Prove or disprove $\sum_{k=1}^{n} \frac{\cos(\frac{2\pi k x}{n})}{n}=1$ iff $n|x$, and equals $0$ otherwise.

Question

I know for sure it must be impossible that the function I put forth to only have all prime and only prime zeroes, but I have to be able to prove it to be untrue.

Below is the summation I have found:

$$f(x) = 2 - \sum_{n=1}^{x} \sum_{k=1}^{n} \frac{\cos(\frac{2\pi k x}{n})}{n}$$

The exact thing I must disprove is that: $f(x) = 0$ iff x is prime, for integer x. If anyone can help me prove this to be false, I would really appreciate it. Thank you.

EDIT: I suppose at this time what I have need to prove or disprove is that:

$$f(n, x) = \sum_{k=1}^{n} \frac{\cos(\frac{2\pi k x}{n})}{n}$$

returns $1$ iff $n | x$, and return $0$ otherwise.

Answer 1

As you remark, it is enough to prove that $$\sum_{k = 1}^n\frac{\cos\left(\frac{2\pi Nk}{n}\right)}{n} = \begin{cases}1, &n\mid N\\0 & n\nmid N.\end{cases}$$ Note that since $n$ is fixed, it is equivalent to show $$\sum_{k = 1}^n\cos\left(\frac{2\pi Nk}{n}\right) = \begin{cases}n, &n\mid N\\0 & n\nmid N.\end{cases}$$

First, if $n\mid N,$ we have $nm = N$ for some integer $N.$ Then it follows that $$ \sum_{k = 1}^n\cos\left(\frac{2\pi Nk}{n}\right) = \sum_{k = 1}^n\cos(2\pi mk) = \sum_{k = 1}^n 1 = n. $$

Now, recall that $\cos(x) + i\sin(x) = e^{ix},$ so that we have $$ \sum_{k = 1}^n\cos\left(\frac{2\pi Nk}{n}\right) = \Re\left(\sum_{k = 1}^n\left(\cos\left(\frac{2\pi Nk}{n}\right) + i\sin\left(\frac{2\pi Nk}{n}\right)\right)\right) = \Re\left(\sum_{k = 1}^n e^{i\frac{2\pi Nk}{n}}\right). $$ Thus, it suffices to prove that more generally that if $n\nmid N,$ $$ \sum_{k = 1}^n e^{i\frac{2\pi Nk}{n}} = 0. $$ Now, set $\zeta_n = e^{2\pi i/n}.$ We know that $\zeta_n$ is a primitive $n$th root of unity, and we have $\zeta_n^{Nk} = e^{i\frac{2\pi Nk}{n}}$. Dividing $N$ by $n,$ we have $N = nq + r$ for some integers $r$ and $q$ with $0 < r < N.$ This implies that $\zeta_n^N = \zeta_n^r.$ If $\gcd(r,n) = 1,$ then $\zeta_n^r$ is a primitive $n$th root of unity as well, and if $\gcd(r,n) = d,$ then $\zeta_n^r$ is a primitive $n/d$-th root of unity.

So, we now need to prove the following:

Lemma: Let $n>2$ be a positive integer. Suppose that $\zeta_n$ is a primitive $n$th root of unity, and let $m$ be a positive multiple of $n.$ Then $$ \sum_{k = 1}^m\zeta_n^k = 0. $$

Proof: Writing $m = na$ for some (positive) integer $a,$ we can split up the sum as follows: \begin{align*} \sum_{k = 1}^m\zeta_n^k &= \sum_{k = 1}^{an}\zeta_n^k\\ &= \sum_{k = 1}^n\zeta_n^k + \sum_{k = n+1}^{2n}\zeta_n^{k} + \dots + \sum_{k = (a-1)n+1}^{an}\zeta_n^{k}\\ &= \sum_{k = 1}^n\zeta_n^k + \sum_{k = 1}^{n}\zeta_n^{n+k} + \dots + \sum_{k = 1}^{n}\zeta_n^{(a-1)n + k}\\ &= \sum_{k = 1}^n\zeta_n^k + \sum_{k = 1}^{n}\zeta_n^n\zeta_n^k + \dots + \sum_{k = 1}^{n}\zeta_n^{(a-1)n}\zeta_n^{k}\\ &= \sum_{k = 1}^n\zeta_n^k + \sum_{k = 1}^{n}\zeta_n^k + \dots + \sum_{k = 1}^{n}\zeta_n^{k}\\ &= a\sum_{k = 1}^n\zeta_n^k.\\ \end{align*} So in fact, it is enough to prove that $\sum_{k = 1}^n\zeta_n^k = 0$ if $\zeta_n$ is a primitive $n$th root of unity. However, we have $$ \zeta_n^n = 1, $$ which implies that $$ \zeta_n^n - 1 = 0. $$ Factoring the left hand side gives $$ (\zeta_n - 1)(\zeta_n^{n - 1} + \zeta_n^{n-2} + \zeta_n^{n-3} + \dots + \zeta_n^2 + \zeta_n + 1) = 0. $$ Since $\zeta_n$ is a primitive $n$th root of unity, $\zeta_n - 1\neq 0,$ and so $$ \zeta_n^{n - 1} + \zeta_n^{n-2} + \zeta_n^{n-3} + \dots + \zeta_n^2 + \zeta_n + 1 = 0. $$ But $1 = \zeta_n^n,$ so $$ \zeta_n^n + \zeta_n^{n - 1} + \zeta_n^{n-2} + \zeta_n^{n-3} + \dots + \zeta_n^2 + \zeta_n = 0. $$ This is exactly what we wanted to prove! $\square$

So, your function does in fact have roots at exactly the primes (supposing that the domain of $f$ is the set of positive integers). However, this doesn't make the Riemann hypothesis unnecessary. We've known formulas for primes for a while, see for example here. I'm not an expert in analytic number theory, but I believe that part of the issue is that the formulas we have for primes are incredibly inefficient. Your formula involves a double sum which has a substantial number of terms, becoming rather computationally heavy as $x$ grows. I would recommend asking an additional question about why formulas for prime numbers (such as yours or the ones in the linked Wikipedia article) don't "make the Riemann hypothesis unnecessary," so that experts on that topic can answer and you can get a better understanding of how the Riemann hypothesis and prime numbers/prime number formulas interact.

Answer 2

Your formula has a closed form. Now first notice that if $n$ divides $x=mn$ we have

$$f(n, mn) = \frac{1}{n}\sum_{k=1}^{n} \cos(\frac{2\pi k mn}{n})$$

but this is simply

$$\frac{1}{n}\sum_{k=1}^{n} \cos(2\pi k m)=1$$

On the other hand the closed form when $n$ does not divide $x$ is

$$f(n, x) = \sum_{k=1}^{n} \frac{\cos(\frac{2\pi k x}{n})}{n}=\frac1{2} \left ( \frac{\sin(\pi(\frac1{n}+2)x)}{\sin(\frac{\pi x}{n})} -1 \right )$$

Since $x$ is an integer, we can remove $2\pi x$ getting

$$\frac1{2} \left ( \frac{\sin(\pi \frac1{n}x) }{\sin(\frac{\pi x}{n})} -1 \right ) = 0$$

So your formula is a form of sieving, linear. Riemann zeta function is based on sieving as well, but it condenses it and technically surpasses its linear nature.