Uniform bounds for the number of powers in arithmetic progressions

Abstract

We give sharp, in some sense uniform bounds for the number of $\ell$-th powers and arbitrary powers among the first N terms of an arithmetic progression, for N large enough.

1 Introduction

The problem of giving (sharp) upper bounds for the number of powers among N consecutive terms of an arithmetic progression is a classical one with many deep results and open problems and conjectures. Here we only give a brief introduction; for a more precise account of the topic the interested reader may consult e.g. the papers [1] and [6].

Let $a,b,\ell$ be integers with $a>0$ and let $\ell \ge 2$. Write $P_{a,b;N}(\ell )$ for the number of $\ell$-th powers among the first N terms of the arithmetic progression $ax+b$ $(x\ge 0)$. Denote by $P_N(\ell )$ the maximum of these values taken over all arithmetic progressions $ax+b$. (Note that this maximum obviously exists.) The case of squares (i.e, $\ell =2$) has been studied by many authors. Erdős [3] conjectured and Szemerédi [10] proved that $P_N(2)=o(N)$. Later, by deep tools (such as e.g. elliptic and higher genus curves, Falting’s theorem, the distribution of primes etc.) Bombieri, Granville and Pintz [1] proved $P_N(2)<O(N^{2/3+o(1)})$, which subsequently was improved to $P_N(2)<O(N^{3/5+o(1)})$ by Bombieri and Zannier [2]. See also Granville [5] for related results and remarks. A strong conjecture of Rudin (see [9], end of paragraph 4.6) predicts that $P_N(2)=O(\sqrt{N})$, or in an even more precise form, that

$$\begin{aligned} P_N(2)=P_{24,1;N}(2)=\sqrt{\frac{8}{3}N}+O(1)\ \ \ (N\ge 6) \end{aligned}$$

(1)

should hold.

In case $\ell \ge 3$ there is hardly anything known. The authors of [1] noted (without proof) that their methods probably make it possible to prove $P_N(3)\ll N^{3/5+\varepsilon }$ and $P_N(\ell )\ll N^{1/2+\varepsilon }$ $(\ell \ge 4)$. Hajdu and Tengely [6] showed that (up to equivalence) for any $\ell \ge 2$ there is a unique arithmetic progression $ax+b$ which contains the most $\ell$-th powers asymptotically, that is, which maximizes the expression

$$\begin{aligned} \lim \limits _{N\rightarrow \infty }\frac{|\{x:ax+b\ \text {is an}\ \ell \text {-th power},\ 0\le x<N\}|}{\root \ell \of {N}}. \end{aligned}$$

(In fact, for $\ell =4$ there are two such progressions.) They could describe these arithmetic progressions $a_\ell x+b_\ell$ explicitly. Based upon their results, they extended Rudin’s conjecture (1) for any $\ell \ge 2$ (by replacing $24x+1$ by $a_\ell x+b_\ell$ and changing the right hand side accordingly), and proved that for $\ell =3,4$ for certain small values of N. Note that this asymptotic (’global’) version of the problem is simpler than the original ’local’ one, namely when we concentrate on a finite part of the progressions. The reason is that the asymptotic approach brings in an ’averaging’ effect, which roughly speaking makes it possible to concentrate on a complete (finite) period of a progression $ax+b$ modulo a.

In this note we prove that for any positive $\varepsilon$ there is an $\ell _0$ depending only on $\varepsilon$ such that for $\ell >\ell _0$ the number of $\ell$-th powers among the first N terms of any integral arithmetic progression is below $(1+\varepsilon )\root \ell \of {N}$, provided that N is large enough in terms of $\varepsilon ,\ell$ and the parameters of the progression. The important feature of $\ell _0$ is that it is uniform in the sense that it depends only on $\varepsilon$, it is independent of the progression. This result is sharp in the sense that for infinitely many $\ell$, one can find a constant $c_1=c_1(\ell )>1$ and an arithmetic progression having more than $c_1\root \ell \of {N}$ $\ell$-th powers among its first N terms, for all N large enough. We also give a uniform, sharp upper bound for the number of powers (with not fixed exponents) among the first N terms of arithmetic progressions. In our proofs we combine a classical result of Wigert [11] concerning the number of divisors of positive integers, a recent result of Hajdu and Tengely [6] concerning arithmetic progressions containing the most $\ell$-th powers asymptotically, and a new assertion answering a question of Hajdu and Tengely from [6].

