# Prime divisors of $\ell$-Genocchi numbers and the ubiquity of Ramanujan-
style congruences of level $\ell$

Pieter Moree  Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn,
Germany<EMAIL_ADDRESS>and  Pietro Sgobba  University of Luxembourg,
Department of Mathematics, 6 Avenue de la Fonte, L-4364 Esch-sur-Alzette,
Luxembourg<EMAIL_ADDRESS>
###### Abstract.

Let $\ell$ be any fixed prime number. We define the $\ell$-Genocchi numbers by
$G_{n}:=\ell(1-\ell^{n})B_{n}$, with $B_{n}$ the $n$-th Bernoulli number. They
are integers. We introduce and study a variant of Kummer’s notion of
regularity of primes. We say that an odd prime $p$ is $\ell$-Genocchi
irregular if it divides at least one of the $\ell$-Genocchi numbers
$G_{2},G_{4},\ldots,G_{p-3}$, and $\ell$-regular otherwise. With the help of
techniques used in the study of Artin’s primitive root conjecture, we give
asymptotic estimates for the number of $\ell$-Genocchi irregular primes in a
prescribed arithmetic progression in case $\ell$ is odd. The case $\ell=2$ was
already dealt with by Hu, Kim, Moree and Sha (2019).

Using similar methods we study the prime factors of $(1-\ell^{n})B_{2n}/2n$
and $(1+\ell^{n})B_{2n}/2n$. This allows us to estimate the number of primes
$p\leq x$ for which there exist modulo $p$ Ramanujan-style congruences between
the Fourier coefficients of an Eisenstein series and some cusp form of prime
level $\ell$.

###### Key words and phrases:

$\ell$-Genocchi numbers, $\ell$-regularity, Ramanujan type congruences,
Artin’s primitive root conjecture

###### 2020 Mathematics Subject Classification:

Primary 11A07, 11B68; secondary 11F33

##  1\. Introduction

Recall that the $n$-th Bernoulli number $B_{n}$ is implicitly defined as the
coefficient of $t^{n}$ in the generating function

$\frac{t}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!}.$ (1)

The Bernoulli numbers are rational. It is easy to see that
$B_{0}=1,B_{1}=-1/2$ and $B_{2n+1}=0$ for $n\geq 1$. By the _von Staudt-
Clausen theorem_ (see for example [2, Chp. 3]) the remaining Bernoulli numbers
satisfy

$B_{2n}+\sum_{p-1\mid 2n}\frac{1}{p}\in\mathbb{Z},$ (2)

where the sum is over the primes $p$ for which $p-1$ divides $2n$, and thus
their denominators are well understood. However, their numerators are far less
so. We say that an odd prime $p$ is $B$-irregular if $p$ divides the numerator
of at least one of the Bernoulli numbers $B_{2},B_{4},\ldots,B_{p-3}$, and
$B$-regular otherwise. This notion has an important application in algebraic
number theory. Let $\mathbb{Q}(\zeta_{p})$ with $\zeta_{p}=e^{2\pi i/p}$ be
the $p$-th cyclotomic field, and $h_{p}$ its class number.

######  Theorem 1 (Kummer).

Let $p$ be an odd prime. If $p\nmid h_{p}$, then the Fermat equation
$x^{p}+y^{p}=z^{p}$ does not have a solution in positive integers $x,y,z$ with
$p$ coprime to $xyz$. An odd prime $p$ is $B$-regular if and only if $p\nmid
h_{p}$.

This result of Kummer is considered to be one of the highlights of 19th
century number theory. In the process of proving it, Kummer developed a lot of
algebraic number theory including the notion of ideals. For a more modern
proof of Theorem 1 using $p$-adic methods, see the book by Borevich and
Shafarevich [4].

In 1915, Jensen (as a student!) proved that there are infinitely many
irregular primes (see, e.g., [4, p. 381] or [28, p. 20]). Unfortunately the
same is not known for regular primes, although numerical evidence indicates
that about 61% of all primes are regular. This number is easily explained
using an heuristical argument due to Siegel [32]. Assuming that the divisor
structure of Bernoulli denominators is random, we expect that $p\nmid B_{k}$
with probability $1-1/p$. We thus might expect that $p$ is regular with
probability $(1-1/p)^{(p-3)/2}$, which as $p$ gets large tends to
$e^{-1/2}=0.60653\ldots$.

######  Conjecture 1 (Siegel).

The number of $B$-regular primes up to $x$ is given asymptotically by

$\frac{x}{\sqrt{e}\log x}.$

In this paper we study the divisibility of some sequences related to Bernoulli
numbers. As usual $\nu_{p}(a)$ denotes the exponent of $p$ in the prime
factorization of the rational number $a$.

######  Definition 1 (Divisibility of a sequence).

Let $A=\\{a_{n}\\}_{n=1}^{\infty}$ be a sequence of rational numbers. We say
that a prime $p$ divides the sequence $A$ if there exists $n\geq 1$ such that
$a_{n}\neq 0$ and $\nu_{p}(a_{n})\geq 1$. The set of prime divisors of $A$ we
denote by ${\mathcal{Q}}_{A}$.

Throughout, if $\mathcal{S}$ is a set of prime numbers, then we denote
$\\#\\{p\leq x:p\in\mathcal{S}\\}$ by $\mathcal{S}(x)$.

It is a consequence of (2) and the Kummer congruences, that a prime $p$
divides the sequence of Bernoulli numbers if and only if it is $B$-irregular.
Recall that the _Kummer congruence_ (cf. Murty [28, pp. 18-19]) implies that

$\frac{B_{j}}{j}\equiv\frac{B_{k}}{k}\mkern 4.0mu({\operator@font mod}\mkern
6.0mup),\leavevmode\nobreak\ {\rm\leavevmode\nobreak\ with\leavevmode\nobreak\
}j\equiv k\not\equiv 0\mkern 4.0mu({\operator@font mod}\mkern 6.0mup-1).$ (3)

The aim of this paper is to study the prime divisors of the sequences
$\\{H_{2n}\\}_{n=1}^{\infty}$, $\\{H_{2n}^{-}\\}_{n=1}^{\infty}$ and
$\\{H_{2n}^{+}\\}_{n=1}^{\infty}$ with

$H_{2n}:=(1-\ell^{2n})\frac{B_{2n}}{2n},\qquad
H_{2n}^{-}:=(1-\ell^{n})\frac{B_{2n}}{2n},\qquad
H_{2n}^{+}:=(1+\ell^{n})\frac{B_{2n}}{2n}.$ (4)

In this context the notion of irregularity will play an important role.

######  Definition 2 ((Ir)regularity).

Given a sequence of rational numbers $\\{A_{k}\\}_{k=1}^{\infty}$, we say that
$p>3$ is $A$-regular if all of $\nu_{p}(A_{2}),\ldots,\nu_{p}(A_{p-3})$ are
non-positive, and $A$-irregular otherwise. The prime $3$ is defined to be
$A$-regular. The set of $A$-irregular primes is denoted by $\mathcal{P}_{A}$.

Sometimes we use $H_{2n}^{-1}$ and $H_{2n}^{1}$ instead of $H_{2n}^{-}$,
respectively $H_{2n}^{+}$. Thus the final two entries in (4) can be more
compactly written as

$H_{2n}^{\varepsilon}:=(1+\varepsilon\ell^{n})\frac{B_{2n}}{2n},\qquad\varepsilon\in\\{-1,1\\}.$

The number $-H_{k}^{\varepsilon}/2$ occurs for even $k$ as constant term in
the Fourier expansion of a generalization $E_{k,\ell}^{\varepsilon}(z)$ (given
by (6)) of the classical weight $k$ Eisenstein series $E_{k}(z)$ to the prime
level $\ell$ setting. In case modulo a prime $p$ this constant is zero, the
Eisenstein series $E_{k,\ell}^{\varepsilon}(z)$ is possibly coefficient wise
congruent to a cusp form leading to a congruence of Ramanujan type (such as
(5)). In Sect. 4 we consider, given $\ell$ and $\varepsilon$, for how many
primes $p\leq x$ there exists at least one Eisenstein series
$E_{k,\ell}^{\varepsilon}(z)$ such that modulo $p$ its constant term
$-H_{k}^{\varepsilon}/2$ is zero. In the next section we discuss this modular
form connection in more detail.

###  1.1. Ramanujan style congruences for prime level $\ell$

Let $E_{k}$ be the Eisenstein series of even weight $k\geq 2$ for the group
$SL_{2}({\mathbb{Z}})$, normalized so that its Fourier series expansion is

$E_{k}(z)=-\frac{B_{k}}{2k}+\sum_{n=1}^{\infty}\sigma_{k-1}(n)e^{2\pi inz},$

where $\sigma_{r}(n)=\sum_{d|n}d^{r}$ is the $r$-th sum of divisors function.
The prototype of a Ramanujan congruence goes back to 1916 and asserts that

$\tau(n)\equiv\sigma_{11}(n)\mkern 4.0mu({\operator@font mod}\mkern
6.0mu691),$ (5)

for every positive integer $n$. This can be viewed as a (coefficient-wise)
congruence between the unique cusp form
$\Delta(z)=\sum_{n=1}^{\infty}\tau(n)e^{2\pi inz}$ of weight $12$ and the
Eisenstein series $E_{12}(z)$, namely $\Delta\equiv E_{12}\mkern
4.0mu({\operator@font mod}\mkern 6.0mu691)$. Note that $691$ divides $B_{12}$.
There are several well-known ways to prove, interpret, and generalize this.
Here we will only focus on a generalization to prime level $\ell$, where the
associated Eisenstein series, for $\varepsilon\in\\{\pm 1\\}$, even weight
$k\geq 2$ and prime level $\ell$, is

$E_{k,\ell}^{\varepsilon}(z)=E_{k}(z)+{\varepsilon}\ell^{k/2}E_{k}(\ell z).$
(6)

Notice that the constant term in the Fourier series of
$E_{k,\ell}^{\varepsilon}(z)$ equals $-H_{k}^{\varepsilon}/2$. Kumar et al.
[16] recently established the following result. Throughout this article, given
a rational number $a$, by $p\mid a$, we mean that the prime number $p$ divides
the reduced numerator of $a$, that is $\nu_{p}(a)\geq 1$.

######  Theorem 2.

Let $k$ be an even natural integer, $\ell$ and $p$ be primes. Let
$S_{k}^{\varepsilon}(\ell)$ denote the $\varepsilon$-eigenspace of the Atkin-
Lehner operator $W_{p}$ inside $S_{k}(\ell)$, the space of modular cusp forms
of weight $k$ for the group $\Gamma_{0}(\ell)$.
i) Suppose that $p\geq 5$ divides $H_{k}$ for some even integer $k\geq 4$.
Then there exists $\varepsilon\in\\{\pm 1\\}$ and a normalized eigenfunction
$f\in S_{k}^{\varepsilon}(\ell)$ for all Hecke operators $T_{q}$ with
$q\neq\ell$ a prime, and a prime ideal $\mathfrak{p}$ over $p$ in the
coefficient field of $f$ such that

$f\equiv E_{k,\ell}^{\varepsilon}\mkern 4.0mu({\operator@font mod}\mkern
6.0mu\mathfrak{p}).$ (7)

ii) Let $\varepsilon\in\\{\pm 1\\}$ be fixed. Suppose that $p\geq 5$ divides
$H_{k}^{\varepsilon}$ for some even integer $k\geq 4$. Then there exists a
normalized eigenfunction $f\in S_{k}^{\varepsilon}(\ell)$ for all Hecke
operators $T_{q}$ with $q\neq\ell$ a prime, and a prime ideal $\mathfrak{p}$
over $p$ in the coefficient field of $f$ such that (7) holds.

If we fix a prime $\ell$, we can wonder about the ubiquity of the primes $p$
for which a Ramanujan-congruence (7) for some even integer $k\geq 4$ exists.
This amounts to estimating the number of prime divisors $p\leq x$ as $x$ gets
large of the sequences $\\{H_{2n}\\}_{n=1}^{\infty}$ and
$\\{H_{2n}^{\varepsilon}\\}_{n=1}^{\infty}$111For various reasons we included
the terms with $n=1$ as well. It is a consequence of the Kummer congruences
that these sequences have the same prime divisors as the ones with the term
$n=1$ left out..

