Significance of two trigonometric function in theory of numbers:

Question

There are two trigonometric function that pops here and there in theory of number ( mostly in conjectural forms):

1.$$g(x)=\frac {\sin(πx)}{πx}$$ 2. $$ f(x) =\frac {xg'(x)}{g(x)}= (1/2)(πx\cot(πx)-1)$$

In the following fields they appear :

1. Sato - Tate measure , Pair correlation of zeroes of zeta, Hilbert - Polya

2. Generating function for $\zeta(2n)$ and other zeta generating function.

The only significance I can see for the given functions is that (1) has all non zero integers as zeroes and (2) has integers as poles ( Euler formula for infinite partial decomposition )

Question : What are some Other non- trivial significant properties of the functions in context of number theory ?