# Function field version

Theorem Let F be a finite field of sufficiently large cardinality. Then there exists a completely multiplicative function $g: F[t]_{monic} \to \{-1,+1\}$ such that the partial sums $\sum_{n \in F[t]_{monic}: \hbox{deg}(n) \leq d} g(n)$ all take values in {0,1}.

Proof Let us write N for $F[t]_{monic}$ (to emphasise the similarity with the natural numbers). We introduce the von Mangoldt function $\Lambda(n)$, defined as $\hbox{deg} p$ when n is a power of an irreducible p, and zero otherwise. As in the integer case, we have the fundamental formula

$\hbox{deg}(n) = \sum_{m|n} \Lambda(m)$.

As a consequence, if we define the quantities

$\alpha_d := \sum_{\hbox{deg}(n)=d} \Lambda(n) g(n)$

and

$\beta_d := \sum_{\hbox{deg}(n)=d} g(n)$

then we have the recursive identity

$d \beta_d = \sum_{j=1}^d \alpha_j \beta_{d-j}$. (1)

Setting g=1 gives $\beta_d = |F|^d$ which then forces

$\alpha_d = \sum_{\hbox{deg}(n)=d} \Lambda(n) = |F|^d$ (2)

(this is the prime number theorem for function fields). For general g, we then have the crude upper bound

$|\alpha_j| \leq |F|^j$ (3).

Now suppose that we have already chosen g(p) for $\hbox{deg}(p) \lt d$ in such a way that the discrepancies $\beta_0+\ldots+\beta_i$ for $i \lt d$ take values in {0,1}, so in particular each $\beta_i$ has magnitude at most 1. Now we want to do the same for $\beta_0+\ldots+\beta_d$, or in other words we need to force $\beta_d$ to take values in either [0,1] or [-1,0].

To do this we need to select $\alpha_d$. From (1) we see that

$d\beta_d = \alpha_d + R$ (4)

where R is independent of $\alpha_d$, and by (3) we have the upper bound

$|R| \leq 2 \sum_{j=1}^{d-1} |F|^{d-j} \leq |F|^d \frac{2}{|F|-1}.$ (5)

On the other hand, the set of possible values of $\alpha_d$ (given our freedom to set g(p) for deg(p)=d) is an arithmetic progression of spacing 2d and length $\pi_d+1$, where $\pi_d$ is the number of primes of degree d. By the prime number theorem, this is $|F|^d/d + O( |F|^{d/2} )$. Since $\alpha_d$ must also range between $|F|^d$ and $|F|^d$, we see from (5) that for |F| large enough, we can always choose $\alpha_d$ so that $\alpha_d + R$ lies within [0,d] or [-d,0], and the claim follows from (4).

It seems that one could also extend the claim to small |F| by doing the first few degrees by hand, and then using the above argument for large degrees.