Mean values of multiplicative functions over function fields

We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors' new proof of Halasz's theorem on mean values to this simpler setting. Several of the technical difficulties that arise over the integers disappear in the function field setting, which helps bring out more clearly the main ideas of the proofs over number fields. We also obtain Lipschitz estimates showing the slow variation of mean values of multiplicative functions over function fields, which display some features that are not present in the integer situation.


Introduction
We begin by introducing multiplicative functions over the polynomial ring F q [x], highlighting the analogy with multiplicative functions over the integers. In the subsequent sections of the introduction we will discuss the new results in this paper.
1.1. An introduction to multiplicative functions over function fields. In the polynomial ring F q [x], where q is a prime power, let M denote the set of monic polynomials and let M n denote the set of monic polynomials of degree n, so that |M n | = q n . Upper case letters like F , G shall denote monic polynomials. Let P denote the set of irreducible monic polynomials, and P n those of degree n, and we reserve the letter P to denote irreducible monic polynomials. We denote the degree of a polynomial F by deg(F ).
We are interested in multiplicative functions f : M → C; that is, functions f satisfying f (F G) = f (F )f (G) for all coprime monic polynomials F and G. The analogous functions over the integers, namely multiplicative functions f : N → C (that is, functions f with f (mn) = f (m)f (n) for all coprime integers m and n), have been extensively investigated. A useful tool in studying multiplicative functions over the integers is the Dirichlet series where the product is over all primes p, and one usually restricts attention to those multiplicative functions for which the series and product are absolutely convergent in Re(s) > 1.
Correspondingly, to study multiplicative functions over function fields, we put where we assume that the series and product converge absolutely in |z| < 1/q. In Section 2 we give several examples of interesting multiplicative functions over F q [x], and for a general introduction to number theory over function fields we refer to [6]. For the moment, it may be helpful to consider the most basic example, the multiplicative function taking the value 1 on all monic polynomials in F q [x]. Here, in the domain |z| < 1/q we have which corresponds to the Riemann zeta-function Taking logarithms above yields that where Λ(F ), in analogy with the von Mangoldt function of prime number theory, is defined to be zero unless F = P k is the power of an irreducible in which case Λ(F ) = deg(P ). Equating coefficients implies that (1.2) F ∈Mn Λ(F ) = q n , and now Möbius inversion gives the the well-known "prime number theorem for F q [x]" |P n | = 1 n d|n µ(d)q n/d = q n n + O q n/2 n .
The analogous relationship in the integers, namely that n≤x Λ(n) = x + O(x 1/2+o (1) ), is an open question, equivalent to the Riemann Hypothesis. For a general multiplicative function f , take logarithms in (1.1), and write for certain coefficients Λ f (F ) with Λ f (F ) = 0 unless F is the power of an irreducible. Differentiating, we may equivalently write (1.3) as For a given positive real number κ we focus on the class of multiplicative functions C(κ) consisting of those f for which f (1) = 1 and for all F . The hypotheses (1.4) and (1.5) ensure the absolute convergence of the series and product in (1.1) for |z| < 1/q. In [2] we studied the analogous class of multiplicative functions f over the integers for which |Λ f (n)| ≤ κΛ(n), where −F ′ (s)/F (s) =: n≥1 Λ f (n)/n s . The bound on |Λ f (n)| guarantees that the Dirichlet series and Euler product defining F (s) are absolutely convergent for Re(s) > 1.
Given a multiplicative function f in C(κ) our aim is to understand (for n ≥ 0) the mean value in terms of the corresponding averages of f over prime powers We have σ(0) = 1, χ(0) = 0 and σ(1) = χ(1), and then we observe (this follows from (1.4), and will be justified in Remark 2 in Section 2 below) that With this notation, we may also write (1.1) and (1.3) as The convolution relation (1.8) is a little more involved in the integer situation. The discrete relation (1.8) is replaced by the continuous integral equation n≤y t Λ f (n) is an average of the multiplicative function f evaluated at prime powers (here y is a suitably large parameter), and then σ(u) approximates (in many situations) the mean-value of the function f evaluated over integers up to y u . Such integral equations were first considered by Wirsing, and are discussed further in [3].

