Primes in arithmetic progressions and nonprimitive roots

TL;DR

This paper investigates the distribution of primes in arithmetic progressions related to near primitive roots, extending known results about primitive roots and their subgroup indices in modular arithmetic.

Contribution

It introduces the concept of t-near primitive roots and proves a new, more complex result about their distribution among primes.

Findings

Each primitive residue class contains a positive density subset of primes without a given t-near primitive root.

Established a density result for primes in arithmetic progressions related to t-near primitive roots.

Extended classical results on primitive roots to the broader context of t-near primitive roots.

Abstract

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a positive natural density subset of primes $p$ not having $g$ as a $t$-near primitive root and prove a more difficult variant.

Equations

$$\delta (S)=\underset{x\to \infty }{\lim }\frac{\#\{p:p\in S,p\le x\}}{\#\{p:p\le x\}},$$

$$A=\prod _p\left(1-\frac{1}{p(p-1)}\right)=0.373955\dots .$$

$${\mathcal{P}}_g(t)=\{p:p\nmid g,p\equiv 1(\mathrm{mod}t),{\mathrm{ord}}_p(g)=(p-1)/t\}.$$

$${\mathcal{P}}_g(t,d,a)=\{p:p\equiv a(\mathrm{mod}d),p\in {\mathcal{P}}_g(t)\}.$$

$$\#\{p:p\le x,p\equiv a(\mathrm{mod}d)\}\sim \frac{x}{\phi (d)\log x},$$

$$\underset{x\to \infty }{\mathrm{limsup}}\frac{\#\{p\le x:p\in {\mathcal{P}}_g(t,d,a)\}}{\pi (x)}\le {\delta }_G({\mathcal{P}}_g(t,d,a)),$$

$${\mathcal{Q}}_g(t,d,a)=\{p:p\equiv a(\mathrm{mod}d),p\notin {\mathcal{P}}_g(t)\}$$

$$\delta ({\mathcal{Q}}_g(t,d,a))=1/\phi (d)-\delta ({\mathcal{P}}_g(t,d,a)).$$

$$\underset{x\to \infty }{\mathrm{liminf}}\frac{\#\{p\le x:p\in {\mathcal{Q}}_g(t,d,a)\}}{\pi (x)}\ge \frac{1}{\phi (d)}-{\delta }_G({\mathcal{P}}_g(t,d,a)).$$

$$t\mid d\text{and}t\mid (a-1).$$

$$\frac{1}{[\mathbb{Q}({\zeta }_d,{\zeta }_q,g^{1/q}):\mathbb{Q}]}.$$

$${\mathcal{R}}_g(t,d,a)=\{p:p\nmid g,p\equiv a(\mathrm{mod}d),p\equiv 1(\mathrm{mod}t),{\mathrm{ord}}_p(g)\mid (p-1)/t\}.$$

$$\frac{1}{[\mathbb{Q}({\zeta }_d,{\zeta }_{qt},g^{1/qt}):\mathbb{Q}]}.$$

$$\delta ({\mathcal{R}}_g(qt,d,a))=\frac{1}{[\mathbb{Q}({\zeta }_d,{\zeta }_{qt},g^{1/qt}):\mathbb{Q}]}.$$

$$17,31,37,41,43,59,67,73,89,97,101,103,109,113,127.$$

$$\frac{1}{\phi (d)q(q-1)}.$$

$$\tau_{n}\big{|}_{{\mathbb{Q}}(\zeta_{d})}=\sigma_{a},\qquad\tau_{n}\big{|}_{{\mathbb{Q}}(\zeta_{n},g^{1/n})}=\text{id},$$

$$\tau_{n}\big{|}_{{\mathbb{Q}}(\zeta_{d})\cap{\mathbb{Q}}(\zeta_{n},g^{1/n})}=\text{id}.$$

$$\mathbb{Q}({\zeta }_d)\cap \mathbb{Q}({\zeta }_n,g^{1/n})=\mathbb{Q},$$

$$\mathbb{Q}({\zeta }_d)\cap \mathbb{Q}({\zeta }_q,g^{1/q})=\mathbb{Q}({\zeta }_d)\cap \mathbb{Q}({\zeta }_q)=\mathbb{Q}({\zeta }_{\gcd (d,q)})=\mathbb{Q},$$

$$[I_{qt}:\mathbb{Q}]=\frac{[\mathbb{Q}({\zeta }_d):\mathbb{Q}]\cdot [\mathbb{Q}({\zeta }_{qt},g^{1/qt}):\mathbb{Q}]}{[\mathbb{Q}({\zeta }_d,{\zeta }_{qt},g^{1/qt}):\mathbb{Q}]}=\frac{\phi (d)[\mathbb{Q}({\zeta }_{qt},g^{1/qt}):\mathbb{Q}]}{[\mathbb{Q}({\zeta }_{qrt},g^{1/qt}):\mathbb{Q}]},$$

$$I_{qt}=I_t.$$

Full text

1. Introduction

1.1. Background

Given a set of primes $S$, the limit

[TABLE]

if it exists, is called the natural density of $S$. (Here and in the sequel the letter $p$ is used to denote a prime number.)