2 New results

Now we give our main results. We use the notation from the introduction.

Theorem 2.1

For every $\varepsilon >0$ there is an $\ell _0$ depending only on $\varepsilon$ such that for any $\ell >\ell _0$ we have $P_{a,b;N}(\ell )\le (1+\varepsilon )\root \ell \of {N}$, whenever $N>N_0$. Here $N_0=N_0(\varepsilon ,\ell ,a,b)$ depends on $\varepsilon ,\ell ,a,b$.

Remarks. The above theorem is sharp in the sense that $1+\varepsilon$ cannot be replaced by 1, and $\ell >\ell _0$ is also necessary. Indeed, Theorem 1 of [6] (see also the Remarks after it) implies that for infinitely many exponents $\ell \ge 2$ there exists a $\delta _\ell >0$ and an arithmetic progression $a_\ell x+b_\ell$ with $P_{a_\ell ,b_\ell ;N}(\ell )>(1+\delta _\ell )\root \ell \of {N}$ for all $N>N_0$. Here $N_0=N_0(\ell )$ depends only on $\ell$.

It is clear that if an arithmetic progression $ax+b$ contains an $\ell$-th power then it contains infinitely many, and we have

$$\begin{aligned} P_{a,b;N}(\ell )>\frac{1}{2a}\root \ell \of {N} \end{aligned}$$

for $N>N_0$, where $N_0$ depends on a, b.

We also mention that on our way to prove Theorem 2.1, we answer a question of Hajdu and Tengely [6] (see Proposition 3.1).

We also give a uniform upper bound for the number of powers in arithmetic progressions. For this, let $P_{a,b;N}(*)$ denote the number of (arbitrary) powers among the first N terms of the arithmetic progression $ax+b$ $(x\ge 0)$.

Theorem 2.2

Let $ax+b$ $(x\ge 0)$ be an arithmetic progression. Then for any $\varepsilon >0$ there exists an $N_0$ such that

$$\begin{aligned} P_{a,b;N}(*)<\left( \sqrt{\frac{8}{3}}+\varepsilon \right) \sqrt{N} \end{aligned}$$

(2)

for any $N>N_0$. Here $N_0=N_0(\varepsilon ,a,b)$ depends only on $\varepsilon ,a,b$.

Remark

One can easily check (see also e.g. Theorem 1 of [6]) that

$$\begin{aligned} \lim \limits _{N\rightarrow \infty } \frac{P_{24,1;N}(2)}{\sqrt{N}}=\sqrt{\frac{8}{3}}. \end{aligned}$$

This shows that the above result is sharp.

Further, it is also easy to see that if $\gcd (a,b)=1$ then there exist infinitely many exponents $\ell$ such that $ax+b$ contains $\ell$-th powers. Note that here the condition $\gcd (a,b)=1$ cannot be dropped: for example, the arithmetic progression $4x+2$ $(x\ge 0)$ contains no powers at all.

3 Proofs

To prove Theorem 2.1 we shall need some known and new assertions. The next lemma is a result of Hajdu and Tengely [6]. For its formulation, we need to introduce some new notions and notation (which will play important roles also later on). For any $\ell \ge 2$ and arithmetic progression $ax+b$ put

$$\begin{aligned} M_{a,b}(\ell ):=|\{u:0\le u<a,\ u^\ell \equiv b\pmod {a}\}| \end{aligned}$$

and $S_{a,b}(\ell ):=M_{a,b}(\ell )a^{\frac{1}{\ell }-1}$.

Lemma 3.1

For any $\ell \ge 2$ and for any arithmetic progression $ax+b$ we have $S_{a,b}(\ell )\le S(\ell )$, where

$$\begin{aligned} S(\ell )= {\left\{ \begin{array}{ll} \sqrt{\frac{8}{3}},&{}\quad \text {if}\ \ell =2,\\ \underset{\frac{\log p}{\log p-\log (p-1)}>\ell }{\prod \limits _{p\ \text {prime},\ p-1\mid \ell ,}} (p-1)p^{\frac{1}{\ell }-1},&{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Proof

The statement is the first half of Theorem 1 of [6]; see also the notation in its proof on p. 970 of [6]. $\square$

Remark

The inequality $S_{a,b}(\ell )\le S(\ell )$ is sharp: for any $\ell$, by an appropriate choice of $ax+b$ (given in [6]) we get equality. Observe that for $\ell$ odd, we have $S(\ell )=1$.