1.2.
Halász's Theorem over function fields. In [2] we show, generalizing a little the pioneering work of Halász, that if x is large, and if |Λ f (n)| ≤ κΛ(n) for all n then If one inserts a trivial bound |F (1+σ +it)| ≪ κ 1/σ κ on the right then one recovers the trivial bound ≪ κ (log x) κ−1 for the left hand side (up to constants), and inserting any non-trivial information about F (1 + σ + it) supplies a non-trivial bound for the left hand side. This lossless quality is the crucial feature of Halász-type theorems. The left-hand side in (1.10) is independent of the values of f (p k ) on the prime powers p k > x. Hence if we define Λ f ⊥ (p k ) = Λ f (p k ) for all prime powers with p k ≤ x, and Λ f ⊥ (p k ) = 0 otherwise, then f ⊥ (n) = f (n) for each n ≤ x, and the Dirichlet series F ⊥ (s) := n≥1 f ⊥ (n)/n s has the finite Euler product p≤x (1 + f ⊥ (p)/p s + f ⊥ (p 2 )/p 2s + . . .) which is analytic for all s with Re(s) > 0. One can replace F in the upper bound in (1.10) with F ⊥ , which is sometimes convenient.
We now describe the corresponding result in the function field setting. Define the multiplicative function f ⊥ by setting Λ f ⊥ (M) = Λ f (M) if deg M < n, and Λ f ⊥ (M) = 0 if deg M ≥ n. Hence χ ⊥ (m) = χ(m) and σ ⊥ (m) = σ(m) for all m < n, whereas σ ⊥ (n) = σ(n) − χ(n)/n in view of (1.8). Now F ⊥ (z) is entire, whereas F (z) might only be analytic in a disc. Paralleling (1.10), we establish the following result. Theorem 1.1. Let f be in the class C(κ) and let σ(n) be defined as in (1.6). Then for all n ≥ 1 we have and therefore Our short proof of Theorem 1.1 is given in Section 3. Note the strong parallel between this theorem and the corresponding estimate (1.10). In our bound for σ(n) above, the term κ/n in the bound arose from the contribution of irreducibles of degree n. Correspondingly, in (1.10) the error term (log log x) κ / log x includes the contribution from primes near x, but also includes contributions from certain other numbers, and from error terms in truncating Perron integrals, and these do not arise in the simpler function field setting.
As in (1.9), and so one can rephrase the estimate in Theorem 1.1 as For every fixed real number θ, the function f θ (M) = f (M)e(−θ deg(M)) is also multiplicative, with χ θ (k) = χ(k)e(−kθ), and correspondingly σ θ (n) = σ(n)e(−nθ) (throughout we define e(t) := e 2πit ). This is analogous to the "twist" f (n)n −iθ of a multiplicative function f on the integers. Note that the inequalities in Theorem 1.1 and (1.13) remain unchanged if we replace χ by χ θ and σ by σ θ .
The integral in Theorem 1.1 can be difficult to work with, and we now give a slightly weaker bound which is simpler to use. By the maximum modulus principle we may bound max |z|= √ t |F ⊥ (z/q)| by max |z|=1 |F ⊥ (z/q)|. Moreover, if |z| ≤ 1 then so that |F ⊥ (z/q)| ≤ (2n) κ . In Section 3.2, using these bounds appropriately in Theorem 1.1 we show the following corollary. Since M ≥ 0 we deduce from Corollary 1.2 the trivial bound |σ(n)| ≪ κ n κ−1 (see Remark 5 in Section 2 for a more precise estimate). Inserting any non-trivial lower bound for M will yield a non-trivial bound for |σ(n)|, which as remarked earlier is the crucial feature of Halász-type theorems.
In the integer situation, the bound corresponding to Corollary 1.2 is This is usually stated with F ⊥ (1 + it) replaced by F (1 + 1/ log x + it); note that these two quantities are of comparable size (up to multiplicative constants).

