I need to prove that the sequence $\{a \log n \}$ is NOT equidistributed for any $a \in \mathbb{R}$.
Now, I think that Weyl's criterion will be a good idea to use here. So, I need to show that $$\frac{1}{N} \sum_{n=1}^{N} e^{2 \pi i k (a \log n)} \rightarrow 0 $$ as $N \rightarrow \infty$ for any $k \in \mathbb{Z} - \{0\}$ doesn't hold.
I have two problems here. Firstly, I don't know whether the base of $\log$ is $10$ or $e$. So, this can be tried for both bases.
Secondly, if I assume that the base is $e$, then I have $$ e^{\log n^{a 2 \pi i k }}$$ inside the summation. This is equal to $n^{a 2 \pi i k}. $ So, I am left with $$\frac{1}{N} \sum_{n=1}^{N} n^{ 2 \pi i k a} $$ Now suppose I take a negative $k$ and positive $a$ or vice-versa, then wouldn't this series converges to $0$ as $N \rightarrow \infty$ ?
Apply the Euler–Maclaurin summation formula. For $f\in C^1\big([1,N]\big)$, it reads $$\sum_{n=1}^N f(n)=\int_1^N f(x)\,dx+\frac{f(1)+f(N)}{2}+\int_1^N\left(\{x\}-\frac12\right)f'(x)\,dx,$$ where $\{x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.
For $f(x)=e^{ic\log x}$ with $c=2\pi ka$, we have $f'(x)=icf(x)/x$ and $$W_N:=\frac1N\sum_{n=1}^N e^{ic\log N}=I_N+F_N+R_N, \\I_N=\frac1N\left.\frac{e^{(1+ic)\log x}}{1+ic}\right|_{x=1}^{x=N}=\frac{e^{ic\log N}-1/N}{1+ic}, \\F_N=\frac{1+e^{ic\log N}}{2N},\quad|R_N|\leqslant\frac{|c|}{N}\int_1^N\frac12\frac{dx}{x}=\frac{|c|}{2}\frac{\log N}{N}.$$
As $N\to\infty$, clearly $F_N\to 0$ and $R_N\to 0$ but $I_N\not\to 0$. Hence, $W_N\not\to 0$.