It is not so difficult to show (see Lemma 4) that a prime $p$ divides the
$H$-sequence if and only if it is $H$-irregular or is in the Wieferich set

${\mathcal{W}}_{\ell}=\\{p>2:\ell^{p-1}\equiv 1\bmod{p^{2}}\\}.$ (8)

Likewise, $p$ divides the $H^{\varepsilon}$-sequence if and only if it is
$H^{\varepsilon}$-irregular or is in the Wieferich set ${\mathcal{W}}_{\ell}$
(if $\varepsilon=-1$) or ${\mathcal{W}}_{\ell}^{+}$ (if $\varepsilon=1$),
where

${\mathcal{W}}_{\ell}^{\varepsilon}=\\{p>2:\ell^{(p-1)/2}\equiv-\varepsilon\bmod{p^{2}}\\},\quad\varepsilon\in\\{-1,1\\}.$
(9)

Note that
${\mathcal{W}}_{\ell}={\mathcal{W}}_{\ell}^{-}+{\mathcal{W}}_{\ell}^{+}$. The
Wieferich sets are believed to be very sparse and thus the congruence ubiquity
problem in essence amounts to estimating ${\mathcal{P}}_{H^{\varepsilon}}(x)$,
the number of $H^{\varepsilon}$-irregular primes up to $x$. The results (too
lengthy to be stated here) are presented in Sect. 4. We show that

${\mathcal{P}}_{H^{\varepsilon}}(x)>\delta_{1}\frac{x}{\log
x},\quad\delta_{1}>0,\quad x\rightarrow\infty,$ (10)

and making heuristical assumptions similar to that of Siegel, we conjecture
that

${\mathcal{P}}_{H^{\varepsilon}}(x)\sim\delta_{2}\frac{x}{\log
x},\quad\delta_{2}>0,\quad x\rightarrow\infty,$

with $\delta_{1}$ and $\delta_{2}>\delta_{1}$ explicit constants. Our
constants $\delta_{2}$ are consistent with numerical data, see Table 2. The
inequality (10) can be compared to the best-known result for Bernoulli
numbers. Namely, Luca, Pizarro-Madariaga, and Pomerance [19] showed that the
number ${\mathcal{P}}_{B}(x)$ of irregular primes up to $x$ satisfies

${\mathcal{P}}_{B}(x)\geq(1+o(1))\frac{\log\log x}{\log\log\log x}.$

In Sec. 7 we extend the above results by restricting to primes in a prescribed
arithmetic progression. Given $1\leq a<d$ coprime integers and a prime $\ell$,
we define

$\mathcal{P}_{H^{\varepsilon}}(d,a):=\\{p:p\equiv a\bmod d\text{ and }p\text{
is $H^{\varepsilon}$-irregular}\\},\,\varepsilon\in\\{-1,1\\}.$

In order to study the primes in these sets we consider the following related
sets:

$\mathcal{A}_{d,a}=\\{p\equiv a\bmod
d:\operatorname{ord}_{p}(\ell)=p-1\\},\,\mathcal{A}_{d,a}^{-}:=\\{p\equiv
a\bmod d:\operatorname{ord}_{p}(\ell)=(p-1)/2\\}.$

To these sets we associate infinite sums $\alpha_{d,a}$ and $\alpha^{-}_{d,a}$
given by (16), respectively (41). Under GRH it follows from general results of
Lenstra [18] that these are the respective relative (inside the set of primes
$p\equiv a\bmod d$) densities of the sets. The infinite sums $\alpha_{d,a}$
and $\alpha^{-}_{d,a}$ are given in Euler product form in Theorem 5 (well-
known), respectively Theorem 7 (new). Both are rational multiples of the
_Artin constant_

$A=\prod_{\textrm{prime
$p$}}\Big{(}1-\frac{1}{p(p-1)}\Big{)}=0.3739558136192022880547280543464\ldots,$
(11)

The set $\mathcal{A}_{d,a}$ has been well-studied, but not so
$\mathcal{A}^{-}_{d,a}$, although some special cases occur in various number
theoretical problems, see, e.g., [5, 7, 27].

######  Theorem 3.

Let $\ell$ be an odd prime. For $\epsilon>0$ arbitrary and
$\varepsilon\in\\{-1,1\\}$ we have

$\mathcal{P}_{H^{\varepsilon}}(d,a)(x)\geq(1-\delta^{\varepsilon}_{d,a}-\epsilon)\frac{x}{\varphi(d)\log
x},$

where $\delta^{-}_{d,a}=\alpha_{d,a}+\alpha^{-}_{d,a}$. We have
$\delta^{+}_{d,a}=\alpha_{d,a}+\rho_{\ell,1}(a,d)$, where $\rho_{\ell,1}(a,d)$
is the density of prime divisors $p\equiv a\bmod{d}$ of the sequence
$\\{\ell^{n}+1\\}_{n\geq 1}^{\infty}$. This is always a rational number and is
explicitly determined in Moree and Sury [26].

The proof is a quite immediate consequence of the results proved in Section 7
and similar to that of Theorem 4 given in Section 6.1.4. The details are left
to the interested reader.

###  1.2. Counting $\ell$-Genocchi irregular primes

We let $G=\\{G_{2n}\\}_{n\geq 1}$ with $G_{2n}:=2\ell
nH_{2n}=\ell(1-\ell^{2n})B_{2n}$ be the sequence of _$\ell$ -Genocchi
numbers_. The inclusion of the factor $\ell$ ensures that they are integers.
In case $\ell=2$ we speak about _Genocchi numbers_. These show up in a result
similar to Kummer’s Theorem 1, see Hu and Kim [13], and it might thus be
reasonable to consider them also for arbitrary $\ell$. An odd prime $p$ is
said to be $\ell$-Genocchi (ir)regular if and only if it is $G$-(ir)regular.
The first twenty $2$-Genocchi irregular primes are

$17,31,\textbf{37},41,43,\textbf{59},\textbf{67},73,89,97,\textbf{101},\textbf{103},109,113,127,\textbf{131},137,\textbf{149},151,\textbf{157},$

where those that are also $B$-irregular are put in boldface. There is a
considerable literature on the classical Genocchi numbers (Sect. 8.1), but
little seems to have been done in the general case (Sect. 8.2). The
$\ell$-Genocchi irregular primes show up in the study of the prime divisors of
$H$. Namely, a prime $p$ divides the $H$-sequence if and only if it is
$\ell$-Genocchi irregular or in the Wieferich set ${\mathcal{W}}_{\ell}$
(Proposition 3). As the Wieferich set is believed to be sparse, the study of
the prime divisors of $H$ is in essence that of $\ell$-Genocchi irregular
primes.

Very little is known about the distribution of irregular primes in a
prescribed arithmetic progression. We will show that for the $\ell$-Genocchi
irregular primes the situation is rather different. This is a consequence of
the divisor structure of $\ell^{2n}-1$ being much better understood than that
of the Bernoulli numerators.

Given $1\leq a<d$ coprime integers and a prime $\ell$, we define

$\mathcal{P}_{G}(d,a):=\\{p:p\equiv a\bmod d\text{ and }p\text{ is
$\ell$-Genocchi irregular}\\}.$

We are interested in the behavior of $\mathcal{P}_{G}(d,a)(x)$ as $x$ gets
large as compared to that of $\pi(x;d,a):=\\#\\{p\leq x:p\equiv a\bmod d\\}$,
which is known to behave asymptotically as $x/(\varphi(d)\log x)$. The latter
function appears in Theorem 4 and Conjecture 2 and can thus be replaced by
$\pi(x;d,a)$. By ${\bigl{(}\frac{*}{*}\bigr{)}}$ we denote the Jacobi symbol.

######  Theorem 4.

Let $\ell$ be an odd prime. For $\epsilon>0$ arbitrary we have

$\mathcal{P}_{G}(d,a)(x)\geq(1-\delta_{d,a}-\epsilon)\frac{x}{\varphi(d)\log
x},$

where $\delta_{d,a}$ is given by (30) and worked out in Euler product form in
Theorem 6. We have $1-\delta_{d,a}>0$. Moreover, either $1-\delta_{d,a}\geq
1/4$, or $4\mid d$ and $a\equiv 3\bmod 4$, or $\ell\mid d$ and
${\bigl{(}\frac{a}{\ell}\bigr{)}}=-1$.

A similar result for $\ell=2$ can be found in Hu et al. [14]. The relative
density $1-\delta_{d,a}$ can be arbitrarily close to $0$, respectively $1$.
For particulars see Sect. 6.1.3.

Regarding the true behavior of $\mathcal{P}_{G}(d,a)(x)$ we make the following
conjecture (consistent with numerical data, see Table 3).

######  Conjecture 2.

Let $\ell$ be an odd prime. Asymptotically one has

$\mathcal{P}_{G}(d,a)(x)\sim\bigg{(}1-\frac{\delta_{d,a}}{\sqrt{e}}\bigg{)}\frac{x}{\varphi(d)\log
x},$

where $\delta_{d,a}$ is given by (30) and worked out in Euler product form in
Theorem 6.

A similar conjecture can be formulated for
$\mathcal{P}_{H^{\varepsilon}}(d,a)(x)$, where one merely replaces
$\delta_{d,a}$ by $\delta^{\varepsilon}_{d,a}$.

By Theorem 6 we have for any odd prime $\ell$ that $\delta_{d,a}=0$ if and
only if $4\ell$ divides $d$, ${\bigl{(}\frac{a}{\ell}\bigr{)}}=1$ and $a\equiv
1\bmod 4$. Thus Conjecture 2 leads to the following weaker conjecture.

######  Conjecture 3.

Let $\ell$ be an odd prime. The set of $\ell$-Genocchi regular primes in the
primitive residue class $a\bmod{d}$ has a positive density, provided we are
not in the case where $4\ell$ divides $d$,
${\bigl{(}\frac{a}{\ell}\bigr{)}}=1$ and $a\equiv 1\bmod 4$. If $\ell=2$ the
density is positive provided we are not in the case where $8$ divides $d$ and
$a\equiv 1\bmod 8$.

The case $\ell=2$ is not covered by our argumentation, but is Conjecture 1.16
of Hu et al. [14].

##  2\. Preliminaries

###  2.1. Basic properties of the numbers $H_{n}$

From (2) we infer that $G_{2n}$ is an integer and $H_{n}$ an $\ell$-integer
(that is satisfies $\nu_{p}(H_{2n})\geq 0$ for every prime $p\neq\ell$). Put
$\zeta_{\ell}=e^{2\pi i/\ell}$. Using (1) we see that

$t\sum^{\ell-1}_{a=1}\frac{\zeta_{\ell}^{a}}{\zeta_{\ell}^{a}-e^{t}}=\frac{t}{e^{t}-1}-\frac{\ell
t}{e^{\ell t}-1}=\sum_{n=1}^{\infty}\frac{H_{n}t^{n}}{(n-1)!},$ (12)

where the first identity follows on noting that, as formal series,

$\displaystyle\sum^{\ell-1}_{a=1}\frac{\zeta_{\ell}^{a}}{\zeta_{\ell}^{a}-e^{t}}$
$\displaystyle=\sum_{n=0}^{\infty}\left(\sum^{\ell-1}_{a=1}\zeta_{\ell}^{-an}\right)e^{tn}=-\sum_{\begin{subarray}{c}n\geq
0\\\ \ell\nmid n\end{subarray}}e^{tn}+(\ell-1)\sum_{\begin{subarray}{c}n\geq
0\\\ \ell\mid n\end{subarray}}e^{tn}$
$\displaystyle=-\sum_{n=0}^{\infty}e^{tn}+\ell\sum_{n=0}^{\infty}e^{\ell
tn}=\frac{1}{e^{t}-1}-\frac{\ell}{e^{\ell t}-1}.$

If $p-1\nmid 2n$, then Voronoi’s congruence (see, for example, Murty [28, Chp.
1]) gives

$H_{2n}\equiv-\ell^{2n-1}\sum_{j=1}^{p-1}j^{2n-1}\big{[}\frac{j\ell}{p}\big{]}\mkern
4.0mu({\operator@font mod}\mkern 6.0mup),$ (13)

where $[y]$ denotes the greatest integer function.

###  2.2. Divisibility of $H,H^{-},H^{+}$ and the $\ell$-Genocchi numbers:
elementary observations

The following trivial result will play an important role. By
$\operatorname{ord}_{p}(\ell)$ we denote the multiplicative order of $\ell$
modulo $p$.

######  Lemma 1.

Let $p$ and $\ell$ be distinct primes.
1) The prime $p$ divides $\ell^{n}+1$ for some integer $n\geq 1$ if and only
if $\operatorname{ord}_{p}(\ell)$ is even.
2) The prime $p$ divides $\ell^{n}+1$ for some $1\leq n\leq(p-3)/2$ if and
only if $\operatorname{ord}_{p}(\ell)$ is even and not equal to $p-1$.
3) The prime $p$ divides $\ell^{n}+1$ for some $1\leq n\leq p-2$ with
$n\neq(p-1)/2$ if and only if $\operatorname{ord}_{p}(\ell)$ is even and not
equal to $p-1$.
4) The prime $p$ divides $\ell^{n}-1$ for some $1\leq n\leq(p-3)/2$ if and
only if $\operatorname{ord}_{p}(\ell)<(p-1)/2$.
5) The prime $p$ divides $\ell^{n}-1$ for some $1\leq n\leq p-2$ with
$n\neq(p-1)/2$ if and only if $\operatorname{ord}_{p}(\ell)<(p-1)/2$.
6) The prime $p$ divides $\ell^{2n}-1$ for some $1\leq n\leq(p-3)/2$ if and
only if $\operatorname{ord}_{p}(\ell^{2})<(p-1)/2$.

