Function field version

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Theorem Let F be a finite field of sufficiently large cardinality. Then there exists a completely multiplicative function [math]g: F[t]_{monic} \to \{-1,+1\}[/math] such that the partial sums [math]\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]F[t]_{monic}[/math] (to emphasise the similarity with the natural numbers). We introduce the von Mangoldt function [math]\Lambda(n)[/math], defined as [math]\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]\hbox{deg}(n) = \sum_{m|n} \Lambda(m)[/math].

As a consequence, if we define the quantities

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

and

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

then we have the recursive identity

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

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

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

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

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

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

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

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