An induction principle for the Bombieri-Vinogradov theorem over ${\mathbb{F}}_{q}[t]$ and a variant of the Titchmarsh divisor problem.

The Bombieri-Vinogradov theorem is one of the most celebrated theorems in analytic number theory, concerned with the equidistribution property of primes in arithmetic progressions. To articulate the theorem precisely, we first set up some notation. Let $a$ and $d$ be coprime integers. Denote

where $p$ represents a prime number. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet $L$-functions, one can get that for any $d\leq x$,

where $\operatorname{Li}(x)$ is the usual logarithmic integral

An unconditional result in this context is the well-known Siegel-Walfisz theorem which asserts that for any $N>0$, there exists $c(N)>0$ such that, if $d\leq(\log x)^{N}$,

uniformly in $d$. In other words, a non-trivial upper bound on the error term

is unconditionally known in the range $d\leq(\log x)^{N}$, for any $N>0$. In 1965, Bombieri and Vinogradov independently proved that for any $A>0$, there exists $B=B(A)>0$ such that

More generally, an arithmetic function $f$ is said to have level of distribution $\theta$, if for any $A>0$, there exists $B=B(A)>0$ such that

The Bombieri- Vinogradov theorem asserts that the level of distribution for the prime indicator function is $\theta=\frac{1}{2}$. Further, it was conjectured by Elliott and Halberstam that the bound (1.2) holds for all $0<\theta<1$ for this function. It is worth noting that Bombieri-Vinogradov theorem yields a bound as strong as what would follow from GRH for Dirichlet $L$-functions. Pushing the level of distribution beyond half has been an active area of research, resulting in important contributions due to Fouvry and Iwaniec [15] [16] , Bombieri, Friedlander and Iwaniec [6], [7], [8], Zhang [43], Maynard [31], [32], [33], Granville and Shao [19] and many others.

In 1976, Y. Motohashi [34] proved an interesting induction principle for the Bombieri-Vinogradov theorem. For any two arithmetic functions $f$ and $g$, their multiplicative convolution or Dirichlet product is defined as

For an arithmetic function $f$, we consider the following three properties:

• (a)

  $f(n)=O(\tau(n)^{C})$ for some fixed $C>0$.

• (b)

  Let $\chi$ be a non-principal character modulo $d$ such that the conductor of $\chi$ is of order $O((\log x)^{D})$, for $D>0$ suitably large. Then

  $$\sum\limits_{n\leq x}f(n)\chi(n)=O\left(\frac{x}{(\log x)^{3D}}\right).$$

• (c)

  The function $f$ satisfies the Bombieri-Vinogradov type equidistribution property, that is, (1.2) holds with $\theta=\frac{1}{2}$.

Motohashi proved that if $f$ and $g$ satisfy properties $(a)$, $(b)$ and $(c)$, then their multiplicative convolution $f\ast g$ also satisfies these three properties.

Recently, Darbar and Mukhopadhyay [10] generalized Motohashi’s result to imaginary quadratic fields. In this paper, we establish an analogous induction principle for equidistribution in arithmetic progressions, in the setting of ${\mathbb{F}}_{q}[t]$. While it is true that the Riemann hypothesis is known over finite fields, theorems of Bombieri-Vinogradov type are relevant as they give information about equidistribution in arithmetic progressions for a variety of functions. For instance, our main result allows us to prove equidistribution in arithmetic progressions for almost primes as well as the $k$-fold divisor function. Pushing the level of distribution beyond half remains an important question in the function field setting as well as demonstrated by recent work due to Sawin [37], [38, Theorem 1.2], as well as Sawin and Shusterman [39, Theorem 1.7].

We proceed to state our main result below after setting up relevant notation. Let ${\mathbb{F}}_{q}$ be a finite field of order $q$ and ${\mathbb{F}}_{q}[t]$ be the polynomial ring defined over ${\mathbb{F}}_{q}$. We denote the degree of a polynomial $f$ in ${\mathbb{F}}_{q}[t]$ by $\text{deg}(f)$, and define the norm of a polynomial $|f|$ as $q^{\text{deg}(f)}$. Throughout the article, we consider $f,g,h$ to be monic polynomials in ${\mathbb{F}}_{q}[t]$. Let $\tau_{k}(f)$ denote the $k$-fold divisor function which counts the number of ways to write $f$ as a product of $k$ monic polynomials. When $k=2$, the function $\tau_{2}(f)$ is the usual divisor function $\tau(f)$ which counts the number of monic polynomials dividing $f$.