###### Proof.

Left to the reader (cf. Moree [21, Prop. 2]). ∎

######  Remark 1.

We will use various times the trivial observation that

$\operatorname{ord}_{p}(\ell^{2})=\begin{cases}\operatorname{ord}_{p}(\ell)&\text{if
}2\nmid\operatorname{ord}_{p}(\ell);\\\
\operatorname{ord}_{p}(\ell)/2&\text{otherwise}.\end{cases}$ (14)

####  2.2.1. The $H$-sequences

With the help of Lemma 1 we will now characterize $H$-, $H^{-}$\- and
$H^{+}$-irregular primes.

######  Lemma 2.

Let $p\neq\ell$ be an odd prime.
1) It is $H$-irregular if and only if it is $B$-irregular or
$\operatorname{ord}_{p}(\ell^{2})<(p-1)/2$.
2) It is $H^{-}$-irregular if and only if it is $B$-irregular or
$\operatorname{ord}_{p}(\ell)<(p-1)/2$.
3) It is $H^{+}$-irregular if and only if it is $B$-irregular or
$\operatorname{ord}_{p}(\ell)$ is even and not equal to $p-1$.
4) It is $H$-irregular if and only if it is either $H^{-}$\- or
$H^{+}$-irregular.
5) It is both $H^{-}$\- and $H^{+}$-regular if it is $B$-regular and satisfies
$p\equiv 3\bmod 4$ and $\operatorname{ord}_{p}(\ell)=(p-1)/2$.

###### Proof.

The claims are clearly true for $p=3$, and so we may assumue $p>3$. It follows
from (2) that the prime factors of the denominator of $B_{2n}$ are precisely
those primes $p$ such that $p-1$ divides $2n$. Therefore,

$\nu_{p}(B_{2n})=\nu_{p}(B_{2n}/2n),\quad 1\leq n\leq(p-3)/2.$ (15)

Suppose that $p$ is $H$-irregular, i.e. $\nu_{p}(H_{2n})\geq 1$ for some
$1\leq n\leq(p-3)/2$. By (15) and Lemma 1.6 this is equivalent with $p$ being
$B$-irregular or $\operatorname{ord}_{p}(\ell^{2})<(p-1)/2$.

The proofs of 2) and 3) are analogous, and follow by a similar argument
involving Lemma 1. Part 4) is a consequence of the observation that
$p\mid\ell^{2n}-1$ if and only if $p\mid\ell^{n}-1$ or $p\mid\ell^{n}+1$.
Finally, part 5) follows from parts 2) and 3) on taking into account the
identity (14). ∎

######  Lemma 3.

Let $p$ be an odd prime and $n$ a positive integer such that $p-1$ divides
$2n$. Then we have

$\nu_{p}(H_{2n})=\nu_{p}(\ell^{p-1}-1)-1.$

Further,

$\nu_{p}(H_{2n}^{-})=\begin{cases}\nu_{p}(\ell^{p-1}-1)-1&\text{if\leavevmode\nobreak\
}(p-1)\mid n;\\\ \nu_{p}(\ell^{(p-1)/2}-1)-1&\text{if\leavevmode\nobreak\
}(p-1)\nmid n\text{\leavevmode\nobreak\ and\leavevmode\nobreak\
}{\bigl{(}\frac{\ell}{p}\bigr{)}}=1;\\\
-1-\nu_{p}(n)&\text{if\leavevmode\nobreak\ }(p-1)\nmid
n\text{\leavevmode\nobreak\ and\leavevmode\nobreak\
}{\bigl{(}\frac{\ell}{p}\bigr{)}}=-1.\end{cases}$

Also,

$\nu_{p}(H_{2n}^{+})=\begin{cases}\nu_{p}(\ell^{(p-1)/2}+1)-1&(p-1)\nmid
n\text{\leavevmode\nobreak\ and\leavevmode\nobreak\
}{\bigl{(}\frac{\ell}{p}\bigr{)}}=-1;\\\ -1-\nu_{p}(n)&otherwise.\end{cases}$

###### Proof.

Write $2n=(p-1)p^{e}m$, with $p\nmid m$. Then, using (2) and the elementary
observation that if $a\equiv 1\bmod p$ and $a\neq 1$, then
$\nu_{p}(a^{j}-1)=\nu_{p}(a-1)+\nu_{p}(j)$ (cf. Beyl [3]), we find that

$\nu_{p}(H_{2n})=\nu_{p}(\ell^{2n}-1)+\nu_{p}(B_{2n})-\nu_{p}(2n)=\nu_{p}(\ell^{p-1}-1)+e-1-e=\nu_{p}(\ell^{p-1}-1)-1.$

The two remaining statements are proved similarly, with the difference that
whereas $\nu_{p}(\ell^{2n}-1)>1$, it can happen that
$\nu_{p}(\ell^{n}+\varepsilon)=0$. For the final claim we use that if
$a\equiv-1\bmod p$ and $a\neq-1$ and $j$ is even, then
$\nu_{p}(a^{j}-1)=\nu_{p}(a+1)+\nu_{p}(j)$. ∎

Recall the definitions (8) and (9) of the Wieferich sets.

######  Lemma 4.

Let $\ell$ and $p>2$ be distinct primes. Then
1) $p$ divides the $H$-sequence if and only if it is $H$-irregular or is in
the Wieferich set ${\mathcal{W}}_{\ell}$;
2) $p$ divides the $H^{-}$-sequence if and only if it is $H^{-}$-irregular or
is in the Wieferich set ${\mathcal{W}}_{\ell}$;
3) $p$ divides the $H^{+}$-sequence if and only if it is $H^{+}$-irregular or
is in the Wieferich set ${\mathcal{W}}_{\ell}^{+}$.

###### Proof.

We will only deal with cases 2) and 3), since case 1) is similar and easier.
It is not difficult to see that only finitely many terms of the
$H^{\varepsilon}$-sequence have to be considered in order to decide whether
$p$ divides the sequence or not. Indeed, the Kummer congruence (3) implies
that if $p-1\nmid 2n$, then

$\frac{B_{2n+r(p-1)}}{2n+r(p-1)}\Big{(}1+\varepsilon\ell^{n+r(p-1)/2}\Big{)}\equiv\frac{B_{2n}}{2n}\Big{(}1+\varepsilon{\bigl{(}\frac{\ell}{p}\bigr{)}}^{r}\ell^{n}\Big{)}\mkern
4.0mu({\operator@font mod}\mkern 6.0mup).$

Thus we have periodicity modulo $p-1$ if ${\bigl{(}\frac{\ell}{p}\bigr{)}}=1$
and modulo $2(p-1)$ otherwise. In particular, if $(p-1)\nmid 2n$ it is enough
to consider the $p$-divisibility of

$H_{2}^{\varepsilon},H_{4}^{\varepsilon}\ldots,H_{p-3}^{\varepsilon},H_{p+1}^{\varepsilon},H_{p+3}^{\varepsilon},\ldots,H_{2p-4}^{\varepsilon}$

The case $p-1\mid 2n$ is not covered, but for this we invoke Lemma 3, which
shows that in this case $p$ divides the $H^{-}$-sequence if and only $p$ is in
${\mathcal{W}}_{\ell}$ and divides the $H^{+}$-sequence if and only $p$ is in
${\mathcal{W}}_{\ell}^{+}$. So we restrict now to the case $p-1\nmid 2n$.

2) Now $\varepsilon=-1$. By Lemma 1.5 the prime $p$ is a divisor if and only
if it is $B$-irregular or $\operatorname{ord}_{p}(\ell)<(p-1)/2$. By Lemma 2.2
this is equivalent with $p$ being $H^{-}$-irregular.

3) Now $\varepsilon=1$. By Lemma 1.3 the prime $p$ is a divisor if and only if
it is $B$-irregular or $\operatorname{ord}_{p}(\ell)$ is even and not equal to
$p-1$. By Lemma 2.3 this is equivalent with $p$ being $H^{+}$-irregular. ∎

####  2.2.2. The $\ell$-Genocchi numbers

Since $H_{n}=G_{n}/\ell n$, it is natural to wonder how $H$-(ir)regular and
$\ell$-Genocchi (ir)regular primes are related.

######  Proposition 1.

An odd prime $p\neq\ell$ is $H$-irregular if and only if $p$ is
$\ell$-Genocchi irregular. If $\ell>3$, then $\ell$ is $\ell$-Genocchi
irregular, and $\ell$ is $H$-irregular if and only if it is $B$-irregular.

###### Proof.

If $p\neq\ell$, then for every $n=1,2,\ldots,(p-3)/2$, since $p$ and $p-1$ do
not divide $2n$, we have that $\nu_{p}(H_{2n})=\nu_{p}(G_{2n})$. If $\ell>3$,
then for every $n=1,2,\ldots,(\ell-3)/2$, since $\ell$ and $\ell-1$ do not
divide $2n$, we have that $\nu_{\ell}(\ell(1-\ell^{2n})B_{2n})\geq 1$, and
$\nu_{\ell}((1-\ell^{2n})B_{2n}/2n)=\nu_{\ell}(B_{2n})$. ∎

In contrast, the prime divisor structure of $G$\- and $H$-sequences are rather
different.

######  Proposition 2.

Let $\ell$ be a fixed prime. Given any prime $p$ (for $\ell\leq 3$ we suppose
$p\neq\ell$), there exists an integer $n$ such that $v_{p}(G_{2n})\geq 1$. If
$\ell\leq 3$, then $v_{\ell}(G_{2n})=0$ for all $n$.

###### Proof.

If $p\neq\ell$, we take $2n=p(p-1)$. Then $\ell^{2n}\equiv 1\bmod p^{2}$ and
$\nu_{p}(B_{2n})=-1$, so that $\nu_{p}(G_{2n})\geq 1$. If $p=\ell$ (and
$\ell>3$), then we take $n=(\ell-3)/2$ and find that $\nu_{\ell}(G_{2n})\geq
1$. If $\ell\leq 3$, then $\ell-1\mid 2n$ and hence $\nu_{\ell}(B_{2n})=-1$,
so that $\nu_{\ell}(G_{2n})=0$. ∎

######  Proposition 3.

A prime $p$ divides the $H$-sequence if and only if it is $\ell$-Genocchi
irregular or in the Wieferich set ${\mathcal{W}}_{\ell}$.

###### Proof.

This follows on combining Lemma 4.1 and Proposition 1. ∎

######  Proposition 4.

Let $\ell$ be a prime. An odd prime $p$, with $p\neq\ell$, is $\ell$-Genocchi
regular if and only if it is $B$-regular and
$\operatorname{ord}_{p}(\ell^{2})=(p-1)/2$.

###### Proof.

For $\ell=2$ it was shown by Hu et al. [14, Theorem 1.8]. For the remaining
$\ell$ it follows from Proposition 1 and Lemma 2.1. ∎

######  Corollary 1.

Let $\ell$ be a prime. If $p\equiv 1\bmod{4}$ and
${\bigl{(}\frac{\ell}{p}\bigr{)}}=1$, then $p$ is $\ell$-Genocchi irregular.

###### Proof.

By contradiction. Suppose that $\operatorname{ord}_{p}(\ell^{2})=(p-1)/2$. The
assumption $p\equiv 1\bmod{4}$ ensures that
$\operatorname{ord}_{p}(\ell)=p-1$, contradicting the assumption that
${\bigl{(}\frac{\ell}{p}\bigr{)}}=1$. ∎

######  Proposition 5.

Let $\ell$ be a prime. If $p$ is congruent to an odd square modulo $4\ell$,
then $p$ is $\ell$-Genocchi irregular.

For $\ell=2$ one verifies this directly. For odd $\ell$ it follows from
Corollary 1 on making use of an alternate form of the law of quadratic
reciprocity, initially conjectured by Euler, cf. Cox [6, p. 15].

######  Lemma 5.

If $p$ and $q$ are distinct odd primes, then ${\bigl{(}\frac{q}{p}\bigr{)}}=1$
if and only if $p\equiv\pm\beta^{2}\bmod{4q}$ for some odd integer $\beta$.

##  3\. Further preliminaries

We recall some relevant results and conjectures from the literature.

###  3.1. Primitive roots in arithmetic progression

Put

$\mathcal{A}_{d,a}=\\{p:p\equiv a\bmod
d,\,\operatorname{ord}_{p}(\ell)=p-1\\}.$

Under GRH the natural density of this set is given by

$\alpha_{d,a}=\sum_{n=1}^{\infty}\frac{\varphi(d)\mu(n)c_{a}(n)}{[\mathbb{Q}(\zeta_{[d,n]},\ell^{1/n}):\mathbb{Q}]},$
(16)

where $c_{a}(n)=1$ if the automorphism $\sigma_{a}$ of $\mathbb{Q}(\zeta_{d})$
determined by $\sigma_{a}(\zeta_{d})=\zeta_{d}^{a}$ is the identity on the
field $\mathbb{Q}(\zeta_{d})\cap\mathbb{Q}(\zeta_{n},\ell^{1/n})$, and
$c_{a}(n)=0$ otherwise. Its Euler product form was first evaluated by Moree
[22] for arbitrary $\ell$. In the relevant case for us where $\ell$ is an odd
prime, this result takes on a rather simpler form.

