A theorem of Bombieri-Vinogradov type with few exceptional moduli

TL;DR

This paper extends the Bombieri-Vinogradov theorem to a specific set of moduli, showing primes in arithmetic progressions behave as expected with few exceptions, under certain size constraints.

Contribution

It establishes a Bombieri-Vinogradov type result for sets of pairwise coprime moduli with size less than x^(9/40), with a bound on exceptional moduli.

Findings

Expected prime distribution in arithmetic progressions for specified moduli

Bound on the number of exceptional moduli by a power of log x

Extension of Bombieri-Vinogradov theorem to new set of moduli

Abstract

If a set S of pairwise coprime moduli q, less than x^(9/40), is considered, one obtains the expected behavior for primes up to x in arithmetic progressions mod q, except for a subset of S whose cardinality is bounded by a power of log x.

Equations

$$\max_{y\leq x}\ \max_{\begin{subarray}{c}a\\ (a,q)=1\end{subarray}}\Bigg{|}\sum_{\begin{subarray}{c}n\equiv a\pmod{q}\\ n\leq y\end{subarray}}\Lambda(n)-\frac{x}{\phi(q)}\Bigg{|}<\frac{x}{\phi(q)(\log x)^{A}}$$

$$\sum _{\begin{array}{c}n\le x\\ n\equiv a(\mathrm{mod}q)\end{array}}\Lambda (n)=\frac{x}{\varphi (q)}\left(1+O_A((\log x{)}^{-A})\right)$$

$$E(x;q,a)=\sum _{\begin{array}{c}n\le x\\ n\equiv a(\mathrm{mod}q)\end{array}}\Lambda (n)-\frac{x}{\varphi (q)},$$

$$E(x,q)=\underset{\begin{array}{c}a\\ (a,q)=1\end{array}}{\max }|E(x;q,a)|,E^{\ast }(x,q)=\underset{y\le x}{\max }|E(y,q)|$$

$$E^{\ast }(x,q)\le \frac{x}{\varphi (q)(\log x{)}^A}$$

$$E^{\ast }(x,q)>\frac{x}{\varphi (q)(\log x{)}^A}$$

$$\mathcal{L}=\log x.$$

$$\text{sideset}{}^{\prime }{\sum }_{\chi (\mathrm{mod}q)},\text{sideset}{}^{\ast }{\sum }_{\chi (\mathrm{mod}q)}$$

$$\psi (y,\chi )=\sum _{n\le y}\Lambda (n)\chi (n),{\psi }_q(n)=\sum _{\begin{array}{c}n\le y\\ (n,q)=1\end{array}}\Lambda (n).$$

$$\sum _{\begin{array}{c}n\le y\\ n\equiv a(\mathrm{mod}q)\end{array}}\Lambda (n)-\frac{1}{\varphi (q)}{\psi }_q(y)=\frac{1}{\varphi (q)}\text{sideset}{}^{\prime }{\sum }_{\chi (\mathrm{mod}q)}\bar{\chi }(a)\psi (y,\chi ).$$

$$S(Q):=\sum _{q<2Q}\text{sideset}{}^{\ast }{\sum }_{\chi (\mathrm{mod}q)}\underset{y\le x}{\max }|\psi (y,\chi )|\ll x{\mathcal{L}}^{34-\delta }.$$

$$\frac{1}{\phi(q)}\Bigg{(}\sum_{n\leq y}\Lambda(n)-\psi_{q}(y)\Bigg{)}\ll\frac{\mathcal{L}^{2}\log\mathcal{L}}{Q}$$

$$\frac{1}{\varphi (q)}(\psi (y,{\chi }_1)-\psi (y,\chi ))\ll \frac{{\mathcal{L}}^2\log \mathcal{L}}{Q}$$

$$E^{\dagger}(x,q)=\max_{y\leq x}\ \max_{(a,q)=1}\Bigg{|}\sum_{\begin{subarray}{c}n\leq y\\ n\equiv a\pmod{q}\end{subarray}}\Lambda(n)-\frac{1}{\phi(q)}\ \sum_{n\leq y}\Lambda(n)\Bigg{|}.$$