Lipschitz-type theorems for mean values of multiplicative functions over function fields. Mean values of multiplicative functions in number fields vary slowly with
x, provided one corrects by the "rotation" of the form n iθ that best approximates f (n); that is, one replaces f (n) by the twist One can show that the mean value of f up to x equals x iθ /(1 + iθ) times the mean value of f θ plus a small error term. Let us restrict for simplicity to the case κ = 1. Building on Elliott's work [1], in [4] (and see also [2]), we obtained the bound In [2] we found examples showing the sharpness of (1.14), up to the factor of log 2/φ. Thus the exponent 1 − 2 π cannot be increased in general, and we say that 1 − 2 π is the Lipschitz exponent for mean values of multiplicative functions over the integers.
We now give analogous "Lipschitz estimates" in the function field case, again restricting attention, for simplicity, to functions in C(1).
3. Let f be in the class C(1), and let σ and χ be defined as in (1.6) and (1.7). Let n ≥ 2 be given, and let f ⊥ = f ⊥,n , σ ⊥ , χ ⊥ , and F ⊥ (z) be defined as before. Select θ ∈ [0, 1) for which |F ⊥ (e(−θ)/q)| is maximized. Then for any integer ℓ with 1 ≤ ℓ ≤ n, we have , and m is the smallest odd integer that does not divide ℓ, with c m := 1/(m sin( π 2m )). Since σ θ (n) = σ(n)e(−nθ), Theorem 1.3 implies that Note that 1 − c m increases to 1 − 2/π as m → ∞ through odd values. The first term of the upper bound in Theorem 1.3 corresponds to the upper bound in (1.14), but the second term indicates a new phenomenon, which has no parallel in the integer situation. In Section 6 we will construct examples, for each odd m > 1, of f for which |σ θ (n + ℓ) − σ θ (n)| ≫ 1/n 1−cm for a positive proportion of integers n. For instance, if m = 3 we take χ(n) = 1 if 3 divides n, and χ(n) = −1 otherwise. Then one can show that σ(3n) = −σ(3n 3 . When ℓ is large, the first term in the Lipschitz bound dominates the second, and one can construct examples in which the exponent 1 − 2/π is attained. Since this situation is in complete analogy with the situation for integers (see [2]), we do not carry out this construction here.