######  Theorem 5.

Let $a$ and $d$ be coprime integers and $\ell$ be an odd prime. Put

$\alpha_{d,a}=A\,c_{1}(d,a)\,R(d,a),$

with

$R(d,a)=\prod_{p\mid(a-1,d)}\left(1-\frac{1}{p}\right)\prod_{p\mid
d}\left(1+\frac{1}{p^{2}-p-1}\right)$ (17)

and

$c_{1}(d,a)=\begin{cases}1-{\bigl{(}\frac{\ell}{a}\bigr{)}}&\text{if
}\ell\equiv 1\bmod 4,\ell\mid d;\\\ 1+\frac{1}{\ell^{2}-\ell-1}&\text{if
}\ell\equiv 1\bmod 4,\ell\nmid d;\\\
1-{\bigl{(}\frac{\ell}{a}\bigr{)}}&\text{if }\ell\equiv 3\bmod 4,4\mid
d,\ell\mid d;\\\
1+{\bigl{(}\frac{-1}{a}\bigr{)}}\frac{1}{\ell^{2}-\ell-1}&\text{if }\ell\equiv
3\bmod 4,4\mid d,\ell\nmid d;\\\ 1&\text{if }\ell\equiv 3\bmod 4,4\nmid
d.\end{cases}$

Let $\epsilon>0$ be any fixed real number. Then, for every $x$ sufficiently
large,

$\mathcal{A}_{d,a}(x)\leq(\alpha_{d,a}+\epsilon)\frac{x}{\varphi(d)\log x}.$
(18)

Assuming GRH, we have

$\mathcal{A}_{d,a}(x)=\frac{\alpha_{d,a}}{\varphi(d)}\frac{x}{\log
x}+O_{d}\left(\frac{x\log\log x}{\log^{2}x}\right).$

The unconditional upper bound (18) is not given by Moree in either [22] or
[23], but is totally standard and for a related problem worked out in detail
in Hu et al. [14].

We note for future use that if $d$ is odd, then

$R(d,a)=\begin{cases}R(2d,a)&\text{\leavevmode\nobreak\ if\leavevmode\nobreak\
}2\nmid a;\\\ R(2d,a+d)&\text{\leavevmode\nobreak\ otherwise}.\end{cases}$
(19)

######  Remark 2.

Let $\Delta$ denote the discriminant of $\mathbb{Q}(\sqrt{\ell})$. In case
$\Delta\mid d$ and ${\bigl{(}\frac{\ell}{a}\bigr{)}}=1$, then using quadratic
reciprocity it is easy to see that $\mathcal{A}_{d,a}$ is empty and so
unconditionally $\alpha_{d,a}=0$. By Theorem 5, under GRH, $\alpha_{d,a}=0$ if
and only if $\Delta\mid d$ and ${\bigl{(}\frac{\ell}{a}\bigr{)}}=1$.

###  3.2. Near-primitive roots

Given integers $t\geq 1$ and $g$, we set

$\mathcal{P}(g,t):=\\{p:p\equiv 1\bmod t,\quad{\rm ord}_{p}(g)=(p-1)/t\\}.$

The primes in $\mathcal{P}(g,t)$ are called _near-primitive roots_. Let
$\epsilon>0$ be fixed. By [14, Theorem 3.1] we have

$\mathcal{P}(g,t)(x)\leq(\delta(g,t)+\epsilon)\frac{x}{\log x},$ (20)

for every $x$ sufficiently large, where

$\delta(g,t)=\sum_{n=1}^{\infty}\frac{\mu(n)}{[\mathbb{Q}(\zeta_{nt},g^{1/nt}):\mathbb{Q}]}.$

Assuming the Riemann Hypothesis for all number fields
$\mathbb{Q}(\zeta_{nt},g^{1/nt})$ with $n$ squarefree, we have the sharper
estimate

$\mathcal{P}(g,t)(x)=\delta(g,t)\frac{x}{\log x}+O_{g,t}\left(\frac{x\log\log
x}{\log^{2}x}\right).$ (21)

Thus, conditionally, the set of primes $\mathcal{P}(g,t)$ has natural density
$\delta(g,t)$. This quantity was explicitly computed for $t=1$ by Hooley [12]
and for general $t$ by Moree [25]. It always equals a rational number times
the Artin constant, where the rational number may depend on both $g$ and $t$.

###  3.3. Divisors of second order recurrences

Let $\alpha$ and $\beta$ be integers. The set of prime divisors
$\mathcal{Q}_{\alpha,\beta}$ of the sequence
$\\{\alpha^{n}+\beta^{n}\\}_{n=1}^{\infty}$ has been well studied. A prime
$p\nmid\alpha\beta$ divides it if and only if
$\operatorname{ord}_{p}(\alpha/\beta)$ is even (Moree [21, Prop. 2]). Let
$\mathcal{Q}_{\alpha,\beta}(a,d)$ be the set of primes $p\equiv a\bmod{d}$ in
$\mathcal{Q}_{\alpha,\beta}$. Moree and Sury [26] showed that in case
$\alpha/\beta>0$, asymptotically,

$\mathcal{Q}_{\alpha,\beta}(a,d)(x)=\rho_{\alpha,\beta}(a,d)\frac{x}{\log
x}+O\Big{(}\frac{x\log\log x}{\log^{7/6}x}\Big{)},\quad x\rightarrow\infty,$
(22)

where $\rho_{\alpha,\beta}(c,d)$ is an explicitly computable rational number.
For an informal proof in case $a=d=1$ and other references see [24, Sect.
9.2].

###  3.4. Wieferich sets

Recall the definitions (8) and (9) of the Wieferich sets. The first case of
Fermat’s Last Theorem (FLTI) is the statement that, for any odd prime $p$, the
equation $x^{p}+y^{p}=z^{p}$ does not have positive integer solutions where
none of $x,y,z$ is divisible by $p$. The _generalized Wieferich criterion_
(for given $q$) is the statement that if FLTI fails for some prime $p$, then
$p\in{\mathcal{W}}_{q}$. This criterion has been proved by Granville and
Monagan [10] for all $q\in\\{2,3,5,7,\ldots,89\\}$, the first 24 primes,
continuing work by a great number of mathematicians, starting with Wieferich
($q=2$) and Mirimanoff ($q=3$). For $q=2$ the only Wieferich numbers known
below $10^{17}$ are $1093$ and $3511$. For more information see
https://en.wikipedia.org/wiki/Wieferich_prime. It is believed that
${\mathcal{W}}_{\ell}(x)=O(\log\log x)$, but it is not even known whether
${\mathcal{W}}_{\ell}(x)=o(x/\log x)$ or not. The same is expected for
${\mathcal{W}}^{\varepsilon}_{\ell}(x)$.

##  4\. Counting $H^{-}$\- and $H^{+}$-irregular primes

In Section 7 we will count $H^{-}$\- and $H^{+}$-irregular primes in
prescribed arithmetic progression, here as a warm-up we consider the problem
of counting _all_ such primes.

Denote by $\mathcal{P}_{H^{\varepsilon}}$ the set of
$H^{\varepsilon}$-irregular primes. By Lemma 2, (20) and (22), we obtain
unconditionally

$\mathcal{P}_{H^{-}}(x)\geq(1-\delta(\ell,1)-\delta(\ell,2)-\epsilon)\frac{x}{\log
x}$ (23)

and

$\mathcal{P}_{H^{+}}(x)\geq(\rho_{\ell,1}(1,1)-\delta(\ell,1)-\epsilon)\frac{x}{\log
x},$ (24)

where $\epsilon>0$ is arbitrary.

######  Conjecture 4.

If we require the primes counted by $\mathcal{P}(g,t)$ and
${\mathcal{Q}}_{\alpha,\beta}$ to be also $B$-regular, then the estimates (21)
and (22) hold with $\delta(g,t)$ and $\rho_{\alpha,\beta}(a,d)$ replaced by
$\delta(g,t)/\sqrt{e}$, respectively $\rho_{\alpha,\beta}(a,d)/\sqrt{e}$.

This conjecture leads to the conjectures that

$\mathcal{P}_{H^{-}}(x)\sim\Big{(}1-\frac{1}{\sqrt{e}}(\delta(\ell,1)+\delta(\ell,2)\Big{)}\frac{x}{\log
x}$ (25)

and

$\mathcal{P}_{H^{+}}(x)\sim\Big{(}1-\frac{1}{\sqrt{e}}(1-\rho_{\ell,1}(1,1)+\delta(\ell,1)\Big{)}\frac{x}{\log
x}.$ (26)

For reasons of space we abstain from writing out (23), (24), (25), and (26)
explicitly, but this can be done easily using the following lemma.

######  Lemma 6.

Let $A$ be the Artin constant defined in (11). If $\ell=2$, then
$\delta(2,1)=A$ and $\delta(2,2)=3A/4$. If $\ell\equiv 1\bmod 4$, then

$\delta(\ell,1)=A\left(1+\frac{1}{\ell^{2}-\ell-1}\right)\qquad\text{ and
}\qquad\delta(\ell,2)=\frac{3A}{4}\left(1-\frac{1}{\ell^{2}-\ell-1}\right).$

If $\ell\equiv 3\bmod 4$, then

$\delta(\ell,1)=A\qquad\text{ and
}\qquad\delta(\ell,2)=\frac{3A}{4}\left(1+\frac{1}{3(\ell^{2}-\ell-1)}\right).$

We have $\rho_{2,1}(1,1)=17/24$ and $\rho_{\ell,1}(1,1)=2/3$ for every odd
prime $\ell$.

The quantities $\delta(\ell,1)$ and $\delta(\ell,2)$ are, for example,
computed in Moree [25]. The results for $\rho_{\ell,1}(1,1)$ were proved by
Hasse [11] for Dirichlet density and by Odoni [29] for natural density (which
is what we use here).

##  5\. Counting $\ell$-Genocchi irregular primes

Recall that by Proposition 1, counting $\ell$-Genocchi irregular primes is,
except possibly for $p=\ell$, the same as counting $H$-irregular primes.
Corollary 1 implies that

$\mathcal{P}_{G}(x)\geq\Big{(}\frac{1}{4}-\epsilon\Big{)}\frac{x}{\log x}.$

By Proposition 4, given $\epsilon>0$ arbitrary and fixed, we have for every
$x$ sufficiently large

$\mathcal{P}_{G}(x)\geq\left(1-\delta(\ell^{2},2)-\epsilon\right)\frac{x}{\log
x},$

Assuming Siegel’s heuristic we conjecture that

$\mathcal{P}_{G}(x)\sim\left(1-\frac{\delta(\ell^{2},2)}{\sqrt{e}}\right)\frac{x}{\log
x}.$ (27)

By [14, Theorem 1.10] for $\ell=2$, and the results of [25] for $\ell$ odd, we
obtain

$\mathcal{P}_{G}(x)\geq\left(1-\frac{3}{2}A-\epsilon\right)\frac{x}{\log x}\
\text{\leavevmode\nobreak\ and\leavevmode\nobreak\ }\
\mathcal{P}_{G}(x)\geq\left(1-\frac{3A}{2}\left(1+\frac{1}{3(\ell^{2}-\ell-1)}\right)-\epsilon\right)\frac{x}{\log
x},$ (28)

respectively, where $\epsilon>0$ is arbitrary and fixed and $x$ sufficiently
large. For $\epsilon$ small enough, the constants involved are in the interval
$(0.4,0.44)$. We conjecture that

$\mathcal{P}_{G}(x)\sim\left(1-\frac{3A}{2\sqrt{e}}\right)\frac{x}{\log
x}\leavevmode\nobreak\ (\ell=2)\ \text{\leavevmode\nobreak\
and\leavevmode\nobreak\ }\
\mathcal{P}_{G}(x)\sim\left(1-\frac{3A}{2\sqrt{e}}\left(1+\frac{1}{3(\ell^{2}-\ell-1)}\right)\right)\frac{x}{\log
x}\leavevmode\nobreak\ (\ell>2),$

with now the constants involved being in $(0.637,0.66)$. See Table 1 for some
numerical examples supporting these conjectures.

##  6\. Irregular primes in arithmetic progression

The goal of this section is proving the main result of this paper, namely
Theorem 4. For $\ell=2$ this was already considered in [14, Sect. 1.3.1].
Further, we study the extremal behavior of $\delta_{d,a}$.