For any integer $g\not\in\{-1,0,1\}$, let ${\mathcal{P}}_{g}$ be the set of primes $p$ such that $g$ is a primitive root modulo $p$, that is $p\nmid g$ and the multiplicative order of $g$ modulo $p,$ $\mathrm{ord}_{p}(g),$ equals $p-1=\#(\mathbb{Z}/p\mathbb{Z})^{*},$ and so $g$ is a generator of $(\mathbb{Z}/p\mathbb{Z})^{*}.$ In 1927, Emil Artin conjectured that the set ${\mathcal{P}}_{g}$ is infinite if $g$ is not a square; moreover he also gave a conjectural formula for its natural density $\delta({\mathcal{P}}_{g})$; see [12] for more details. There is no explicit value of $g$ known for which ${\mathcal{P}}_{g}$ can be unconditionally proved to be infinite. However Heath-Brown [3], building on earlier fundamental work by Gupta and Murty [2], showed that, given any three distinct primes $p_{1},p_{2}$ and $p_{3},$ there is at least one $i$ such that ${\mathcal{P}}_{p_{i}}$ is infinite.

In 1967, Hooley [4] established Artin’s conjecture under the Generalized Riemann Hypothesis (GRH) and determined $\delta({\mathcal{P}}_{g})$. Ten years later, Lenstra [7] considered a wide class of generalizations of Artin’s conjecture. For example, under GRH he showed that the primes in ${\mathcal{P}}_{g}$ that are in a prescribed arithmetic progression have a natural density and gave a Galois theoretic formula for it. This was worked out explicitly by the first author [9, 11], who showed that $\delta({\mathcal{P}}_{g})=r_{g}A,$ with $r_{g}$ an explicit rational number and the Artin constant

[TABLE]

Using a powerful and very general algebraic method, this result was rederived in a very different way by Lenstra et al. [8].

For any integer $t\geq 1$, let

[TABLE]

If $p$ is in ${\mathcal{P}}_{g}(t)$, then it is said to have $g$ as a $t$-near primitive root. Assuming GRH, the first author [13] determined $\delta({\mathcal{P}}_{g}(t))$ in case $g>1$ is square-free.

A more refined problem is how the primes in ${\mathcal{P}}_{g}(t)$ are distributed over arithmetic progressions. To this end, let $a,d\geq 1$ be coprime integers and define

[TABLE]

By the prime number theorem for arithmetic progressions,

[TABLE]

where $\varphi$ denotes Euler’s totient function. A straightforward combination of the ideas used in the study of near-primitive roots and those for primitive roots in arithmetic progression, allows one to show, assuming GRH, that $\delta({\mathcal{P}}_{g}(t,d,a))$ exists and derive a Galois theoretic expression $\delta_{G}({\mathcal{P}}_{g}(t,d,a))$ for it (see Hu et al. [6, Theorem 3.1]). Moreover, it can be unconditionally shown (see [6, Equation (3.7)]) that

[TABLE]

where as usual $\pi(x)$ denotes the prime counting function. The idea of the proof is to apply the simple asymptotic sieve up to a range in which the unconditional Chebotarev density theorem is valid.

On the basis of insights from [8], we know that $\delta_{G}({\mathcal{P}}_{g}(t,d,a))$ equals a rational multiple of the Artin constant $A,$ where the rational multiple can be worked out in full generality. However, this is likely to produce a result involving several case distinctions (as in the restricted case where $t=1$ and in the case where $t$ is arbitrary and $g$ is square-free). In the much less general case $g=4$ and $t=2$, the expression was explicitly worked out in [6]; see Section 1.3 for more background.

1.2. Our considerations

In this paper we study, motivated by the following questions, the distribution of primes not having a prescribed near-primitive root in arithmetic progressions.

Questions 1.1**.**

Let $t\geq 1$ and $g\not\in\{-1,0,1\}$ be integers. Let $a,d$ be positive coprime integers.

(A) Is the set

[TABLE]

infinite?

