The Riemann zeta-function, which encodes information about the integers and the prime numbers, has been studied extensively. Its values at 2,4,... are well-known, but much less is known about its values at 3,5,... . This difference can be explained to an extent by the different behaviour of certain groups (algebraic K-groups) of the rationals.
In this talk, we discuss some basic examples of such K-groups, and some links between them and arithmetic.

To be rational, or not to be, that is the internal conflict a cubic faces; its linear destiny provides a way to rationalize its life in a rich world and its whimsical action shows a way for a cubic to refuse its rational life in an impoverished field.

In his death bed letter to Hardy, Ramanujan introduced mock theta functions, which are now prototypes of mock modular forms. The coefficients of mock modular forms encode the number of certain combinatorial objects and we will discuss how mock modularity works to investigate arithmetic properties for these counting functions. On the other hand, generating functions for certain unimodal sequences are now becoming prototypes of quantum modular forms. We will discuss how they are related and what we expect from quantum modularity.