I really need help with two problems that I can't solve.
- I have the following function: $$R(N) = \sum_{n=1}^N\exp(2\pi i\sqrt{n})$$ I was able to show that $R(n)=O(\sqrt{n})$, but I am also being asked if the following is true $R(n)=o(\sqrt{n})$. I don't have any ideas here yet.
- Now I have a function $$R_q(N)=\sum_{n\leq N} \tau(n)\exp(\frac{2\pi i n}{q})$$ I need to calculate the limit $$\lim\limits_{N\to+\infty} \frac{R_q(N)}{N\ln N}$$
I'll be glad of any help, because I'm stuck.
For the limit question, you can solve it by first expanding with the definition of $\tau(n)$ and interchanging to give
$$R_q(N) = \sum_{d\le N}{\sum_{n\le N \\\ d\mid n}{\exp(2\pi i\cdot n/q)}}$$
but often you'll see the exponential written as $e_q(n) := \exp(2\pi i n/q)$ for notational convenience. Then $R_q(N)$ becomes
$$\sum_{d\le N}{\sum_{n\le N \\\ d\mid n}{e_q(n)}} = \sum_{d\le N}{\sum_{k\le \frac{N}{d}}{e_q(kd)}}.$$
The idea now is to split the sum into the case when $q$ divides $d$ meaning $e_q(kd) = 1$ and when $q$ doesn't divide $d$. For the latter case, we can split $N/d$ up into intervals of length $q$ as this matches the period of $e_q$. Thus,
$$R_q(N) = \sum_{d\le N\\\ q\mid d}{\left[\frac{N}{d}\right]} + \sum_{d\le N\\\ q\nmid d}{\left(\frac{N}{dq} \sum_{k = 1}^q{e_q(kd)} + O(q)\right)}$$
where the $O(q)$ represents an error caused by being off by at most $1$ interval. Conveniently, the inner sum is just equal to $0$ using the formula for a geometric series, and this works because we've excluded the case when $e_q(d) = 1$. Therefore, we get
$$R_q(N) = N\sum_{d\le N\\\ q\mid d}{\frac{1}{d}} + O(qN) = \frac{N\log(N)}{q} + O(qN)$$
meaning the limit is $1/q$.