###  6.1. The $\ell$-Genocchi case

Let $\ell$ be an odd prime number and $1\leq a<d$ be coprime integers. In this
section we consider the set of rational (odd) primes

$\mathcal{P}_{d,a}:=\\{p\equiv a\bmod
d:\operatorname{ord}_{p}(\ell^{2})=(p-1)/2\\}.$ (29)

By Proposition 4 the primes in $\mathcal{P}_{d,a}$ are irregular. Under GRH we
have (see [14, Theorem 3.1])

$\lim_{x\rightarrow\infty}\frac{\mathcal{P}_{d,a}(x)}{\pi(x;d,a)}=\delta_{d,a},$

with

$\delta_{d,a}=\sum_{n=1}^{\infty}\frac{\varphi(d)\mu(n)c_{a}(n)}{[\mathbb{Q}(\zeta_{[d,2n]},\ell^{1/n}):\mathbb{Q}]},$
(30)

where $c_{a}(n)=1$ if the automorphism $\sigma_{a}$ of $\mathbb{Q}(\zeta_{d})$
determined by $\sigma_{a}(\zeta_{d})=\zeta_{d}^{a}$ is the identity on the
field $\mathbb{Q}(\zeta_{d})\cap\mathbb{Q}(\zeta_{2n},\ell^{1/n})$, and
$c_{a}(n)=0$ otherwise.

We will now express $\delta_{d,a}$ as an Euler product in two different ways:
one more starting from first principles and another shorter one relying more
heavily on existing results.

######  Theorem 6.

Let $a$ and $d$ be coprime integers, $\mathcal{P}_{d,a}$ and $\delta_{d,a}$ as
in (29), respectively (30), and $\epsilon>0$ be arbitrary and fixed. Then for
every $x$ sufficiently large we have

$\mathcal{P}_{d,a}(x)\leq(\delta_{d,a}+\epsilon)\frac{x}{\varphi(d)\log x},$
(31)

where

$\delta_{d,a}=A\,c(d,a)\,R(d,a),$

with $R(d,a)$ as in (17) and

$c(d,a)=\begin{cases}\frac{1}{2}\left(3+\frac{1}{\ell^{2}-\ell-1}\right)&\text{if
}\ell\nmid d,4\nmid d;\\\ 1+\frac{1}{\ell^{2}-\ell-1}&\text{if }\ell\nmid
d,4\mid d,a\equiv 1\bmod 4;\\\ 1&\text{if }\ell\mid d,4\nmid
d,\left(\frac{a}{\ell}\right)=1;\\\ 2&\text{if }4\mid d\text{ and }a\equiv
3\bmod 4,\text{ or }\ell\mid d\text{ and }\left(\frac{a}{\ell}\right)=-1;\\\
0&\text{if }4\ell\mid d,\left(\frac{a}{\ell}\right)=1,a\equiv 1\bmod
4.\end{cases}$

Assuming GRH we have

$\mathcal{P}_{d,a}(x)=\frac{\delta_{d,a}}{\varphi(d)}\frac{x}{\log
x}+O_{d}\left(\frac{x\log\log x}{\log^{2}x}\right).$

######  Remark 3.

Note that alternatively we can write

$c(d,a)=\begin{cases}\frac{1}{2}\left(3+\frac{1}{\ell^{2}-\ell-1}\right)&\text{if
}4\nmid d,\ell\nmid d;\\\
\frac{1}{2}(3-{\bigl{(}\frac{a}{\ell}\bigr{)}})&\text{if }4\nmid d,\ell\mid
d;\\\ 1+\frac{1}{\ell^{2}-\ell-1}&\text{if }4\mid d,a\equiv 1\bmod 4,\ell\nmid
d;\\\ 1-{\bigl{(}\frac{a}{\ell}\bigr{)}}&\text{if }4\mid d,a\equiv 1\bmod
4,\ell\mid d;\\\ 2&\text{if }4\mid d\text{ and }a\equiv 3\bmod 4.\\\
\end{cases}$

######  Remark 4.

From Theorem 6 we infer that the analogue of identity (19) holds for
$\delta_{d,a}$ as well, which is consistent with the fact that the analogue of
this identity also holds for $\mathcal{P}_{d,a}$.

####  6.1.1. The proof of Theorem 6

The proof requires a few preliminary lemmas. The first is merely a special
case of Lemma 3.1 of [23].

######  Lemma 7.

Put

$\omega_{d}(n):=\frac{n\varphi([d,n])}{\varphi(d)}.$

It is a multiplicative function in $n$. For $m\geq 1$, let

$S(m)=\sum_{\begin{subarray}{c}n\geq 1,\ m\mid n\\\ a\equiv
1\bmod(d,n)\end{subarray}}\frac{\mu(n)}{\omega_{d}(n)},\qquad
S_{2}(m)=\sum_{\begin{subarray}{c}n\geq 1,\ [2,m]\mid n\\\ a\equiv
1\bmod(d,n)\end{subarray}}\frac{\mu(n)}{\omega_{d}(n)}.$

We have $S_{2}(m)=-S(m)$. Further, $S(1)=AR(d,a)$ and

$S(\ell)=\begin{cases}-\frac{A}{\ell^{2}-\ell-1}R(d,a)&\text{ if }\ell\nmid
d;\\\ -\frac{A}{\ell-1}R(d,a)&\text{ if }\ell\mid d,\end{cases}$

where $\ell$ is an odd prime.

We recall the law of quadratic reciprocity for the Jacobi symbol: if $j$ and
$k$ are odd coprime positive integers, then

${\bigl{(}\frac{j}{k}\bigr{)}}{\bigl{(}\frac{k}{j}\bigr{)}}=(-1)^{(j-1)(k-1)/4}.$
(32)

In the following we set $\ell^{*}=(-1)^{(\ell-1)/2}\ell$.

######  Lemma 8.

Let $\sigma_{a}$ be the automorphism of $\mathbb{Q}(\zeta_{d})$ uniquely
determined by $\sigma_{a}(\zeta_{d})=\zeta_{d}^{a}$ with $a$ a positive
integer coprime to $d$. If $4\ell\mid d$, then
$\sigma_{a}(\sqrt{\ell})={\bigl{(}\frac{\ell}{a}\bigr{)}}\sqrt{\ell}$. If
$\ell\mid d$, then
$\sigma_{a}(\sqrt{\ell^{*}})={\bigl{(}\frac{a}{\ell}\bigr{)}}\sqrt{\ell^{*}}$.

###### Proof.

The quadratic Gauss sum expresses $\sqrt{\ell^{*}}$ as an element in
$\mathbb{Q}(\zeta_{d})$, which allows one to determine
$\sigma_{a}(\sqrt{\ell^{*}})$ and from this, using $\sigma_{a}(i)=i^{a}$, also
$\sigma_{a}(\sqrt{\ell})$. Invoking (32) we can formulate the outcome in a
more compact way. ∎

######  Remark 5.

This can also be proved using that $\sigma_{a}(\sqrt{\ell})/\sqrt{\ell}$ is a
character, see [23, Lemma 2.1].

######  Lemma 9.

Let $n$ be a squarefree integer. If $n$ is odd, or $\ell\mid n$, or $n$ is
even and $\ell\nmid nd$, then we have: $c_{a}(n)=1$ if and only if $a\equiv
1\bmod(d,2n)$. If $n$ is even, $\ell\nmid n$ and $\ell\mid d$, then we have:
$c_{a}(n)=1$ if and only if $a\equiv 1\bmod(d,2n)$ and
$\left(\frac{a}{\ell}\right)=1$.

###### Proof.

Let us set $I:=\mathbb{Q}(\zeta_{d})\cap\mathbb{Q}(\zeta_{2n},\ell^{1/n})$. By
Kummer theory we may argue that, since
$\mathbb{Q}(\zeta_{\infty})\cap\mathbb{Q}(\zeta_{2n},\ell^{1/n})$ is a finite
abelian extension of $\mathbb{Q}(\zeta_{2n})$, it is of the form
$\mathbb{Q}(\zeta_{2n},\ell^{e/n})$ for some $e\geq 0$, and hence by
Schinzel’s Theorem [31, Theorem 2] it is either $\mathbb{Q}(\zeta_{2n})$ if
$n$ is odd, or $\mathbb{Q}(\zeta_{2n},\sqrt{\ell})$ if $n$ is even. The latter
extension equals $\mathbb{Q}(\zeta_{2n})$ if $\ell\mid n$ (as
$\mathbb{Q}(\sqrt{\ell})\subseteq\mathbb{Q}(\zeta_{4\ell})$ and we already
have that $n$ is even).

Thus, if $n$ is odd or $\ell\mid n$, we deduce that
$I=\mathbb{Q}(\zeta_{(d,2n)})$. For $n$ even with $\ell\nmid n$, it suffices
to notice that $\mathbb{Q}(\sqrt{\ell^{*}})$ is contained in
$\mathbb{Q}(\zeta_{d})$ if and only if $\ell\mid d$. If $\ell\nmid d$, then we
have $I=\mathbb{Q}(\zeta_{(d,2n)})$. If $\ell\mid d$, then noticing that
$\mathbb{Q}(\zeta_{(d,2n)})\subseteq
I\subseteq\mathbb{Q}(\zeta_{(d,2n)},\zeta_{\ell})$, we deduce that
$I=\mathbb{Q}(\zeta_{(d,2n)},\sqrt{\ell^{*}})$ (where
$\mathbb{Q}(\sqrt{\ell^{*}})$ is not contained in
$\mathbb{Q}(\zeta_{(d,2n)})$).

If $I=\mathbb{Q}(\zeta_{(d,2n)})$, then the automorphism $\sigma_{a}$ fixes
$I$ if and only if $a\equiv 1\bmod(d,2n)$. Suppose now that
$I=\mathbb{Q}(\zeta_{(d,2n)},\sqrt{\ell^{*}})$ and
$\mathbb{Q}(\sqrt{\ell^{*}})\not\subseteq\mathbb{Q}(\zeta_{(d,2n)})$. By Lemma
8 the automorphism $\sigma_{a}$ fixes
$\mathbb{Q}(\sqrt{\ell^{*}})\subseteq\mathbb{Q}(\zeta_{\ell})$ if and only if
$a$ is a square modulo $\ell$, i.e. $\left(\frac{a}{\ell}\right)=1$. ∎

######  Remark 6.

Let $n$ be a squarefree number. We collect here some technical details on the
numbers $(d,2n)$ and $[d,2n]$, and on the condition $a\equiv 1\bmod(d,2n)$.

  * •

If $d$ is odd, or $4\nmid d$ and $n$ is even, then $(d,2n)=(d,n)$ and
$[d,2n]=2[d,n]$.

  * •

If $d$ is even and $n$ is odd, or $4\mid d$, then $(d,2n)=2(d,n)$ and
$[d,2n]=[d,n]$.

If $d$ is even and $n$ is odd, then we have $a\equiv 1\bmod 2(d,n)$ if and
only if $a\equiv 1\bmod(d,n)$, because $a$ must be odd. If $4\mid d$ and $n$
is even, then $a\equiv 1\bmod 2(d,n)$ holds only if $a\equiv 1\bmod 4$, and in
this case $a\equiv 1\bmod 2(d,n)$ is equivalent to $a\equiv 1\bmod(d,n)$.

###### Proof of Theorem 6.

Recall that the degree $[\mathbb{Q}(\zeta_{[d,2n]},\ell^{1/n}):\mathbb{Q}]$
equals $\varphi([d,2n])n/2$ if $n$ is even and $\ell\mid[d,n]$, and it equals
$\varphi([d,2n])n$ otherwise. For the computation of the density
$\delta_{d,a}$ we distinguish the cases $\ell\nmid d$ and $\ell\mid d$.

_Case 1: $\ell\nmid d$._ Using Lemma 9 we see that the expression (30)
simplifies to

$\delta_{d,a}=\sum_{\begin{subarray}{c}n\geq 1\\\ a\equiv
1\bmod(d,2n)\end{subarray}}\frac{\varphi(d)\mu(n)}{[\mathbb{Q}(\zeta_{[d,2n]},\ell^{1/n}):\mathbb{Q}]}.$

In view of the degree formulas, we obtain

$\delta_{d,a}=\Bigg{(}\sum_{\begin{subarray}{c}n\geq 1\\\ a\equiv
1\bmod(d,2n)\end{subarray}}+\sum_{\begin{subarray}{c}2\ell\mid n\\\ a\equiv
1\bmod(d,2n)\end{subarray}}\Bigg{)}\frac{\mu(n)}{\omega_{d}(n)},$ (33)

with $\omega_{d}(n)$ as in Lemma 7.