Let $m$ be a non-constant polynomial in ${\mathbb{F}}_{q}[t]$. We may denote the ideal $(m)$ by $m$ without mentioning this explicitly if the usage is clear from the context. A multiplicative character $\chi$ modulo $m$ is a complex valued multiplicative function on $\left({\mathbb{F}}_{q}[t]/(m)\right)^{\ast}$. We extend $\chi$ to ${\mathbb{F}}_{q}[t]$ by putting $\chi(f)=\chi(\overline{f})$, where $f\equiv\overline{f}\pmod{m}$, and zero otherwise. A principal character $\chi_{0}$ mod $m$ is defined by the property that $\chi_{0}(f)=1$ if $(f,m)=1$ and $0$ otherwise. A character $\chi$ mod $m$ is called primitive multiplicative character if there is no $d|m$ such that $(d)\neq(m)$ and $\chi$ is induced by a character mod $d$. For a character $\chi$ modulo $m$, a monic polynomial $d|m$ is called the conductor of $\chi$ if there is no $d^{\prime}|d$ such that $\chi$ is induced by a primitive character mod $d^{\prime}$. For more details we refer the reader to [36, ch. 4] and [27].

For arithmetic functions $\psi_{1},\psi_{2}:{\mathbb{F}}_{q}[t]\rightarrow\mathbb{C}$, their multiplicative convolution denoted by $\psi_{1}\ast\psi_{2}$ is defined as

For an arithmetic function $\psi:{\mathbb{F}}_{q}[t]\rightarrow\mathbb{C}$, consider the following properties:

1. (1)

   (growth condition) $\psi(f)=O(\tau(f)^{C})$ for some $C>0$.

2. (2)

   (Siegel-Walfisz bound) For a non-principal character $\chi$ with conductor of degree $\leq D\log N$, we have

   $$\sum\limits_{\deg(f)=N}\psi(f)\chi(f)=O\left(\frac{q^{N}}{N^{3D}}\right),$$

   for $D>0$ sufficiently large.

3. (3)

   (Bombieri-Vinogradov type equidistribution property) For $m,\ell\in{\mathbb{F}}_{q}[t]$ with $m$ monic, and $(m,\ell)=1$, let

   $$E(M;m,l;\psi):=\sum\limits_{\begin{subarray}{c}f\equiv l\mkern 4.0mu({\operator@font mod}\mkern 6.0mum)\\ \deg(f)=M\end{subarray}}\psi(f)-\frac{1}{\phi(m)}\sum\limits_{\begin{subarray}{c}(f,m)=1\\ \deg(f)=M\end{subarray}}\psi(f).$$

   (1.3)

   Then for any $A>0$, there exists $B=B(A)>0$ such that

   $$\sum\limits_{\deg(m)\leq\frac{N}{2}-B\log N}\max\limits_{M\leq N}\max\limits_{(m,l)=1}|E(M;m,l;\psi)|\ll_{A}\frac{q^{N}}{N^{A}}.$$

An important distinction that presents itself in our function-field analogue is that we derive the corresponding version of the large sieve inequality required for our proof. Classically, it is well-known that the large sieve inequality plays a pivotal role in proving results of this type. In the function field case, various versions of the large sieve inequality have been developed, for instance by Hsu [23] as well as Baier and Singh ([3], [4]). While versions of the large sieve inequality with additive characters are known in the function field case, for our purpose, we need a large sieve estimate involving multiplicative characters. A version of this has recently been given by Klurman, Mangerel and Teräväinen ([28, Lemma 4.2]). As we shall see in Section 3 (see the remark following Theorem 3.4), the estimate in [28] does not suffice for us and we require an upper bound which is better on average over Dirichlet characters $\chi$. We proceed to prove this in Section 3 (see Theorem 3.5).

Finally, as an application, we also obtain an asymptotic for the average behaviour of the number of divisors over shifts of products of two primes in ${\mathbb{F}}_{q}[t]$. It is worth noting that this requires us to invoke a function field analogue of the Brun-Titchmarsh inequality, proved by Hsu in [23]. It is conceivable that the method should extend to shifts of products of $k$-primes, where $k$ is fixed, though we do not do this here.

A direct consequence is the following corollary which we obtain upon using Theorem 1.1 iteratively.

Motohashi’s generalization is significant in terms of yielding a family of arithmetic functions for which equidistribution results now become available. In the setting of ${\mathbb{F}}_{q}[t]$ as well, we find such interesting applications.

