As a generalization of the Euler-Mascheroni constant, Ihara introduced the Euler-Kronecker constant attached to a number field. These constants are closely related to various problems in number theory. In the course of his study, Ihara proposed several conjectures regarding the behaviour of these constants. In this talk, we will discuss the connection between these constants and certain arithmetic properties of number fields. We will also describe recent progress towards Ihara's conjectures. This is joint work with Ram Murty.