_Case 1.1: $4\nmid d$._ By Remark 6, from (33) we have

$\displaystyle\delta_{d,a}$ $\displaystyle=\Bigg{(}\sum_{2\nmid n\atop a\equiv
1\bmod(d,n)}+\frac{1}{2}\sum_{2\mid n\atop a\equiv
1\bmod(d,n)}+\frac{1}{2}\sum_{2\ell\mid n\atop a\equiv
1\bmod(d,n)}\Bigg{)}\frac{\mu(n)}{\omega_{d}(n)}$
$\displaystyle=\Bigg{(}\sum_{n\geq 1\atop a\equiv
1\bmod(d,n)}-\frac{1}{2}\sum_{2\mid n\atop a\equiv
1\bmod(d,n)}+\frac{1}{2}\sum_{2\ell\mid n\atop a\equiv
1\bmod(d,n)}\Bigg{)}\frac{\mu(n)}{\omega_{d}(n)}.$

Then using Lemma 7 we obtain

$\delta_{d,a}=S(1)-\frac{1}{2}S_{2}(1)+\frac{1}{2}S_{2}(\ell)=\frac{1}{2}\left(3S(1)-S(\ell)\right)=\frac{A}{2}\left(3+\frac{1}{\ell^{2}-\ell-1}\right)R(d,a).$

_Case 1.2: $4\mid d$._ In view of Remark 6, if $a\equiv 1\bmod 4$, then (33)
becomes

$\delta_{d,a}=\Bigg{(}\sum_{n\geq 1\atop a\equiv 1\bmod(d,n)}+\sum_{2\ell\mid
n\atop a\equiv
1\bmod(d,n)}\Bigg{)}\frac{\mu(n)}{\omega_{d}(n)}=S(1)+S_{2}(\ell)=A\left(1+\frac{1}{\ell^{2}-\ell-1}\right)R(d,a).$

If $a\equiv 3\bmod 4$, then from (33) we are left with

$\delta_{d,a}=\sum_{2\nmid n\atop a\equiv
1\bmod(d,2n)}\frac{\mu(n)}{\omega_{d}(n)}=2S(1)=2AR(d,a).$ (34)

_Case 2: $\ell\mid d$._ We distinguish the two cases: $a$ is a square modulo
$\ell$ or not.

_Case 2.1: $\left(\frac{a}{\ell}\right)=-1$._ Notice that the condition
$a\equiv 1\bmod(d,2n)$ implies in particular that $\ell\nmid n$, otherwise we
would have $a\equiv 1\bmod\ell$ and hence a contradiction with the assumption.
Thus, by Lemma 9 and Remark 6 we have

$\delta_{d,a}=\sum_{2\nmid n\atop a\equiv
1\bmod(d,n)}\frac{\mu(n)}{\omega_{d}(n)}\\\ =S(1)-S_{2}(1)=2AR(d,a).$

_Case 2.2: $\left(\frac{a}{\ell}\right)=1$._ By Lemma 9 we obtain

$\delta_{d,a}=\sum_{n\geq 1\atop a\equiv
1\bmod(d,2n)}\frac{\varphi(d)\mu(n)}{[\mathbb{Q}(\zeta_{[d,2n]},\ell^{1/n}):\mathbb{Q}]}=\Bigg{(}\sum_{2\nmid
n\atop a\equiv 1\bmod(d,2n)}+2\sum_{2\mid n\atop a\equiv
1\bmod(d,2n)}\Bigg{)}\frac{\mu(n)}{\omega_{d}(n)}.$

In the following we take Remark 6 into account. If $4\nmid d$, then

$\delta_{d,a}=S(1)=AR(d,a).$

If $4\mid d$ and $a\equiv 1\bmod 4$, then

$\delta_{d,a}=S(1)+S_{2}(1)=0.$

If $4\mid d$ and $a\equiv 3\bmod 4$, then

$\delta_{d,a}=S(1)-S_{2}(1)=2AR(d,a).\qed$

####  6.1.2. Alternative proof of Theorem 6

###### Proof of Theorem 6.

We start by noting, cf. (14), that

$\mathcal{P}_{d,a}=\\{p:p\equiv a\bmod
d,\,\operatorname{ord}_{p}(\ell)=p-1\text{\leavevmode\nobreak\
or\leavevmode\nobreak\ }p\equiv 3\bmod 4\text{\leavevmode\nobreak\
and\leavevmode\nobreak\ }\operatorname{ord}_{p}(\ell)=(p-1)/2\\}.$ (35)

Without loss of generalization we may assume that $4$ divides $d$: if $4\nmid
d$ we split the progression into two, according to whether $a\equiv 1\bmod 4$
or $a\equiv 3\bmod 4$, and add the results. Thus, if $a\equiv 1\bmod 4$, then
we just have

$\mathcal{P}_{d,a}=\\{p:p\equiv a\bmod
d,\,\operatorname{ord}_{p}(\ell)=p-1\\}.$

Using Theorem 5 and the law of quadratic reciprocity we conclude that

$\delta_{d,a}=AR(d,a)c(d,a)\text{\leavevmode\nobreak\ with\leavevmode\nobreak\
}c(d,a)=\begin{cases}1-\left(\frac{a}{\ell}\right)&\text{if }a\equiv 1\bmod
4,\,\ell\mid d;\\\ 1+\frac{1}{\ell^{2}-\ell-1}&\text{if }a\equiv 1\bmod
4,\,\ell\nmid d.\\\ \end{cases}$ (36)

If $a\equiv 3\bmod 4$, then by (34) we arrive at

$\delta_{d,a}=AR(d,a)c(d,a),\text{\leavevmode\nobreak\
with\leavevmode\nobreak\ }c(d,a)=2.$

Now let us suppose that $4\nmid d$. We put $d_{1}=\operatorname{lcm}(4,d)$. We
let $a_{1}$ and $a_{3}$ be integers such that $a_{j}\equiv a\bmod d$ and
$a_{j}\equiv j\bmod 4$. Noting that $R(d_{1},a_{j})=R(d,a)$ and
$\varphi(d_{1})=2\varphi(d)$, we conclude that

$\delta_{d,a}=\frac{\delta_{d_{1},a_{1}}+\delta_{d_{1},a_{3}}}{2}=\frac{A}{2}(c(d_{1},a_{1})+c(d_{1},a_{3}))R(d,a).$

We find that $c(d_{1},a_{3})=2$ and noticing that if $\ell\mid d$, then we
have ${\bigl{(}\frac{a_{1}}{\ell}\bigr{)}}={\bigl{(}\frac{a}{\ell}\bigr{)}}$,
we obtain from (36) that

$c(d_{1},a_{1})=\begin{cases}1-\left(\frac{a}{\ell}\right)&\text{if }\ell\mid
d;\\\ 1+\frac{1}{\ell^{2}-\ell-1}&\text{if }\ell\nmid d.\end{cases}$

The proof (with the reformulation of $c(d,a)$ as given in Remark 3) is now
easily completed. ∎

######  Example 1.

Take $d=4$ and $\ell=3$. Moree and Zumalacárregui [27] crucially made use of
the sets $\mathcal{P}_{4,1}$ and $\mathcal{P}_{4,3}$ in their solution of a
conjecture of Salajan. By a simple direct computation they showed that
$\delta_{4,1}=6A/5$ and $\delta_{4,3}=2A$ [27, Appendix A], in agreement with
our results. These sets play also an important role in the resolution of
Browkin’s generalization of the Salajan conjecture by Ciolan and Moree [5].

####  6.1.3. The extremal behavior of $\delta_{d,a}$

Theorem 4 gives a lower bound for $\mathcal{P}_{G}(d,a)(x)$. In this section
we will study how small and large this lower bound can be. For $\ell=2$ this
was done in [14, Sect. 2.2]. This amounts to bounding $\delta_{d,a}$, which a
priori satisfies $0\leq\delta_{d,a}\leq 1$, as it is a relative density.

Small $\delta_{d,a}$. We put

$G(d)=A\prod_{p\mid d}\left(1+\frac{1}{p^{2}-p-1}\right).$

Note that

$G(d)=\prod_{p\nmid d}\left(1-\frac{1}{p(p-1)}\right)<1.$

Recall that

$AR(d,a)=G(d)\prod_{p\mid(a-1,d)}\left(1-\frac{1}{p}\right).$

If $d$ is even, then $(a-1,d)$ is even and we infer that $AR(d,a)<1/2$. If $d$
is odd, then $G(d)<1/2$ and again $AR(d,a)<1/2$. As $c(d,a)\leq 2$, we infer
that

$\delta_{d,a}=AR(d,a)c(d,a)<1.$

Clearly

$AR(d,a)\geq AR(d,1)=\frac{\varphi(d)}{d}G(d).$

The ratio $\varphi(d)/d$ takes on local minima on products of consecutive
primes. For these products we see that $G(d)$ tends to 1, on noting that

$\prod_{p\geq
q}\Big{(}1-\frac{1}{q(q-1)}\Big{)}=1+O\Big{(}\frac{1}{q}\Big{)}.$ (37)

Using this and Mertens’ theorem (see [1, Theorem 13.13]), with $\gamma$
Euler’s constant,

$\prod_{p\leq x}\Big{(}1-\frac{1}{p}\Big{)}\sim\frac{e^{-\gamma}}{\log x},$
(38)

we can then infer that

$\liminf_{d\to\infty}AR(d,1)\log\log d=e^{-\gamma}.$ (39)

The argument is similar to that of the proof of the classical result (see, for
instance, [1, Theorem 13.14])

$\liminf_{d\to\infty}\frac{\varphi(d)}{d}\log\log d=e^{-\gamma}.$

######  Proposition 6.

Let $\ell$ be a prime. We have

$\liminf_{d\to\infty}\min_{\begin{subarray}{c}1\leq a<d\\\ (a,d)=1\\\
\delta_{d,a}>0\end{subarray}}\delta_{d,a}\log\log d=e^{-\gamma}.$

###### Proof.

For $\ell=2$ this result is Proposition 2.3 of [14]. Our proof for odd $\ell$
is in the same spirit.

If $\delta_{d,a}>0$, then $c(d,a)\geq 1$. Hence $\delta_{d,a}\log\log d\geq
AR(d,a)\log\log d\geq AR(d,1)\log\log d$. It now follows from (39) that the
limes inferior is $\geq e^{-\gamma}$. In order to show that this bound is
actually sharp we will show that
$\lim_{n\rightarrow\infty}\delta_{d_{n},1}=e^{-\gamma}$, where
$d_{n}=\prod_{3\leq p\leq n}p$.

Let $n\geq\ell$ be arbitrary. We have $c(d_{n},1)=1$ and

$\delta_{d_{n},1}=\Big{(}1+O\Big{(}\frac{1}{n}\Big{)}\Big{)}\prod_{2\leq p\leq
n}\Big{(}1-\frac{1}{p}\Big{)},$

where we used that

$\prod_{p\nmid
d_{n}}\Big{(}1-\frac{1}{p(p-1)}\Big{)}=\frac{1}{2}\prod_{p>n}\Big{(}1-\frac{1}{p(p-1)}\Big{)}=\frac{1}{2}+O\Big{(}\frac{1}{n}\Big{)},$

with the last equality following from (37). Using Mertens’ theorem (38) and
the prime number theorem in the form $\log d_{n}\sim n$, we deduce that, as
$n$ tends to infinity,

$\delta_{d_{n},1}\sim\frac{e^{-\gamma}}{\log n}\sim\frac{e^{-\gamma}}{\log\log
d_{n}},$

completing the proof. ∎

Large $\delta_{d,a}$. Proposition 7 gives information on how large
$\delta_{d,a}$ can be. In its proof we make use of the elementary concepts of
$a$\- and $d$-sequence, which we now introduce. Let $q_{1},q_{2},\ldots$ be a,
possibly finite, sequence of pairwise coprime integers and
$\alpha_{1},\alpha_{2},\ldots$ any integer sequence of equal length. Put
$d_{k}=\prod_{j=1}^{k}q_{j}$. By the Chinese remainder theorem the system of
congruences

$x\equiv\alpha_{1}\bmod{q_{1}},\,x\equiv\alpha_{2}\bmod{q_{2}},\ldots,\,x\equiv\alpha_{k}\bmod{q_{k}},$
(40)

is equivalent with

$x\equiv a_{k}\bmod{d_{k}},\quad 0\leq a_{k}<d_{k},$

with $a_{k}$ unique (which is a consequence of the Chinese remainder theorem).
Thus to the system (40) we can associate the $(a,d)$-sequence
$(a_{1},d_{1}),(a_{2},d_{2}),\ldots$. For example, the system of congruences
$x\equiv 3\bmod 4$ and $x\equiv 2\bmod{p_{i}}$, with $p_{i}$ running through
the consecutive odd primes, leads to the $a$-sequence
$\\{3,11,47,107,3467,45047,\ldots\\}$.

