Show that ${a log n }$ is not equidistributed for any $a$

Question

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 $N rightarrow infty$ ?

metamorphy · Answer

Apply the Euler–Maclaurin summation formula. For $fin C^1big([1,N]big)$, it reads $$sum_{n=1}^N f(n)=int_1^N f(x),dx+frac{f(1)+f(N)}{2}+int_1^Nleft({x}-frac12right)f'(x),dx,$$ where ${x}=x-lfloor xrfloor$ is the fractional part of $x$.
For $f(x)=e^{iclog x}$ with $c=2pi ka$, we have $f'(x)=icf(x)/x$ and $$W_N:=frac1Nsum_{n=1}^N e^{iclog N}=I_N+F_N+R_N,
\I_N=frac1Nleft.frac{e^{(1+ic)log x}}{1+ic}right|_{x=1}^{x=N}=frac{e^{iclog N}-1/N}{1+ic},
\F_N=frac{1+e^{iclog N}}{2N},quad|R_N|leqslantfrac{|c|}{N}int_1^Nfrac12frac{dx}{x}=frac{|c|}{2}frac{log N}{N}.$$
As $Ntoinfty$, clearly $F_Nto 0$ and $R_Nto 0$ but $I_Nnotto 0$. Hence, $W_Nnotto 0$.

Show that ${a log n }$ is not equidistributed for any $a$

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