(B) Does the set ${\mathcal{Q}}_{g}(t,d,a)$ have a natural density and can it be computed? **

Since ${\mathcal{P}}_{g}(t,d,a)\cup{\mathcal{Q}}_{g}(t,d,a)=\{p:\,p\equiv a~{}({\rm mod~{}}d)\}$, if $\delta({\mathcal{P}}_{g}(t,d,a))$ exists, then using (1.1) we have

[TABLE]

Question B can currently be answered only assuming GRH. However, in this approach it is far from evident under which conditions on the parameters $g,t,d$ and $a$ we have $\delta({\mathcal{Q}}_{g}(t,d,a))>0$, thus guaranteeing the infinitude of the set ${\mathcal{Q}}_{g}(t,d,a).$

Unconditionally using (1.2) we infer that

[TABLE]

If there exists a prime $p_{0}\nmid t$ satisfying both $p_{0}\equiv a~{}({\rm mod~{}}d)$ and $p_{0}\not\equiv 1~{}({\rm mod~{}}t)$, then all the primes $p\equiv p_{0}~{}({\rm mod~{}}dt)$ are in ${\mathcal{Q}}_{g}(t,d,a)$ (due to $t\nmid(p-1)$). By (1.1), there are infinitely many primes $p\equiv p_{0}~{}({\rm mod~{}}dt)$, and they have a positive natural density. Thus, Question A is only nontrivial when $p\equiv a~{}({\rm mod~{}}d)$ implies $p\mid t$ or $p\equiv 1~{}({\rm mod~{}}t),$ which is true if and only if

[TABLE]

In this note we will see that answering Question A is actually also rather easy in case (1.3) is satisfied. The answer to Question A is yes, and we can be even a little bit more precise on using Kummerian extensions of cyclotomic number fields $\mathbb{Q}(\zeta_{n})$ with $\zeta_{n}=e^{2\pi i/n}$.

Proposition 1.2**.**

Let $g\not\in\{-1,0,1\}$ and $t\geq 1$ be integers. Let $a,d$ be positive coprime integers. Then, for any integer $q>2$ coprime to $2dt$, the set ${\mathcal{Q}}_{g}(t,d,a)$ contains a positive natural density subset of primes $p$ having natural density

[TABLE]

The field degree $[{\mathbb{Q}}(\zeta_{d},\zeta_{q},g^{1/q}):{\mathbb{Q}}]=[{\mathbb{Q}}(\zeta_{\text{lcm}(d,q)},g^{1/q}):{\mathbb{Q}}]$ is not difficult to compute for any given $g,d$ and $q$; see [10, Lemma 1] for the general result (which is a direct consequence of [15, Proposition 4.1]). Using this computation the maximum density of the $q$-dependent subsets arising in Proposition 1.2 can be determined; see the next section for an example. If $\ell$ is a prime factor of $q$, then ${\mathbb{Q}}(\zeta_{d},\zeta_{\ell},g^{1/\ell})\subseteq{\mathbb{Q}}(\zeta_{d},\zeta_{q},g^{1/q})$, and so a priori the maximum occurs in an odd prime.

We will also establish a more difficult variant of Proposition 1.2. Letting $g,t,d,a$ be as in Proposition 1.2, we define the set

[TABLE]

Clearly, we have ${\mathcal{P}}_{g}(t,d,a)\subseteq{\mathcal{R}}_{g}(t,d,a)$. Our purpose is to show that if ${\mathcal{R}}_{g}(t,d,a)$ is not empty, then ${\mathcal{R}}_{g}(t,d,a)$ contains a positive density subset of primes not contained in ${\mathcal{P}}_{g}(t,d,a)$.

Theorem 1.3**.**

Let $g\not\in\{-1,0,1\}$ and $t\geq 1$ be integers. Let $a,d$ be positive coprime integers. Suppose the set ${\mathcal{R}}_{g}(t,d,a)$ is not empty. Then, for any integer $q>2$ coprime to $2dgt$, the set ${\mathcal{R}}_{g}(t,d,a)$ contains a subset of primes $p$ for which $g$ is a non $t$-near primitive root modulo $p$ having natural density

[TABLE]

Again, given $d,g$ and $t,$ the maximum density of the $q$-dependent subsets arising in the theorem can be determined, and for this it suffices to consider primes $q\nmid 2dgt$.

Note that for any integer $q\geq 2$, each prime in ${\mathcal{R}}_{g}(qt,d,a)$ is not contained in ${\mathcal{P}}_{g}(t,d,a)$. So, Theorem 1.3 is derived directly from the following proposition, which might be of independent interest.