In the proofs of Theorems 2.1 and 2.2 we shall need the following new assertion. This answers a question of Hajdu and Tengely concerning the limit of the sequence $S(\ell )$ (see the ’concrete question’ on p. 966 in the Remarks after Theorem 1 in [6]), and we find it of possible independent interest.

Proposition 3.1

By the notation of Lemma 3.1, for any $\gamma >0$ there exists an $\ell _1=\ell _1(\gamma )$ depending only on $\gamma$ such that for $\ell >\ell _1$ we have

$$\begin{aligned} S(\ell )<\exp \left( \ell ^{-1+\gamma }\right) . \end{aligned}$$

(3)

In particular, $\lim \limits _{\ell \rightarrow \infty } S(\ell )=1$ holds.

Remark

One can easily check that (3) implies that

$$\begin{aligned} S(\ell )<1+2\ell ^{-1+\gamma } \end{aligned}$$

for $\ell$ large enough.

To prove the above statement, we need the next classical theorem concerning the number of divisors d(n) of a positive integer n.

Lemma 3.2

If $\varepsilon >0$, $X>X_0(\varepsilon )$ then we have

$$\begin{aligned} \max \limits _{n\le X} d(n) <\exp \left( (\log 2+\varepsilon )\frac{\log X}{\log \log X}\right) . \end{aligned}$$

Proof

This is a classical result of Wigert [11]. Note that in [7], p. 56 a stronger form of this assertion is given, however, the above inequality is sufficient for our present purposes. $\square$

Proof of Proposition 3.1

As one can easily check by a direct calculation, the function $(t-1)t^{1/\ell -1}$ is strictly monotone increasing for $t>0$, for any fixed $\ell \ge 3$. Thus, as for $\ell \ge 3$ the product appearing in $S(\ell )$ has at most $d(\ell )$ terms and in every term $p\le \ell +1$ holds, we have

$$\begin{aligned} 1\le S(\ell ) \le \left( \ell (\ell +1)^{1/\ell -1}\right) ^{d(\ell )}<\left( \root \ell \of {\ell } \right) ^{d(\ell )}. \end{aligned}$$

Here we also used that by the condition $\frac{\log p}{\log p-\log (p-1)}>\ell$, the terms appearing in $S(\ell )$ are greater than 1. Now by Lemma 3.2 we get that

$$\begin{aligned} S(\ell )< & {} \exp \left( \frac{d(\ell )\log \ell }{\ell }\right)<\exp \left( \frac{\exp \left( {\frac{\log \ell }{\log \log \ell }}\right) \log \ell }{\ell }\right) \\= & {} \exp \left( \exp \left( {\frac{\log \ell }{\log \log \ell }+\log \log \ell -\log \ell }\right) \right) <\exp (\ell ^{-1+\gamma }) \end{aligned}$$

hold, for any $\gamma >0$ with $\ell >\ell _1$, where $\ell _1=\ell _1(\gamma )$ depends only on $\gamma$. Thus the first part of the statement is proved. The second part of the claim, taking any $\gamma$ with $0<\gamma <1$, from this immediately follows. $\square$

Now we can give the

Proof of Theorem 2.1

To bound $P_{a,b;N}(\ell )$, we need to give an upper bound for the number of $\ell$-th powers among the numbers

$$\begin{aligned} b,a+b,\dots ,a(N-1)+b. \end{aligned}$$

In view of that $N_0$ depends on a, b, we may assume that $a(N-1)+b\ge 0$. An $\ell$-th power $u^\ell$ belongs to the above terms if its size is ’between’ b and $a(N-1)+b$, and $u^\ell \equiv b\pmod {a}$. Thus we see that

$$\begin{aligned} P_{a,b;N}(\ell )\le \left( \root \ell \of {aN+|b|}+\root \ell \of {|b|}\right) \frac{M_{a,b}(\ell )}{a}+M_{a,b}(\ell ). \end{aligned}$$