$$\sum _{q\in \mathcal{S}}{E}^{†}(x,Q)$$

$$+\sum _{3\le q_1\le Q}\text{sideset}{}^{\ast }{\sum }_{{\chi }_1(\mathrm{mod}{q}_1)}\underset{y\le x}{\max }|\psi (y,{\chi }_1)|\sum _{\begin{array}{c}\frac{Q}{q_1}\le k<\frac{2Q}{q_1}\\ q_{1}k\in \mathcal{S}\end{array}}\frac{1}{\varphi (k{q}_1)}$$

$$\ll Q{\mathcal{L}}^3+\frac{\log \mathcal{L}}{Q}\sum _{3\le q_1\le Q}\text{sideset}{}^{\ast }{\sum }_{{\chi }_1(\mathrm{mod}{q}_1)}\underset{y\le x}{\max }|\psi (y,{\chi }_1)|\sum _{\begin{array}{c}\frac{Q}{q_1}\le k<\frac{2Q}{q_1}\\ q_{1}k\in \mathcal{S}\end{array}}1.$$

$$\sum _{q\in \mathcal{S}}{E}^{†}(x,Q)\ll Q{\mathcal{L}}^3+\frac{\log \mathcal{L}}{Q}S(Q)\ll \frac{x{\mathcal{L}}^{34}}{Q}.$$

$$E^{†}(x,q)>\frac{x}{2\varphi (q)}{\mathcal{L}}^{-A}$$

$$|\mathcal{A}|\ll {\mathcal{L}}^{34+A}.$$

$$\displaystyle\Bigg{|}\sum_{\begin{subarray}{c}n\leq y\\ n\equiv a\pmod{q}\end{subarray}}$$

$$\displaystyle\leq\frac{x\mathcal{L}^{-A}}{2\phi(q)}+\Bigg{|}\sum_{n\leq y}\Lambda(n)-\frac{y}{\phi(q)}\Bigg{|}$$

$$\le \frac{x}{\varphi (q)}{\mathcal{L}}^{-A}$$

$$\Bigg{|}\sum_{n\leq y}\Lambda(n)\chi(n)\Bigg{|}\quad(y\leq x)$$

$$a(\chi)\sum_{n\leq y(\chi)}\Lambda(n)\chi(n)=\Bigg{|}\sum_{n\leq y(\chi)}\Lambda(n)\chi(n)\Bigg{|}.$$

$$S(Q)=\sum _{q<2Q}\text{sideset}{}^{\ast }{\sum }_{\chi (\mathrm{mod}q)}a(\chi )\sum _{n\le y(\chi )}\Lambda (n)\chi (n).$$

$$S:=\sum _{q<2Q}\text{sideset}{}^{\ast }{\sum }_{\chi (\mathrm{mod}q)}a(\chi )\sum _{\begin{array}{c}{n}_1\dots n_8\le y(\chi )\\ n_i\in I_i\end{array}}(\log n_1)\mu (n_5)\dots \mu (n_8)\chi (n_1)\dots \chi (n_8)$$

$$S=\sum _{q,\chi }{\int }_1^{N_1}\frac{1}{v}\sum _{\begin{array}{c}{n}_i\in I_i^{\prime }\\ n_1\dots n_8\le y(\chi )\end{array}}\mu (n_5)\dots \mu (n_8)\chi (n_1)\dots \chi (n_8)dv,$$

$$F_j(s,\chi )=F_j(s,\chi ,v)=\sum _{n\in I_j^{\prime }}{a}_j(n)\chi (n)n^{-s}$$

$$\sum _{\begin{array}{c}{n}_i\in I_i^{\prime }\\ n_1\dots n_8\le y(\chi )\end{array}}\mu (n_5)\dots \mu (n_8)\chi (n_1)\dots \chi (n_8)$$

$$=\frac{1}{2\pi i}{\int }_{1+{\mathcal{L}}^{-1}-ix}^{1+{\mathcal{L}}^{-1}+ix}{F}_1(s,\chi )\dots F_8(s,\chi )\frac{y(\chi {)}^s}{s}dx+O({\mathcal{L}}^2).$$

$$\left|\prod _{j=1}^8{F}_j(\sigma \pm ix,\chi )\right|\le \prod _{j=1}^8{N}_j^{1-\sigma }\le x^{1-\sigma },$$

$$S$$

Full text

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holds for all $q$ in $\mathcal{S}$ with $O((\log x)^{34+A})$ exceptions.