Some Examples and Remarks
In this section we collect together some remarks on our class C(κ) and offer various motivating examples of functions in this class. The proofs of our theorems are deferred to the subsequent sections.
. The function f 1 × f 2 matches the product f 1 f 2 on squarefree M, but the two functions differ on prime powers P k with k > 1 with the Rankin-Selberg convolution being the more natural choice.
Remark 2. Let f ∈ C(κ), and let the averages σ and χ be as in (1.6) and (1.7). Note that χ(0) = 0 and |χ(n)| ≤ κ for all n. We now prove the convolution relation (1.8) satisfied by σ and χ. First note that which follows upon comparing the two sides of the relation zF ′ (z) = (zF ′ /F (z))F (z). Taking the average over F ∈ M n gives In other words we have the convolution identity As discussed in the introduction, this is a simpler, discrete version of the integral equation that occurs for number field mean values, which was first considered by Wirsing, and discussed further in [3].
Remark 3. The convolution identity (1.8) shows that σ(n), the average of f over elements of M n , depends only on the χ-values, which are the average of f taken over suitable prime powers, but not on the individual f (P ℓ ). Thus there is no loss in generality in assuming that f (P ℓ ) = χ(k) whenever deg(P ℓ ) = k. We will use this observation repeatedly in the sequel when discussing and constructing examples. Further, this observation means that we can view Theorem 1.1 purely as a result in analysis: given information about the coefficients of the power series ∞ k=1 χ(k)z k /k, it obtains information about the coefficients of the series In the introduction, we saw that if f (P k ) = 1 for all prime powers P k then F (z) = (1 − qz) −1 , and χ(k) = 1 for all k ≥ 1, and σ(n) = 1 for all n ≥ 0.
Generalizing this, we consider the construction where χ(k) = α for some fixed α ∈ C, and all k ≥ 1. We find (either by solving the recurrence (1.8), or by noting that the corresponding Therefore σ(n) ∼ n α−1 /Γ(α) for large n. This is analogous to the Selberg-Delange theorem, which gives asymptotics for n≤x d z (n) where ζ(s) z = n≥1 d z (n)/n s . When α = k ∈ N, the example above deals with the k-divisor function over F q [x], and (2.1) gives the average number of ways of writing a polynomial F of degree n as the product F 1 · · · F k . The case α = −1 deals with the analog of the Möbius function: f (P ) = −1 for irreducibles P , and f (P ℓ ) = 0 for ℓ ≥ 2. Finally, note that when α = −k is a negative integer then σ(n) = 0 for all n ≥ k + 1. for some positive constant c. This implies that in the function field case, the Dickman function is not a good approximation to N(n, m) if n ≈ m 2 (whereas it is a good approximation in the corresponding range u ≈ log y for the y-smooth integer counting problem). A similar phenomenon occurs when we count "smooth permutations"; that is, elements of S n composed of cycles of length at most m.
Remark 7. In Example 4 we saw that if χ(ℓ) = −k is a negative integer for all ℓ ≥ 1 then σ(n) equals zero for all n ≥ k + 1. We now consider the converse situation: if σ(n) = 0 for all n ≥ k + 1 (where k ≥ 0), what can we conclude about χ(ℓ)? Our assumption implies that is a polynomial of degree k. Factoring F into its roots we obtain for some complex numbers α j . Since log F (z) = ∞ ℓ=1 χ(ℓ) ℓ (qz) ℓ is holomorphic in the region |z| < 1/q, we see that |α j | ≤ q, or in other words all the zeros of F lie in |z| ≥ 1/q. Further we have If |χ(ℓ)| ≤ κ for all ℓ, it follows that there can be at most κ values of α j with |α j | = q, and the rest are strictly smaller than q in magnitude.
If f ∈ C(κ) with κ < 1 satisfies σ(n) = 0 for n ≥ k + 1, then from the above we conclude that χ(ℓ) must decrease exponentially for large ℓ. If κ = 1, then either |α j | < q for all j, in which case χ(ℓ) once again decreases exponentially for large ℓ, or |α j | = q for some j (say j = 1). In the latter case, we may use Dirichlet's theorem to find ℓ such that all the α ℓ j (for 1 ≤ j ≤ k) have argument in (−π/8, π/8) say, and this forces k = 1 (else one would find an ℓ with |χ(ℓ)| > 1). Thus in this case one must have F (z) = (1 − qze(θ)) for some θ; in other words, the only possibility for f is a twist of the Möbius function by some θ. This is a simple analog of a striking converse theorem of Koukoulopoulos [5] for multiplicative functions over the integers. It may be interesting to work out a precise analog of his result, which would involve imposing (given f ∈ C(1)) the weaker restriction |σ(n)| ≪ n −2−δ for some δ > 0, and deriving a similar dichotomy for the behavior of χ(ℓ).
We proved that for any κ > 0, at most ⌊κ⌋ of the α j can have size q. Here too, one would like to replace the condition that σ(n) = 0 for large n, by a weaker condition like σ(n) ≪ n −A for some A = A(κ). Koukoulopoulos and the third author have taken some first steps in this direction for multiplicative functions over the integers.
Remark 8. One of the main results in [3] states that if f : N → {−1, 1} is a completely multiplicative function, then for large x one has n≤x f (n) ≥ (δ 1 + o(1))x, where a j e(−nα j ).
Here, by (1.9), ∞ n=0 σ(n)z n =: By matching up the coefficients of the (1 − ze(−α j )) −a j , we may construct a function such that F (z/q) − G(z), and its first derivative are bounded uniformly in |z| < 1 (the bound may depend on the e(α j )'s and a j , but remains uniform as |z| → 1). Here, for example, for each j, and the C 1 (j) are given by a similar but more complicated expression. Since the first derivative of F (z/q) − G(z) is bounded uniformly in |z| < 1, it follows that the n-th since each |a j | ≤ 1.

Proofs of Theorem 1.1 and Corollary 1.2
The key to our proof of Halász's Theorem over function fields, as well as over number fields, is an identity, given in Lemma 3.1 below. As discussed in [2], the crucial feature of this identity is the presence of three generating functions in the integral on the right-hand side, which will allow us to bound that integral efficiently. There is a strong analogy with additive number theory, where ternary problems are accessible to harmonic analysis techniques (such as the circle method) but binary problems are usually not.
Lemma 3.1. Let f be any multiplicative function in the class C(κ), and let F (z) be as in (1.1). Let r be a positive real number with r < 1/q. Then Proof. By Cauchy's formula we may write, for any 0 < r < 1/q, and use this expression in (3.1). The first term above gives matching the first term in the right-hand side of the lemma. The second term gives, upon interchanging the integrals over z and t, the other term in the right-hand side of the lemma.
In [2], we use an analogous "triple convolution" identity in our proof of Halász's Theorem over number fields, but the key analytic technique there is Perron's formula rather than Cauchy's formula, which leads to several additional complications.
3.1. Proof of Halász's Theorem in function fields. Theorem 1.1 clearly holds when n = 1, and so we suppose below that n ≥ 2. Since σ(n) depends only on the values of f on prime powers with degree at most n, we are motivated to use the multiplicative function f ⊥ (= f ⊥,n ) as described in the introduction. We recall that σ ⊥ (j) = σ(j) for all j ≤ n − 1, and that σ ⊥ (n) = σ(n) − χ(n)/n, and note that F ⊥ (z) is an entire function for all z ∈ C. Now use Lemma 3.1 with F replaced by F ⊥ there. From our observations above, we obtain (with F ⊥′ (z) denoting the derivative of F ⊥ (z)) By (1.12) we see uF ⊥′ /F ⊥ (u) = n−1 j=1 χ(j)(qu) j is a finite sum, so we may take the inner integral over z in (3.2) to be over the circle with radius 1/(q √ t), and obtain