Let $\pi(N)$ denote the number of monic irreducible polynomials of degree $N$ in ${\mathbb{F}}_{q}[t]$. The prime number theorem (cf. [36], Theorem 2.2) in ${\mathbb{F}}_{q}[t]$ gives

We also have the prime number theorem for arithmetic progressions (cf. [36], Theorem 4.8) stated as follows. Let $d,a\in{\mathbb{F}}_{q}[t]$, $d$ has positive degree and $(a,d)=1$. Then the number of monic irreducible polynomials of degree $N$ in ${\mathbb{F}}_{q}[t]$ in the arithmetic progression $a\mkern 4.0mu({\operator@font mod}\mkern 6.0mud)$ is given by

Thus, the primes are equidistributed in arithmetic progressions with a level of distribution $\theta=\frac{1}{2}$. A natural question that arises is about the level of distribution of products of two primes in arithmetic progressions. Let $\mathbbm{1}_{\mathcal{P}}$ denote the prime indicator function. Applying Theorem 1.1 on $\mathbbm{1}_{\mathcal{P}}$, we have the following result for the indicator function of the product of two primes.

Now, taking $\psi\equiv 1$, it is easy to see that $\psi$ has level of distribution $1$ (see Section 5), and satisfies properties (1), (2). So, our Theorem 1.1 gives that $\tau_{k}$ has Bombieri-Vinogradov type inequality, for each $k$. We state this as the following corollary.

One of the significant applications of the classical Bombieri-Vinogradov theorem is to the celebrated Titchmarsh divisor problem. In 1930, Titchmarsh [41] proved that

for a fixed integer $a$ and some constant $C_{1}>1$ under the assumption of GRH. It was only after more than three decades that an unconditional proof of this result was obtained by Linnik [30]. In [6], Bombieri, Friedlander and Iwaniec proved a version of the theorem with arbitrary $\log x$ savings. The error term in the dispersion method was further improved by Drappeau [11] to obtain power savings under GRH. Several variants of the above sum have been studied by Rodriques [35], Halberstam [20], Fouvry [14], Akbary and Ghioca [1], Felix [13], Vatwani and Wong [42] and others. Using the Bombieri-Vinogradov theorem, one can show the following (cf. [9], Theorem $9.3.1$). For a fixed $a$, there exists a positive constant $c$ such that

In [34], Motohashi generalized this divisor problem to products of $k$-primes. In the same year, Fuji [17] proved that

where $\zeta(s)$ denotes the Riemann zeta function. Drappeau and Topacogullari [12] studied analogous sums over integers with a fixed number of distinct prime divisors. More precisely, letting $N\geq 1$ and $\epsilon>0$, they showed that there exists a constant $\delta>0$ and polynomials $P_{h,\ell}^{k}(X)$ of degree $k-1$ such that, for $1\leq k\ll\log\log x$ and $|h|\leq x^{\delta}$,

Here $\omega(n)$ denotes the number of distinct prime divisors of an integer $n$ and the implicit constants depend only on $N$ and $\epsilon$. Taking $k=2$ in their result gives a close analogue of (1.6). Darbar and Mukhopadhyay extended (1.6) to imaginary quadratic number fields (cf. [10], Theorem 1.6). Analogues of this for function fields have been studied extensively in the literature. Let $P$ denote a monic irreducible polynomial in ${\mathbb{F}}_{q}[t]$ and let $a$ be a fixed non-zero polynomial in ${\mathbb{F}}_{q}[t]$. Then for a fixed $q$, as $N\rightarrow\infty$, Hsu [24] proved that

where the implied constant depends only on $a$. Here $\zeta_{q}(s)$ is the zeta function over ${\mathbb{F}}_{q}[t]$, defined by

for $\mathrm{Re}(s)>1$, where the product runs over all the monic irreducible polynomials in ${\mathbb{F}}_{q}[t]$.

If we keep $N$ fixed and let $q\rightarrow\infty$, then Andrade, Bary-Soroker and Rudnick [2] obtained the asymptotic formula

As an application of the induction principle in the setting of ${\mathbb{F}}_{q}[t]$, we can obtain a generalization of Hsu’s result to products of $m$-primes where $m$ is fixed. In particular, we obtain the following analogue of the Titchmarsh divisor problem over ${\mathbb{F}}_{q}[t]$.

In case we allow both $N$ and $q$ to go to infinity, we get the following version of the above result.

Moreover, letting $q$ be fixed and $N\to\infty$, it is possible to extend the above formulas for products of $m$ primes to get

where $c(m)$ is a constant independent of $N$. This is a tedious but straightforward modification of the proof of Theorem 1.5 which we leave to the reader.

The paper is organized as follows. In Section 2, we prove some basic results on the divisor function and state a version of Perron’s formula over ${\mathbb{F}}_{q}[t]$. In Section 3, we establish a large sieve inequality for multiplicative characters which will be needed to prove our main theorem. The proofs of Theorem 1.1 and Corollary 1.4 are contained in Sections 4 and 5 respectively. By invoking an ${\mathbb{F}}_{q}[t]$-analogue of the Brun-Titchmarsh inequality, we prove Theorem 1.5 and Theorem 1.6 in Section 6.

