara's Euler-Kronecker constants for certain families of ray class fields of imaginary quadratic fields. Near the end of the talk, we will also touch upon some applications of counting prime ideals. This talk will range over several research papers, some written jointly with senior professors, some with graduate students and some single author papers.