Here the term in brackets on the right hand side provides an upper bound for the number of (consecutive) integers (forming an interval I) with $\ell$-th power of the ’appropriate’ size, the factor $M_{a,b}(\ell )/a$ is the ratio of $\ell$-th powers in the residue class $b \pmod {a}$, while the last term is to bound the number of possible $\ell$-th powers in the progression coming from the last part of I (having less than a elements). This yields

$$\begin{aligned} P_{a,b;N}(\ell )\le M_{a,b}(\ell )a^{\frac{1}{\ell }-1}\root \ell \of {N}\left( \root \ell \of {1+\frac{|b|}{aN}}+\root \ell \of {\frac{|b|}{aN}}+\frac{a}{\root \ell \of {aN}}\right) . \end{aligned}$$

(4)

Let $\varepsilon >0$ arbitrary. Clearly, there exists an $N_1=N_1(\varepsilon ,\ell ,a,b)$ depending on $\varepsilon ,\ell ,a,b$ such that for $N>N_1$ we have

$$\begin{aligned} \root \ell \of {1+\frac{|b|}{aN}}+\root \ell \of {\frac{|b|}{aN}}+\frac{a}{\root \ell \of {aN}}<1+\frac{\varepsilon }{2}. \end{aligned}$$

By Lemma 3.1 this together with (4) implies

$$\begin{aligned} P_{a,b;N}(\ell )<\left( 1+\frac{\varepsilon }{2}\right) S(\ell )\root \ell \of {N}. \end{aligned}$$

(5)

In view of Proposition 3.1 we can take an $\ell _0$ such that

$$\begin{aligned} S(\ell )<\frac{2+2\varepsilon }{2+\varepsilon } \end{aligned}$$

for $\ell >\ell _0$. This by (5) yields that

$$\begin{aligned} P_{a,b;N}(\ell )<(1+\varepsilon )\root \ell \of {N} \end{aligned}$$

under the assumptions made for $\ell$ and N. Hence our claim follows. $\square$

Now we give the

Proof of Theorem 2.2

Throughout the proof we use the phrase ’N is large enough’ to express that N is larger than an appropriate bound depending only on $\varepsilon ,a,b$.

Combining (4) and Lemma 3.1 we obtain

$$\begin{aligned} P_{a,b;N}(2)<\left( \sqrt{\frac{8}{3}}+\frac{\varepsilon }{2}\right) \sqrt{N} \end{aligned}$$

(6)

for $\ell =2$ and

$$\begin{aligned} P_{a,b;N}(\ell )<S(\ell )(a+3)\root \ell \of {N} \end{aligned}$$

for $\ell \ge 3$, respectively, for N large enough. In view of Proposition 3.1, the latter assertion implies that there exists an absolute constant C such that

$$\begin{aligned} P_{a,b;N}(\ell )<C(a+3)\root \ell \of {N} \end{aligned}$$

(7)

for any $\ell \ge 3$, for N large enough. Further, if N is large enough then we have $aN+b\ge |b|$. Hence if $u^\ell$ with $|u|>1$ belongs to $ax+b$ $(0\le x<N)$ then we have

$$\begin{aligned} \ell \le \frac{\log (aN+|b|)}{\log 2}. \end{aligned}$$

(If $u\in \{-1,0,1\}$, then we may assume that $\ell \le 3$.) This together with (6) and (7) gives

$$\begin{aligned} P_{a,b;N}(*)\le & {} \sum \limits _{2\le \ell \le \frac{\log (aN+|b|)}{\log 2}}P_{a,b;N}(\ell )=P_{a,b;N}(2)+\sum \limits _{3\le \ell \le \frac{\log (aN+|b|)}{\log 2}}P_{a,b;N}(\ell )\\< & {} \left( \sqrt{\frac{8}{3}}+\frac{\varepsilon }{2}\right) \sqrt{N}+C(a+3)\frac{\log (aN+|b|)}{\log 2}\root 3 \of {N}<\left( \sqrt{\frac{8}{3}}+\varepsilon \right) \sqrt{N} \end{aligned}$$

for N large enough. This proves the statement. $\square$