1. Introduction

Let $\Lambda(n)$ denote the von Mangoldt function. The prime number theorem in the form

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for every $A>0$, holds for $q\leq(\log x)^{A}$, $(a,q)=1$. The best-known average result for a set of moduli $q$ is the Bombieri-Vinogradov theorem. Let

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It is easy to deduce from the presentation of the Bombieri-Vinogradov theorem in [2] that

[TABLE]

for all integers $q$ in $[Q,2Q)$ with at most $O(Q(\log x)^{-A})$ exceptions, provided that $Q\leq x^{1/2}(\log x)^{-2A-6}$.

It is of interest to restrict the size of this exceptional set further. Following Cui and Xue [1] we find that provided only prime moduli $q$ are considered, the exceptional set has cardinality $O(\mathcal{L}^{C+A})$ for some absolute constant $C$ when $Q\leq x^{1/5}$.

Glyn Harman has pointed out to me that one can obtain the result of [1] directly from Vaughan [9, Theorem 1] with $C=3$.

In the present paper, the constant 1/5 is increased to 9/40 by adding a ‘Halasz-Montgomery-Huxley’ bound to the tools employed in [1]; see Lemma 3 below. We also relax the primality condition a little.

Theorem**.**

Let $Q\leq x^{9/40}$. Let $\mathcal{S}$ be a set of pairwise relatively prime integers in $[Q,2Q)$. The number of $q$ in $\mathcal{S}$ for which

[TABLE]

is $O((\log x)^{34+A})$.

As a simple example, we may take $\mathcal{S}$ to be the set of prime powers in $[Q,2Q)$. The constant 34 could be reduced with further effort. Constants implied by ‘$O$’, ‘$\ll$’ are absolute constants throughout the paper. We write $|\mathcal{E}|$ for the cardinality of a finite set $\mathcal{E}$ and

[TABLE]

We suppose, as we may, that $x$ is large.

2. A proposition which implies the theorem

We write

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for a sum respectively over non-principal characters and primitive characters $\pmod{q}$. For $y\leq x$, let

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We note the identity, for $(a,q)=1$,

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For brevity, we write $\delta=1/20$.

Proposition**.**

Let $Q\leq x^{9/40}$. Then

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The Proposition implies the Theorem. For if $q\in[Q,2Q)$, an argument on page 163 of [2] yields, for $y\leq x$,

[TABLE]

and

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where $\chi$ is induced by the primitive character $\chi_{1}$. Let

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We combine all contributions to $E^{\dagger}(x,q)$ made by an individual primitive character. We see from (2.1)–(2.3) that

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The inner sum is 0 or 1 by our hypothesis on $\mathcal{S}$, and we obtain

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The set $\mathcal{A}$ of $q\in[Q,2Q)$ for which

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thus has cardinality

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For $q\in\mathcal{S}_{Q}-\mathcal{A}$, $y\leq x$, $(a,q)=1$ we have

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by the prime number theorem. This completes the proof of the theorem.

We now explain the initial stage of the proof of the proposition. For $\chi$ $\pmod{q}$ a primitive character, $Q\leq q<2Q$, choose $y(\chi)$ to maximize

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and $a(\chi)$ so that $|a(\chi)|=1$,

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Thus

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From the discussion in Heath-Brown [3], $S(Q)$ is a linear combination, with bounded coefficients, of $O(\mathcal{L}^{8})$ sums of the form

[TABLE]

in which $I_{i}=(N_{i},2N_{i}]$, $\prod\limits_{i}N_{i}\leq x$ and $2N_{i}\leq x^{1/4}$ if $i>4$. Some of the intervals $I_{i}$ may contain only the integer 1, and we replace these by $[1,2)$ without affecting the upper bound $\prod\limits_{i}N_{i}\leq x$. Now we need only bound $S$ by $O(x\mathcal{L}^{26-\delta}/Q)$.

It is convenient to get rid of the factor $\log n_{1}$ in $S$. We have

[TABLE]

where $I_{1}=(\max(v,N_{1}),2N_{1}]$ and $I_{i}^{\prime}=I_{i}$ for $i>1$.