######  Proposition 7.

Let $\ell$ be an odd prime. We have

$\delta_{d,a}<\begin{cases}\frac{1}{4}\Big{(}3-\frac{2}{\ell(\ell-1)}\Big{)}&\text{if
}\ell\nmid d,4\nmid d;\\\ \frac{1}{2}&\text{if }\ell\nmid d,4\mid d,a\equiv
1\bmod 4;\\\ \frac{1}{3}&\text{if }3\mid d,\ell=3,4\nmid d,a\equiv 1\bmod
3;\\\ \frac{1}{2}&\text{if }\ell\mid d,\ell>3,4\nmid
d,\left(\frac{a}{\ell}\right)=1;\\\ 1&\text{if }4\mid d\text{ and }a\equiv
3\bmod 4;\\\ 1&\text{if }\ell\mid d\text{ and
}\left(\frac{a}{\ell}\right)=-1,\end{cases}$

and $\delta_{d,a}=0$ in the remaining cases. All of the upper bounds are sharp
in the sense that they do not always hold if an arbitrary $\epsilon>0$ is
subtracted from them.

###### Proof.

Starting point is the formula

$\delta_{d,a}=c(d,a)\prod_{p\mid(a-1,d)}\left(1-\frac{1}{p}\right)\prod_{p\nmid
d}\left(1-\frac{1}{p(p-1)}\right)=c(d,a)\,\Pi_{1}\,\Pi_{2},$

say. The six subcases we denote by respectively a,b,c,d,e and f. For each of
them the conditions imposed on $a$ and $d$ ensure that $(a-1,d)$ has certain
prime factors, e.g., if $4\mid d$, then $2\mid(a-1,d)$. These factors are
indicated in the $\Pi_{1}$ column of Table 1. Likewise certain factors have to
appear in the $\Pi_{2}$ column. An entry $e$ in the $\Pi_{1}$ column leads to
a factor $1-\frac{1}{e}$ in $\delta_{d,a}$, in the $\Pi_{2}$ column $e$ leads
to $1-\frac{1}{e(e-1)}$. Further, in absence of an entry, we put a $1$ as
factor. Clearly multiplying everything and also multiplying by $c(d,a)$ (which
has a fixed value in each subcase), leads to an upper bound for
$\delta_{d,a}$. For example, in subcase a we obtain

$\frac{1}{2}\Big{(}3+\frac{1}{\ell^{2}-\ell-1}\Big{)}\cdot
1\cdot\frac{1}{2}\Big{(}1-\frac{1}{\ell(\ell-1)}\Big{)}=\frac{1}{4}\Big{(}3-\frac{2}{\ell(\ell-1)}\Big{)},$

with the factor before the dot being $c(d,a)$, $1$ being the contribution to
$\Pi_{1}$ and the rest being the contribution to $\Pi_{2}$. This explains the
upper bound for $\delta_{d,a}$ in subcase a, and the other upper bounds are
read off similarly from Table 1.

It remains to establish the sharpness of the upper bounds. We do this by
indicating six families $(a_{j},d_{j})$ such that $\delta_{d_{j},a_{j}}$ tends
to the indicated upper bound. The $a_{j}$ are solutions of a certain system of
congruences, the default system being

$x\equiv 2\bmod{3},\,x\equiv 2\bmod{5},\,x\equiv 2\bmod{7},\ldots,\,x\equiv
2\bmod{11},\ldots$

where the moduli run over the consecutive odd primes. In each of the six cases
we make some small modifications to the default system involves at most the
moduli $3$ and $\ell$, and we possibly add a congruence modulo $4$.
Construction of the $(a,d)$-sequences
a) We remove the congruence $x\equiv 2\bmod\ell$. Trivially now the
$a$-sequence is $2,2,2,2,\ldots$.
b) We start with the congruence $x\equiv 1\bmod 4$ and remove the congruence
$x\equiv 2\bmod\ell$. Thus for $\ell=5$ we find, for example, the $a$-sequence
$1,5,65,233,3005,51053,\ldots$.
c) We start with $x\equiv 1\bmod 3$ and obtain the $a$-sequence
$1,7,37,772,10012,85087\ldots$.
d) Here we need $\ell>3$. The congruence $x\equiv 2\bmod\ell$ is changed to
$x\equiv 4\bmod\ell$. Thus for $\ell=5$ we find, for example, the $a$-sequence
$2,14,44,464,12014,102104,\ldots$.
e) We start with the congruence $x\equiv 3\bmod 4$, leading to an $a$-sequence
$3,11,47,107,3467,45047,\ldots$
f) We change $x\equiv 2\bmod\ell$ to $x\equiv n_{0}\bmod\ell$, with $n_{0}$
the smallest non-residue modulo $\ell$. For $\ell=7$ we obtain $a$-sequence
$2,2,17,332,10727,145862,\ldots$. (Note that we get the default congruence
system if and only if $\ell\equiv\pm 3\bmod 8$.)

Let $k\geq 2$. In each of the above subcases the constructed $(a,d)$-sequence
has the property that $(a_{k},d_{k})=1$ and in addition $p\mid(a_{k}-1,d_{k})$
if and only if $p$ is in the $\Pi_{1}$ column. If $k$ is large enough, the
primes that appear in the $\Pi_{2}$ column are precisely those indicated in
that column, with in addition all prime $p\geq p_{0}$ for some $p_{0}$ tending
to infinity with $k$. We conclude that, as $k$ gets larger,

$\delta_{a_{k},d_{k}}=(\text{upper bound})\prod_{p\geq
p_{0}}\left(1-\frac{1}{p(p-1)}\right)\rightarrow\text{upper bound},$

concluding the proof. ∎

Table 1

subcase | $c(d,a)$ | $\Pi_{1}$ | $\Pi_{2}$
---|---|---|---
a | $\frac{1}{2}(3+\frac{1}{\ell^{2}-\ell-1})$ |  | $2,\ell$
b | $1+\frac{1}{\ell^{2}-\ell-1}$ | $2$ | $\ell$
c | $1$ | $3$ | $2$
d | $1$ |  | $2$
e | $2$ | $2$ |
f | $2$ |  $2$ if $2\mid d$ |  $2$ if $2\nmid d$

######  Remark 7.

Our choice of the six sequences $(a_{k},d_{k})$ was very canonical, but in
fact given $d_{k}$ many choices of $a_{k}$ are allowed. E.g., in subcase c
there are $\varphi(d_{k})\prod_{3\leq p\leq p_{k}}(p-2)/(p-1)$ choices
allowed, with $p_{k}$ the $k$th odd prime. This number is asymptotically equal
to $c_{1}\varphi(d_{k})/\log\log d_{k}$, for some positive constant $c_{1}$.
The same conclusion, with possibly different $c_{1}$, is valid for the other
five sequences.

It remains to deal with the case $\ell=2$. Proceeding as above one deduces
from Proposition 1.12 of [14] the following result.

######  Proposition 8.

Let $\ell=2$. We have

$\delta_{d,a}=\begin{cases}\frac{3}{4}&\text{if\leavevmode\nobreak\ }4\nmid
d;\\\ \frac{1}{2}&\text{if\leavevmode\nobreak\ }4\mid d,8\nmid
d,\leavevmode\nobreak\ a\equiv 1\leavevmode\nobreak\ ({\rm
mod\leavevmode\nobreak\ }4);\\\ 1&\text{if\leavevmode\nobreak\ }4\mid d,8\nmid
d,\leavevmode\nobreak\ a\equiv 3\leavevmode\nobreak\ ({\rm
mod\leavevmode\nobreak\ }4);\\\ 1&\text{if\leavevmode\nobreak\ }8\mid
d,\leavevmode\nobreak\ a\not\equiv 1\leavevmode\nobreak\ ({\rm
mod\leavevmode\nobreak\ }8);\end{cases}$

and $\delta_{d,a}=0$ in the remaining cases. All of the upper bounds are sharp
in the sense that they do not always hold if an arbitrary $\epsilon>0$ is
subtracted from them.

####  6.1.4. The proof of Theorem 4

###### Proof.

Let $\epsilon>0$ be arbitrary. Recalling the definition (29) of
$\mathcal{P}_{d,a}$, we find that, for all $x$ sufficiently large,

$\mathcal{P}_{G}(d,a)(x)\geq\pi(x;d,a)-\mathcal{P}_{d,a}(x)+O(1)\geq(1-\delta_{d,a}-\epsilon)\frac{x}{\varphi(d)\log
x},$

where we used the upper bound (31) for $\mathcal{P}_{d,a}(x)$, and the well-
known asymptotic $\pi(x;d,a)\sim x/(\varphi(d)\log x)$. The claims regarding
$1-\delta_{d,a}$ are an immediate consequence of Proposition 7. ∎

######  Example 2.

Setting $a=d=1$ we have $\mathcal{P}_{G}(1,1)(x)=\mathcal{P}_{G}(x)$ in our
earlier notation. Theorem 4 then yields the second inequality in (28).

##  7\. Prime divisors of the $H^{\varepsilon}$-sequences

###  7.1. The $H^{-}$-sequence

In this section we obtain the results on the growth behavior of
$\mathcal{P}_{H^{\varepsilon}}(d,a)(x)$ that are needed in order to prove
Theorem 3.

####  7.1.1. Statement of results

Let $\ell$ be an odd prime number and $1\leq a<d$ be coprime integers. In this
section we consider the set of rational (odd) primes

$\mathcal{P}_{d,a}^{-}:=\\{p\equiv a\bmod
d:\operatorname{ord}_{p}(\ell)=(p-1)/2\text{\leavevmode\nobreak\
or\leavevmode\nobreak\ }\operatorname{ord}_{p}(\ell)=p-1\\}.$

As Theorem 5 provides all the information we need on the set $\\{p\equiv
a\bmod d:\operatorname{ord}_{p}(\ell)=p-1\\}$, it suffices to consider the set

$\mathcal{A}_{d,a}^{-}:=\\{p\equiv a\bmod
d:\operatorname{ord}_{p}(\ell)=(p-1)/2\\},$

which, under GRH, has density

$\alpha^{-}_{d,a}=\sum_{n=1}^{\infty}\frac{\varphi(d)\mu(n)c_{a}^{-}(n)}{[\mathbb{Q}(\zeta_{[d,2n]},\ell^{1/2n}):\mathbb{Q}]},$
(41)

where $c_{a}^{-}(n)=1$ if the automorphism $\sigma_{a}$ of
$\mathbb{Q}(\zeta_{d})$ determined by $\sigma_{a}(\zeta_{d})=\zeta_{d}^{a}$ is
the identity on the field
$\mathbb{Q}(\zeta_{d})\cap\mathbb{Q}(\zeta_{2n},\ell^{1/2n})$, and
$c_{a}^{-}(n)=0$ otherwise.

######  Theorem 7.

Let $\ell$ be an odd prime, $a$ and $d$ coprime positive integers,
$\delta_{d,a}^{-}$ as in (41) and $\epsilon$ be arbitrary and fixed. Then for
every $x$ sufficiently large we have

$\mathcal{A}_{d,a}^{-}(x)\leq(\alpha^{-}_{d,a}+\epsilon)\frac{x}{\varphi(d)\log
x},$

where

$\alpha^{-}_{d,a}=A\,c^{-}(d,a)\,R(d,a),$

with $R(d,a)$ as in (17). If $4\mid d$, then

$c^{-}(d,a)=\begin{cases}\frac{1}{2}\left(1-\frac{1}{\ell^{2}-\ell-1}\right)&\text{if
}\ell\nmid d,a\equiv 1\bmod 4;\\\
1-\left(\frac{-1}{\ell}\right)\frac{1}{\ell^{2}-\ell-1}&\text{if }\ell\nmid
d,a\equiv 3\bmod 4;\\\ 0&\textit{if }\ell\mid
d,\left(\frac{\ell}{a}\right)=-1;\\\
\frac{1}{2}\left(3-\left(\frac{-1}{a}\right)\right)&\textit{if }\ell\mid
d,\left(\frac{\ell}{a}\right)=1.\end{cases}$ (42)

If $4\nmid d$, then

$c^{-}(d,a)=\begin{cases}\frac{3}{4}\left(1-\frac{1}{\ell^{2}-\ell-1}\right)&\text{if
}\ell\nmid d,\ell\equiv 1\bmod 4;\\\
\frac{1}{4}\left(3+\frac{1}{\ell^{2}-\ell-1}\right)&\text{if }\ell\nmid
d,\ell\equiv 3\bmod 4;\\\
\frac{3}{4}\left(1+\left(\frac{a}{\ell}\right)\right)&\textit{if }\ell\mid
d,\ell\equiv 1\bmod 4;\\\
\frac{1}{8}\left(5+\left(\frac{a}{\ell}\right)\right)&\textit{if }\ell\mid
d,\ell\equiv 3\bmod 4.\end{cases}$

Assuming GRH, we have

$\mathcal{A}_{d,a}^{-}(x)=\frac{\alpha^{-}_{d,a}}{\varphi(d)}\frac{x}{\log
x}+O_{d}\left(\frac{x\log\log x}{\log^{2}x}\right).$

######  Remark 8.

If $4\ell\mid d$ and ${\bigl{(}\frac{\ell}{a}\bigr{)}}=-1$, then
$\alpha^{-}_{d,a}=0$ for trivial reasons and even $\mathcal{A}^{-}(d,a)$ is
empty. Namely, if $p\in\mathcal{A}^{-}(d,a)$, then $p\equiv a\bmod{\ell}$ and
$\operatorname{ord}_{p}(\ell)=(p-1)/2$. It follows that
${\bigl{(}\frac{\ell}{p}\bigr{)}}=1$ and hence
${\bigl{(}\frac{\ell}{a}\bigr{)}}=1$.

