the zero-counting function
If you want to talk about how the zeros of an entire function are spread out, you first need to count them. The zero-counting function is simply the running tally: as you grow a disk of radius r centered at the origin, n(r) is the number of zeros of f that have fallen inside so far, each counted as many times as its multiplicity. It is a staircase that jumps up by one (or by m, at a zero of order m) every time the expanding circle sweeps past a zero. It turns the cloud of zeros into a single increasing function of the radius.
There are two standard versions. The plain count n(t) is the number of zeros with |z| <= t. The smoothed, integrated count N(r) is the integral from 0 to r of (n(t) - n(0)) / t dt, plus n(0) log r — this averaged form is what appears naturally in Jensen's formula and Nevanlinna theory, because it is gentler and matches log-scaled growth. To picture n(t): list the zeros by increasing modulus, |a_1| <= |a_2| <= ...; then n(t) equals the index k of the last zero whose modulus does not exceed t. So n(t) holds steady between consecutive zero-radii and steps up as each new shell of zeros is enclosed.
The counting function is the precise object that Jensen's formula bounds: rearranging Jensen gives that N(r) is controlled by the average of log|f| on the boundary, hence by log M(r). Combined with the definition of order, this yields the headline result of the field — for an entire function of finite order rho, n(r) grows at most like r^(rho + epsilon), so the order caps the density of zeros. A common confusion to avoid: n(r) counts with multiplicity, so a zero of order 3 adds 3 to the tally, not 1; and n(r) tells you how many zeros lie inside radius r, never where they sit on that circle.
For f(z) = sin(pi z), the zeros are the integers 0, plus or minus 1, plus or minus 2, .... Inside |z| <= r there are about 2r of them, so n(r) is roughly 2r, growing linearly. This matches sin z having order 1: the zero density is consistent with the bound n(r) <= r^(1 + epsilon).
An order-1 function like sin(pi z) has its zero count grow linearly in r — the order caps the density.
Always count zeros with multiplicity — a zero of order m adds m to n(r). And n(r) is a counting function only: it knows how many zeros are inside radius r, not their angular positions, so two functions with identical n(r) can have completely different zero patterns.