Function field version

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

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

[math]\displaystyle{ \hbox{deg}(n) = \sum_{m|n} \Lambda(m) }[/math].

As a consequence, if we define the quantities

[math]\displaystyle{ \alpha_d := \sum_{\hbox{deg}(n)=d} \Lambda(n) g(n) }[/math]

and

[math]\displaystyle{ \beta_d := \sum_{\hbox{deg}(n)=d} g(n) }[/math]

then we have the recursive identity

[math]\displaystyle{ d \beta_d = \sum_{j=1}^d \alpha_j \beta_{d-j} }[/math]. (1)

Setting g=1 gives [math]\displaystyle{ \beta_d = |F|^d }[/math] which then forces

[math]\displaystyle{ \alpha_d = \sum_{\hbox{deg}(n)=d} \Lambda(n) = |F|^d }[/math] (2)

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

[math]\displaystyle{ |\alpha_j| \leq |F|^j }[/math] (3).

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

To do this we need to select [math]\displaystyle{ \alpha_d }[/math]. From (1) we see that

[math]\displaystyle{ d\beta_d = \alpha_d + R }[/math] (4)

where R is independent of [math]\displaystyle{ \alpha_d }[/math], and by (3) we have the upper bound

[math]\displaystyle{ |R| \leq 2 \sum_{j=1}^{d-1} |F|^{d-j} \leq |F|^d \frac{2}{|F|-1}. }[/math] (5)

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