Next we use Perron’s formula [8, Lemma 3.12]. Let

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where $a_{j}(n)=1$ $(j\leq 4)$, $a_{j}(n)=\mu(n)$ $(j>4)$. Then

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We shift the path of integration to $\operatorname{Re}(s)=1/2$. We have

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so that the integral on the horizontal segments is $O(1)$. Thus

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where $T$ takes the values $2^{k}$, $1\leq k\leq\mathcal{L}/\log 2$. Here $v$ is now fixed in $[1,N_{1}]$. Since $Q^{4}<x$, we need only show that

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This is done by grouping $F_{1}\ldots F_{8}$ into two or three subproducts. It is time to state the lemmas we need on Dirichlet polynomials. For the rest of this section, let

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where $1\leq N\leq x$, $N\leq N^{\prime}\leq cN$ with an absolute constant $c$, and let

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Lemma 1**.**

Let $1\leq T$, $Q\leq x$. For a primitive character $\chi\pmod{q}$ let $J_{\chi}$ be a set of numbers in $[-T,T]$ such that $|t-t^{\prime}|\geq 1$ for distinct $t$, $t^{\prime}$ in $J_{\chi}$. Then

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Proof.

This follows at once from Theorem 7.3 of Montgomery [7]. ∎

Lemma 2**.**

Let $a_{n}=1$ $(N\leq n\leq N^{\prime})$ in (2.4). Let $J_{\chi}$ be as in Lemma 1. Then

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Proof.

Following the argument of Liu and Liu [6], proof of Proposition 5.3, we find that

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and, for the derivative $S^{\prime}$,

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We now appeal to Lemma 1.4 of [4] with $S^{2}$, $2SS^{\prime}$ in place of $S^{\prime}$ This gives for the left-hand side of (2.7) the bound

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By the Cauchy-Schwarz inequality, the product contributes at most

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which by Hölder’s inequality is at most $M_{1}^{3/4}M_{2}^{1/4}$. The proof is now completed using (2.8), (2.9). ∎

Lemma 3**.**

Let $\mathcal{B}$ be the set of $(q,\chi,t)$ with $q\leq Q$, $\chi\pmod{q}$, $t\in J_{\chi}$ and

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in Lemma 1. Then

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Proof.

This is a very slight variant of Iwaniec and Kowalski [5, Theorem 9.18]. ∎

Let $\tau_{b}(n)$ be the number of factorizations $n=n_{1}\ldots n_{b}$.

If $S(it,\chi)$ is the product of $b$ of the above functions $F_{j}\left(\frac{1}{2}+it,\chi\right)$, it is clear that

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The last step is a standard application of Perron’s formula to

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which we can write as $F(s)\zeta(s)^{b^{2}}$ with $F$ analytic and bounded in $\operatorname{Re}(s)\geq 2/3$. It follows that, with $J_{\chi}$ as in Lemma 1,

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for $N_{j}\ll x^{9/40}$ (using Lemma 1 and (2.11)) and for $N_{j}>x^{1/4}$ (using Lemma 2). This explains the role of the ‘difficult interval’ $(9/40,1/4)$ in Lemma 4 below.

3. Proof of the Proposition

Lemma 4**.**

Let $u_{1}\geq\cdots\geq u_{8}\geq 0$, $u_{1}+\cdots+u_{8}\leq 1$. Then either

(a) there is a partition $\{i\}$, $\mathcal{A}_{1}$, $\mathcal{A}_{2}$ of $\{1,\ldots,8\}$ with $\max(|\mathcal{A}_{1}|),|\mathcal{A}_{2}|)\leq 5$,

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or

(b) there is a partition $\mathcal{A}_{1}$, $\mathcal{A}_{2}$ of $\{1,\ldots,8\}$ with $\max(|\mathcal{A}_{1}|,|\mathcal{A}_{2}|)\leq 6$,

[TABLE]

Proof.

If $u_{1}+\cdots+u_{5}\leq 11/20$ we have (b) with $\mathcal{A}_{1}=\{1,\ldots,5\}$ since $u_{6}+u_{7}+u_{8}\leq\frac{3}{8}$. Assume $u_{1}+\cdots+u_{5}>11/20$. Let $k$ be the least integer such that

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One of the following cases must occur.

1. (i)

$u_{1}\not\in(9/40,1/4)$ , $u_{2}+u_{4}+u_{6}+u_{8}>9/20$. 2. (ii)

$u_{1}\not\in(9/40,1/4)$ , $u_{2}+u_{4}+u_{6}+u_{8}\leq 9/20$. 3. (iii)

$u_{1}\in(9/40,1/4)$ , $u_{1}+\cdots+u_{k}\leq 11/20$. 4. (iv)

$u_{1}\in(9/40,1/4)$ , $u_{1}+\cdots+u_{k}>11/20$.

