Counting Lattice Points

Let $a$ and $n$ be positive integers. The number of those of the numbers $1, 2, 3, \ldots , n$ that are divisible by $a$ is $\left[n/a\right]$.

How many zeros does the function $\sin x$ have in the interval $a < x \leq b$? How many does it have in the interval $a \leq x < b$?

Let $0 \leq \alpha \leq \pi$. Denote by $V_n(\alpha )$ the number of changes of sign in the sequence

Then we have

Let $\vartheta$ be an irrational number, $0 < \vartheta < 1$ and $g_n = 0$ or $1$, according as $[n\vartheta ]$ and $[(n-1)\vartheta ]$ are equal or different. Show that

What is the number $N(r,a,\alpha )$ of zeros of the entire function $e^z - ae^{i\alpha }$ in the circular disc $\left|z\right| \leq r$? ($r, a, \alpha$ real constants, $r>0$, $a>0$.)

Let $a$ and $b$ be integers, $f(x)$ a function defined in $a \leq x \leq b$, $f(x) > 0$. Using the symbol $[\ ]$ find an expression for the number of lattice points (points with integral coordinates) that are situated in the part of the plane bounded by the inequalities

Let $p$ and $q$ be positive integers with no common divisor different from 1, i.e. $p, q$ relatively prime. By counting lattice points prove the formula

If $p$ and $q$ are odd positive integers with no common divisor, then with

Let $p$ be a prime number of the form $4n+1$. We then have