Proposition 1.4**.**

Let $g\not\in\{-1,0,1\}$ and $t\geq 1$ be integers. Let $a,d$ be positive coprime integers. Suppose the set ${\mathcal{R}}_{g}(t,d,a)$ is not empty. Then, for any positive integer $q$ coprime to $2dgt$, we have

[TABLE]

1.3. An application

Proposition 1.2 has an application to Genocchi numbers $G_{n},$ which are defined by $G_{n}=2(1-2^{n})B_{n},$ where $B_{n}$ is the $n^{\text{th}}$ Bernoulli number. The Genocchi numbers are actually integers. As introduced in [6], if a prime $p>3$ divides at least one of the Genocchi numbers $G_{2},G_{4},\ldots,G_{p-3},$ it is said to be $G$-irregular and $G$-regular otherwise. The first fifteen G-irregular primes [1] are

[TABLE]

The G-regularity of primes can be linked to the divisibility of certain class numbers of cyclotomic fields. Let $S$ be the set of infinite places of ${\mathbb{Q}}(\zeta_{p})$ and $T$ the set of places above the prime 2. Denote by $h_{p,2}$ the $(S,T)$-refined class number of ${\mathbb{Q}}(\zeta_{p})$ and $h_{p,2}^{+}$ be the refined class number of ${\mathbb{Q}}(\zeta_{p}+\zeta_{p}^{-1})$ with respect to its infinite places and places above the prime 2 (for the definition of the refined class number of global fields, see for example Hu and Kim [5, Section 2]). Define $h_{p,2}^{-}=h_{p,2}/h_{p,2}^{+}.$ It turns out that $h_{p,2}^{-}$ is an integer (see [5, Proof of Proposition 3.4]). Recall that a Wieferich prime is an odd prime $p$ such that $2^{p-1}\equiv 1~{}({\rm mod~{}}p^{2}).$

Theorem 1.5**.**

[6, Theorem 1.5]*.

Let $p$ be an odd prime. Then, if $p$ is G-irregular, we have $p\mid h_{p,2}^{-}$. If furthermore $p$ is not a Wieferich prime, the converse is also true.*

It is easy to show that if $\mathrm{ord}_{p}(4)\neq(p-1)/2,$ then $p$ is G-irregular; see [6, Theorem 1.6]. Hence, taking $g=4$ and $t=2$ in Proposition 1.2 and noting that we have $[{\mathbb{Q}}(\zeta_{d},\zeta_{q},4^{1/q}):{\mathbb{Q}}]=\varphi(d)q(q-1)$ for any prime $q\nmid 2d$, we arrive at the following result.

Proposition 1.6**.**

Let $a,d$ be positive coprime integers. Let $q$ be the smallest prime not dividing $2d$. The set of G-irregular primes $p$ satisfying $p\equiv a~{}({\rm mod~{}}d)$ contains a subset having natural density

[TABLE]

This result is a weaker version of Theorem 1.11 in [6], however its proof is much more elementary, and it still shows that each coprime residue class contains a subset of G-irregular primes having positive natural density.

2. Preliminaries

Given any integers $d,n\geq 1$ put $K_{n}={\mathbb{Q}}(\zeta_{d},\zeta_{n},g^{1/n}).$ For $a$ coprime to $d,$ let $\sigma_{a}$ be the endomorphism of ${\mathbb{Q}}(\zeta_{d})$ over ${\mathbb{Q}}$ defined by $\sigma_{a}(\zeta_{d})=\zeta_{d}^{a}$. Let $C_{n}$ be the conjugacy class of elements of the Galois group $G_{n}={\mathrm{Gal}}(K_{n}/{\mathbb{Q}})$ such that for any $\tau_{n}\in C_{n}$,

[TABLE]

where ‘id’ stands for the identity map. Note that either $C_{n}$ is empty, or $C_{n}$ is nonempty and $|C_{n}|=1$. The latter case occurs if and only if

[TABLE]

If this condition is satisfied, then by the Chebotarev density theorem (in its natural density form, cf. Serre [14], the original form being for Dirichlet density), the primes unramified in $K_{n}$ and with Frobenius $C_{n}$ have natural density $1/[K_{n}:{\mathbb{Q}}]$. Note that the primes unramified in $K_{n}$ are exactly the primes $p\nmid dgn$. The first condition on $\tau_{n}$ ensures that the primes $p\nmid dgn$ having $\tau_{n}$ as Frobenius satisfy $p\equiv a~{}({\rm mod~{}}d)$. Likewise the second condition ensures that such primes satisfy $\mathrm{ord}_{p}(g)\mid(p-1)/n$.