In Case (i) we have

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and (b) holds with $\mathcal{A}_{1}=\{2,4,6,8\}$.

In Case (ii) we have $u_{3}+u_{5}+u_{7}\leq u_{2}+u_{4}+u_{6}$,and (a) holds with $i=1$, $\mathcal{A}_{1}=\{3,5,7\}$.

In Case (iii), (b) holds with $\mathcal{A}_{1}=\{1,\ldots,k\}$.

In Case (iv), we have $k\geq 3$. Now (a) holds with $i=2$, $\mathcal{A}_{1}=\{1,3,\ldots,k\}$. For $u_{2}\leq\frac{u_{1}+u_{2}}{2}<9/40$ and $u_{1}+u_{3}+\cdots+u_{k}\leq u_{1}+u_{2}+\cdots+u_{k-1}<9/20$. ∎

Proof of the Proposition.

In place of (2.4), it clearly suffices to show that, with $J_{\chi}$ as in Lemma 1,

[TABLE]

We reorder $N_{1},\ldots,N_{8}$ so that $N_{1}\geq\cdots\geq N_{8}$ and write $N_{j}=x^{u_{j}}$ with $u_{1}\geq\cdots\geq u_{8}\geq 0$, $u_{1}+\cdots+u_{8}\leq 1$. Suppose we are in Case (b) of Lemma 4. Let us write

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We bound $E$ using Cauchy’s inequality and (2.6), (2.11) for $S=S_{1},S_{2}$:

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Now

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This is acceptable in (3.1)

Suppose now we are in Case (a) of Lemma 4. The argument mimics one due to Iwaniec [4]. We retain the notation (3.2), and write

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The contribution to $E$ from those $t$ with

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is at most

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By a simple splitting-up argument, there is a subset $\mathcal{B}$ of the set of triples $(q,\chi,t)$, $q<2Q$, $\chi\pmod{q}$, $t\in J_{\chi}$ in (3.1) such that, for $(q,\chi,t)\in\mathcal{B}$, we have

[TABLE]

for positive numbers $U$, $V$, $W$ with

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while

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Here

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and we have used (2.6), (2.7), (2.10), (2.11) in the second step in (3.3). It remains to show that

[TABLE]

We consider four cases.

Case 1. $P\leq\dfrac{2M}{V^{2}}\,\mathcal{L}^{(7-b)^{2}+5}$, $P\leq\dfrac{2N}{W^{2}}\,\mathcal{L}^{b^{2}+5}$.

In this case

[TABLE]

Case 2. $P>\dfrac{2M}{V^{2}}$ $\mathcal{L}^{(7-b)^{2}+5}$, $P>2\,\dfrac{N}{W^{2}}$ $\mathcal{L}^{b^{2}+5}$. In this case,

[TABLE]

where

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and

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We have, for a constant $K_{1}$,

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We bound the last minimum by $(x^{-9/80}T^{-1/4}L^{1/2})^{1/8}$, obtaining

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which is acceptable in (3.4). Now

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where

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We bound the last minimum by $\mathcal{L}^{22-3/40}(T^{1/4}L^{1/6}(MN)^{-1/24})^{3/10}$, obtaining

[TABLE]

which is acceptable in (3.4).

Case 3. $P>2V^{-2}M\mathcal{L}^{(7-b)^{2}+5}$, $P\leq 2W^{-2}N\mathcal{L}^{b^{2}+5}$. In this case, for a constant $K_{4}$,

[TABLE]

where

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Now

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We bound the last minimum by $(x^{-9/80}T^{-1/4}L^{1/2})^{1/4}$, obtaining

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(using $N\leq x^{9/20}$), which is acceptable. Further,

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We bound the last minimum by $(L^{1/6}M^{-1/12})^{1/2}$. Similarly to the bound for $A_{2}$,

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which is acceptable.

Case 4. $P\leq 2V^{-2}M\mathcal{L}^{(7-b)^{2}+5}$, $P>2W^{-2}N\mathcal{L}^{b^{2}+5}$. We proceed as in Case 3, interchanging the roles of $S_{1}$ and $S_{2}$.

This establishes (3.4) and completes the proof of the Proposition. ∎