In this section, we state some lemmas which will be useful to prove the main theorems in this paper.

Let $k\geq 2$. For the $k$-fold divisor function, we have ([2, Lemma 2.2])

For our purpose we will use the following inequality which is a direct consequence of the above result.

Further, the following lemma allows us to bound $(\tau(f))^{d}$ by $\tau_{k}(f)$ for suitably large values of $k$.

In [24], Hsu proved an asymptotic bound on the sum of $\frac{1}{\phi(f)}$ over monic polynomials. Let $\zeta_{q}(s)$ be the zeta function over ${\mathbb{F}}_{q}[t]$ as in (1.7). Using the identity

we record Hsu’s result here.

We will be using this lemma with $g=1$ or with $g$ being a fixed polynomial throughout this paper. For the sake of convenience of the reader, we remark that there is a minor typo in Lemma 3.1 of [24]. In the main term on the right-hand side, the product should be over $p\in I,p\nmid g$, where $I$ denotes the set of monic irreducible polynomials in ${\mathbb{F}}_{q}[t]$.

We will also use the following ${\mathbb{F}}_{q}[t]$ analogue of the classical Brun-Titchmarsh inequality concerning the number of primes in an arithmetic progression, derived by Hsu in [23].

Next, we derive what can be thought of as some version of Perron’s formula over ${\mathbb{F}}_{q}[t]$. This is derived using Cauchy’s residue theorem and will play a crucial role in proving our main theorem.

We end the current section by stating the orthogonality property of multiplicative characters. For more details, the reader may refer to [36, ch. 4].

The large sieve inequality has proved to be a versatile and powerful tool in number theory. The Bombieri-Vinogradov theorem can be considered as one of the finest applications of the large sieve method. The classical large sieve was first introduced by Linnik [29] around 1941 in the context of solving Vinogradov’s hypothesis related to the size of the least quadratic non-residue modulo a prime. It was further developed by contributions of Bombieri, Davenport, Halberstam, Gallagher, and many others. Analogous results on the large sieve inequality over number fields have been proved by Huxley ([25], [26]) and Hinz ([21], [22]). In 1971, Johnsen [27] established an analogue of the large sieve inequality for additive characters, and using similar techniques as introduced by Gallagher in [18], extended the theory to multiplicative characters as well. However, it appears that one requires to impose additional conditions on the set of moduli (see [27], p. 173). More generally, Hsu [23] proved a function field analogue of the large sieve inequality in arbitrary dimension. Recently Baier and Singh ([3], [5]) extended the large sieve inequality to square moduli and power moduli. In particular, [3] yields results on the large sieve inequality with additive characters in arbitrary dimension with a restricted set of moduli. For our purpose, we concentrate on the dimension one case (cf. [3], Corollary 6.5).

Continuing with the same notation and definitions as in [23] and [3], let ${\mathbb{F}}_{q}(t)$ be the rational function field and ${\mathbb{F}}_{q}(t)_{\infty}$ be the completion of ${\mathbb{F}}_{q}(t)$ at the prime at infinity denoted by $\infty$. The absolute norm denoted by $|.|_{\infty}$ is defined as

when $0\neq a_{n}\in{\mathbb{F}}_{q}$. Over ${\mathbb{F}}_{q}[t]$, this defines the usual norm of a polynomial. Let $\mathrm{Tr}:\,{\mathbb{F}}_{q}\rightarrow{\mathbb{F}}_{p}$ be the usual trace map, where $p$ is the characteristic of the field ${\mathbb{F}}_{q}$. Consider the non-trivial group homomorphism $E:{\mathbb{F}}_{q}\rightarrow\mathbb{C}^{\ast}$ defined as

and define a map $e:{\mathbb{F}}_{q}(t)_{\infty}\rightarrow\mathbb{C}^{\ast}$ as

Using the additivity property of the trace function we see that, $E$ is a nontrivial additive character on ${\mathbb{F}}_{q}$ and consequently $e$ becomes an additive character on ${\mathbb{F}}_{q}[t]$. In particular, for some fixed $f\in{\mathbb{F}}_{q}[t]$, let $\sigma_{f}:\,{\mathbb{F}}_{q}[t]\rightarrow\mathbb{C}^{\ast}$, be the additive character

With the above notation in mind, we record the following result by Baier and Singh for dimension one. Considering the $n=1$ case of Corollary $6.1$ of [3], we are able to obtain a version of Corollary $6.5$ of [3] with $q+1$ replaced by $q$ as noted below. This is significant to us since we will keep $q$ fixed and $N\rightarrow\infty$.