In particular, in case ${\mathbb{Q}}(\zeta_{d})$ and ${\mathbb{Q}}(\zeta_{n},g^{1/n})$ are linearly disjoint over ${\mathbb{Q}}$, that is,

[TABLE]

we have $|C_{n}|=1$, and the primes $p\nmid dgn$ with Frobenius $C_{n}$ satisfy $p\equiv a~{}({\rm mod~{}}d)$ and $\mathrm{ord}_{p}(g)\mid(p-1)/n$, and they have natural density $1/[K_{n}:{\mathbb{Q}}]$.

3. Proofs

3.1. Proof of Proposition 1.2

Since $q$ is odd, the extension ${\mathbb{Q}}(\zeta_{q},g^{1/q})$ of ${\mathbb{Q}}(\zeta_{q})$ is nonabelian and

[TABLE]

as $\gcd(q,d)=1$. Thus (2.3) is satisfied and consequently there is a set with natural density $1/[K_{q}:{\mathbb{Q}}]$ of primes $p$ satisfying $p\equiv a~{}({\rm mod~{}}d)$ and $\mathrm{ord}_{p}(g)\mid(p-1)/q.$ Since by assumption $q\nmid t$, it follows that for these primes $p$, $\mathrm{ord}_{p}(g)\neq(p-1)/t$, and so for them $g$ is a non $t$-near primitive root. This completes the proof.

3.2. Proof of Proposition 1.4

From now on we assume that $g,t,a$ and $d$ are as in Proposition 1.4. The proof of Proposition 1.4 rests on the Chebotarev density theorem and the following lemma. Recall that $K_{n}={\mathbb{Q}}(\zeta_{d},\zeta_{n},g^{1/n})$.

Lemma 3.1**.**

Put $I_{n}={\mathbb{Q}}(\zeta_{d})\cap{\mathbb{Q}}(\zeta_{n},g^{1/n}).$ Then, for any positive integer $q$ coprime to $2dgt$, we have $I_{qt}=I_{t}$.

Proof.

Since $I_{t}\subseteq I_{qt}$, it suffices to show that $[I_{qt}:{\mathbb{Q}}]=[I_{t}:{\mathbb{Q}}]$. Obviously $[d,t]=rt$ for some positive integer $r$. By elementary Galois theory and noticing that $\gcd(q,dt)=1$, we see that

[TABLE]

and, similarly, $[I_{t}:{\mathbb{Q}}]=\varphi(d)[{\mathbb{Q}}(\zeta_{t},g^{1/t}):{\mathbb{Q}}]/[{\mathbb{Q}}(\zeta_{rt},g^{1/t}):{\mathbb{Q}}].$ Then, by Lemma 1 of [10] and noticing $\gcd(q,2dgt)=1$, it is straightforward to deduce that $[I_{qt}:{\mathbb{Q}}]=[I_{t}:{\mathbb{Q}}]$. ∎

Remark 3.2**.**

We remark that the condition $\gcd(q,2dgt)=1$ cannot be removed. For example, choosing $g=21,d=3,t=10,q=7$ and using [11, Lemma 2.4], we have $I_{t}={\mathbb{Q}}$ and $I_{qt}={\mathbb{Q}}(\zeta_{d})={\mathbb{Q}}(\sqrt{-3})\neq I_{t}$. **

Proof of Proposition 1.4.

By Lemma 3.1 it follows that

[TABLE]

By assumption, ${\mathcal{R}}_{g}(t,d,a)$ is not empty. Then, this implies that the two automorphisms in (2.1) are compatible and hence (2.2) is satisfied, which leads to the conclusion that ${\mathcal{R}}_{g}(t,d,a)$ is not only nonempty, but even has a positive natural density, moreover $\delta({\mathcal{R}}_{g}(t,d,a))=[K_{t}:{\mathbb{Q}}]^{-1}$ by the discussions in Section 2. So, there must be a $\tau_{t}\in C_{t}$ such that $\tau_{t}|_{I_{t}}=\text{id},$ which by (3.1) implies the existence of an automorphism $\tau_{qt}\in C_{qt}$ such that $\tau_{qt}|_{I_{qt}}=\text{id}$. Then, it follows from the discussions in Section 2 that $\delta({\mathcal{R}}_{g}(qt,d,a))=[K_{qt}:\mathbb{Q}]^{-1}$. ∎

3.3. Proof of Theorem 1.3

Proof of Theorem 1.3.

A direct consequence of Proposition  1.4. ∎

Acknowledgement

The research of Min Sha was supported by a Macquarie University Research Fellowship. The authors thank Christine McMeekin and Peter Stevenhagen for very helpful discussions and feedback. The latter pointed out a serious oversight in an earlier version. The feedback of the referee is also gratefully acknowledged.