Now consider the inner integral in (3.3). Using Cauchy-Schwarz we see that
By Parseval, and since |χ(j)| ≤ κ for all j, we have

3.2.
Proof of Corollary 1.2. As mentioned in the introduction, the maximum modulus principle gives Therefore (3.6) max Taking t = (1 − u) 2 , and using (3.6), we obtain Substituting this bound into Theorem 1.1 yields Corollary 1.2.

Lipschitz estimates: A key proposition
Throughout this section, we restrict attention to f ∈ C(1), and prove an appropriate modification of Corollary 1.2 to bound the difference |σ θ (n + ℓ) − σ θ (n)|.
We will apply this result for a suitable choice of θ in the next section, so as to deduce Theorem 1.3.

5.1.
Determining what is to be optimized.
Lemma 5.1. Select θ so as to maximize Re( n−1 j=1 χ θ (j)/j). Then Proof. If z = e(α) then, by the definition of θ as the maximizer, This proves the lemma. Note also that equality holds in the last step above when χ(k) = e(k(θ − α/2)) sign(cos(πkα)), and so the lemma is sharp in general.

5.2.
Upper bounds for a given α. An alternative expression for c m is Proof. The function | cos(πt)| is periodic with period 1, and a little computation gives the Fourier expansion The first sum is = log n + O(1), and all subsequent sums over k are ≪ log n, so we may truncate the r-sum at r ≤ R, with an error of O(1). Let S(x) := k≤x cos(2πkrα), which is easily seen to be ≪ 1/ rα . By partial summation, we deduce that if K ≫ 1/ rα then For each r ≤ R we have |rα − rb/m| ≤ r/2mR ≤ 1/2m. Therefore if m ∤ r then rα ≥ 1/2m, and so log(min{1/ rα , n}) = log(1/ rb/m ) + O(1). Since this is ≪ log m we see in particular that the terms in the r-sum, for which R ≥ r > log m and m ∤ r, contribute O(1). Therefore the contribution of the terms r ≤ R with m ∤ r to (5.1) is Note also that if r ≤ R and m|r then rα = (r/m) mα , and so the contribution of these terms to (5.1) is Using these observations in (5.1), we deduce the lemma.
Remark 10. From (5.2), we see that the m O(1) term in Lemma 5.2 can be replaced by the more precise expression where (t) m is the least residue of t (mod m), in absolute value. This can be shown to be ≪ m 1− 2 π (which is attained when b = 1) and ≫ m 1 2 − 2 π (which is attained when 2b ≡ 1 (mod m)).
Proof. Observe first, for use later, that c 1 = 1 and c 2 = 1/2, and that the c m tend upwards to 2/π as m varies over even values, and they tend downwards to 2/π as m varies over odd values. Note also that if m is odd we have 2 π + C m 2 > c m > 2 π + c m 2 , for certain absolute constants C, c > 0.
Let 0 ≤ α < 1, and correspondingly choose 1 ≤ m ≤ 2⌈log n⌉ as in Lemma 5.2. First we shall establish the upper bound implicit in Corollary 5.3.
Arguing as in Example 9, we may find asymptotics for σ(n). We now consider the special case where m > 1 is odd, and χ(k) = sign(cos(2πk/m)) for all k ≥ 1. Thus χ(k) = 1 when k/m < 1 4 and χ(k) = −1 otherwise. Recall from Remark 3 above that we are free to construct examples simply by specifying the behaviour of χ(k). Since χ(k) = χ(m − k) for all 1 ≤ k ≤ m − 1 we see that