######  Remark 9.

Note that $\mathcal{P}_{d,a}\subseteq\mathcal{P}^{-}_{d,a}$. Comparison with
(35) shows that $\mathcal{P}^{-}_{d,a}=\mathcal{P}_{d,a}$ if $4\mid d$ and
$a\equiv 3\bmod 4$. Hence in this case, under GRH, we have
$c^{-}(d,a)=c(d,a)-c_{1}(d,a)$, with $c(d,a)$ and $c_{1}(d,a)$ explicitly
given in Theorem 6, respectively Theorem 5. By definition, we have
$\mathcal{P}^{-}_{d,a}=\mathcal{P}_{d,a}$ if and only if the density of primes
$p$ such that $p\equiv a\bmod d$, $p\equiv 1\bmod 4$ and
$\operatorname{ord}_{p}(\ell)=(p-1)/2$ is zero. This happens trivially if the
two modular congruences are not compatible, namely when $4\mid d$ and
$a\not\equiv 1\bmod 4$. If they are compatible, then by Theorem 7 under GRH
the considered density is zero if and only if one of the following two
conditions holds:

  * •

$4\ell\mid d$, $a\equiv 1\bmod 4$ and ${\bigl{(}\frac{\ell}{a}\bigr{)}}=-1$;

  * •

$4\nmid d$, $\ell\mid d$, $\ell=1\bmod 4$ and
${\bigl{(}\frac{a}{\ell}\bigr{)}}=-1$.

Combination of Theorem 7 and Theorem 5 yields the following result.

######  Theorem 8.

We have

$\mathcal{P}^{-}_{d,a}(x)\leq(\delta^{-}_{d,a}+\epsilon)\frac{x}{\varphi(d)\log
x},$

where $\delta^{-}_{d,a}=\alpha^{-}_{d,a}+\alpha_{d,a}=AR(d,a)c_{2}(d,a)$ with

$c_{2}(d,a)=c^{-}(d,a)+c_{1}(d,a)=\begin{cases}\frac{1}{2}\left(3+\frac{1}{\ell^{2}-\ell-1}\right)&\text{if
}\ell\nmid d,\ell\equiv 1\bmod 4;\\\ 2&\text{if }\ell\nmid d,\ell\equiv 3\bmod
4;\\\ 2&\text{if }\ell\mid d,\ell\equiv 1\bmod 4;\\\
\frac{1}{2}\left(3-\left(\frac{-1}{a}\right)\right)&\text{if }\ell\mid
d,\ell\equiv 3\bmod 4,4\mid d;\\\ \frac{3}{2}&\textit{if }\ell\mid
d,\ell\equiv 3\bmod 4,4\nmid d.\end{cases}$

Assuming GRH we have

$\mathcal{P}_{d,a}^{-}(x)=\frac{\delta^{-}_{d,a}}{\varphi(d)}\frac{x}{\log
x}+O_{d}\left(\frac{x\log\log x}{\log^{2}x}\right).$

####  7.1.2. Proofs

We start by determining the coefficients $c^{-}_{a}(n)$ that occur in the
infinite sum (41).

######  Lemma 10.

Let $\ell$ be an odd prime, $n$ a squarefree integer, and $d$ a natural number
such that $4\mid d$. Let $\Delta$ be the discriminant of
$\mathbb{Q}(\sqrt{\ell})$. We have

$\mathbb{Q}(\zeta_{d})\cap\mathbb{Q}(\zeta_{2n},\ell^{1/2n})=\mathbb{Q}(\zeta_{(d,2n)},\alpha)$

with

$\alpha=\begin{cases}\sqrt{\ell}&\text{if\leavevmode\nobreak\ }\ell\mid
d,\Delta\nmid 2n;\\\ i&\text{if\leavevmode\nobreak\ }\ell\nmid d,\ell\mid
n,2\nmid n,\ell\equiv 3\bmod 4;\\\ 1&\text{otherwise}.\end{cases}$

In the first two cases $\mathbb{Q}(\zeta_{(d,2n)},\alpha)$ is a quadratic
extension of $\mathbb{Q}(\zeta_{(d,2n)})$.

###### Proof.

Let us set $I:=\mathbb{Q}(\zeta_{d})\cap\mathbb{Q}(\zeta_{2n},\ell^{1/2n})$.
By Kummer theory we may argue that, since
$\mathbb{Q}(\zeta_{\infty})\cap\mathbb{Q}(\zeta_{2n},\ell^{1/2n})$ is a finite
abelian extension of $\mathbb{Q}(\zeta_{2n})$, it is of the form
$\mathbb{Q}(\zeta_{2n},\ell^{e/2n})$ for some $e\geq 0$, and hence by
Schinzel’s theorem [31, Theorem 2] it is $\mathbb{Q}(\zeta_{2n},\sqrt{\ell})$.
The latter extension equals $\mathbb{Q}(\zeta_{2n})$ if $\Delta\mid 2n$.

Therefore, if $\ell\nmid dn$ or $\Delta\mid 2n$, then
$I=\mathbb{Q}(\zeta_{(d,2n)})$. If $\Delta\nmid 2n$ and $\ell\mid d$, then
noticing that $\mathbb{Q}(\zeta_{(d,2n)})\subseteq
I\subseteq\mathbb{Q}(\zeta_{(d,2n)},\zeta_{4\ell})$ and
$\mathbb{Q}(\zeta_{4\ell})\subseteq\mathbb{Q}(\zeta_{d})$, we deduce that
$I=\mathbb{Q}(\zeta_{(d,2n)},\sqrt{\ell})$. If $\ell\mid n$, $n$ is odd,
$\ell\equiv 3\bmod 4$ and $\ell\nmid d$, then we have
$\mathbb{Q}(\zeta_{(d,2n)})\subseteq I\subseteq\mathbb{Q}(\zeta_{(d,4n)})$,
yielding $I=\mathbb{Q}(\zeta_{4(d,n)})$ as $\zeta_{4}\in I$ but
$\zeta_{4}\notin\mathbb{Q}(\zeta_{(d,2n)})$. ∎

######  Corollary 2.

Let $\ell$ be an odd prime, $n$ a squarefree integer, and $a,d$ natural
numbers such that $4\mid d$ and $(a,d)=1$. If $a\not\equiv 1\bmod(d,2n)$, then
$c^{-}_{a}(n)=0$. If $a\equiv 1\bmod(d,2n)$, then

$c^{-}_{a}(n)=\begin{cases}\frac{1}{2}\left(1+\left(\frac{\ell}{a}\right)\right)&\text{if\leavevmode\nobreak\
}\ell\mid d,\Delta\nmid n;\\\
\frac{1}{2}\left(1+\left(\frac{-1}{a}\right)\right)&\text{if\leavevmode\nobreak\
}\ell\nmid d,\ell\mid n,2\nmid n,\ell\equiv 3\bmod 4;\\\
1&\text{otherwise}.\end{cases}$

###### Proof.

An immediate consequence of Lemma 10 on noting that
$\sigma_{a}(\zeta_{(d,2n)})=\zeta_{(d,2n)}^{a}$,
$\sigma_{a}(i)=i^{a}={\bigl{(}\frac{-1}{a}\bigr{)}}i$, and, if $\ell\mid d$,
then $\sigma_{a}(\sqrt{\ell})={\bigl{(}\frac{\ell}{a}\bigr{)}}\sqrt{\ell}$ (by
Lemma 8). ∎

######  Remark 10.

A different proof of Lemma 10 is obtained on using that if $K/\mathbb{Q}$ is
Galois, then

$I:=[K\cap L:\mathbb{Q}]=\frac{[K:\mathbb{Q}][L:\mathbb{Q}]}{[K\cdot
L:\mathbb{Q}]}.$

Applying this equality with $K=\mathbb{Q}(\zeta_{d})$ and
$L=\mathbb{Q}(\zeta_{2n},\ell^{1/2n})$, computing all the degree occurring,
and using that $\varphi((d,2n))\varphi([d,2n])=\varphi(d)\varphi(2n)$, then
shows that in the first two cases $I$ is a quadratic extension of
$\mathbb{Q}(\zeta_{(d,2n)})$ and $I=\mathbb{Q}(\zeta_{(d,2n)})$ otherwise. The
proof is then easily completed.

###### Proof of Theorem 7.

Our starting point for is formula (41), which expresses $\alpha^{-}_{d,a}$ as
an infinite sum, which we will rewrite as an Euler product using Lemma 7 (the
notation of which we will use). Throughout $n$ will be a squarefree integer.
We first assume $4\mid d$, and so $[d,2n]=[d,n]$. Recall that the degree
$[\mathbb{Q}(\zeta_{[d,2n]},\ell^{1/2n}):\mathbb{Q}]$ equals $\varphi([d,n])n$
if $\ell\mid[d,n]$ and $\varphi([d,n])2n$ otherwise. We put

$\Sigma_{1}=\sum_{a\equiv
1\bmod(d,2n)}\frac{\varphi(d)\mu(n)}{[\mathbb{Q}(\zeta_{[d,2n]},\ell^{1/2n}):\mathbb{Q}]},\quad\Sigma_{2}=\sum_{a\equiv
1\bmod(d,2n)\atop\ell\mid d,\,\Delta\nmid
n}\frac{\varphi(d)\mu(n)}{[\mathbb{Q}(\zeta_{[d,2n]},\ell^{1/2n}):\mathbb{Q}]},$

and

$\Sigma_{3}=\sum_{a\equiv 1\bmod(d,2n)\atop\ell\nmid d,\,\ell\mid n,\,2\nmid
n,\,\ell\equiv
3\bmod{4}}\frac{\varphi(d)\mu(n)}{[\mathbb{Q}(\zeta_{[d,2n]},\ell^{1/2n}):\mathbb{Q}]}.$

Notice that the three sums $\Sigma_{2},\Sigma_{3}$ and $\Sigma_{1}$
(respectively) reflect the three cases distinguished in Corollary 2. Making
the values of $c^{-}_{a}(n)$ in (41) explicit using Corollary 2 we obtain

$\alpha^{-}_{d,a}=\Sigma_{1}+\frac{1}{2}\Big{(}{\bigl{(}\frac{\ell}{a}\bigr{)}}-1\Big{)}\Sigma_{2}+\frac{1}{2}\Big{(}{\bigl{(}\frac{-1}{a}\bigr{)}}-1\Big{)}\Sigma_{3}.$

_Case 1: $\ell\mid d$._ Now $\Sigma_{3}=0$ and so

$\alpha^{-}_{d,a}=\Sigma_{1}+\frac{1}{2}\Big{(}{\bigl{(}\frac{\ell}{a}\bigr{)}}-1\Big{)}\Sigma_{2}.$

We have

$\Sigma_{1}=\sum_{a\equiv 1\bmod(d,2n)}\frac{\mu(n)}{\omega_{d}(n)},$

which equals $S(1)=AR(d,a)$ for $a\equiv 1\bmod 4$, and equals
$S(1)-S_{2}(1)=2S(1)=2AR(d,a)$ if $a\equiv 3\bmod 4$.

_Subcase 1.1: ${\bigl{(}\frac{\ell}{a}\bigr{)}}=1$._ Then
$\alpha^{-}_{d,a}=\Sigma_{1}$ and we find
$c^{-}(d,a)=\frac{1}{2}(3-{\bigl{(}\frac{-1}{a}\bigr{)}})$.

_Subcase 1.2: ${\bigl{(}\frac{\ell}{a}\bigr{)}}=-1$._ Suppose $\ell\equiv
1\bmod{4}$. Then by quadratic reciprocity we have
${\bigl{(}\frac{a}{\ell}\bigr{)}}=-1$ and so all $n$ that contribute to
$\Sigma_{1}$ satisfy $\ell\nmid n$. As $\Delta=\ell$ we infer that
$\Sigma_{1}=\Sigma_{2}$ and hence $\alpha^{-}_{d,a}=\Sigma_{1}-\Sigma_{2}=0$.
If $\ell\equiv 3\bmod{4}$, the condition $\Delta\nmid n$ is automatically
satisfied and we obtain $\Sigma_{2}=\Sigma_{1}$, and hence again
$\alpha^{-}_{d,a}=0$. (For a different argument why $\alpha^{-}_{d,a}=0$ see
Remark 8.) We conclude that $c^{-}(d,a)=0$.
_Case 2: $\ell\nmid d$._ Now

$\alpha^{-}_{d,a}=\Sigma_{1}+\frac{1}{2}\Big{(}{\bigl{(}\frac{-1}{a}\bigr{)}}-1\Big{)}\Sigma_{3}.$

We have

$\Sigma_{1}=\Bigg{(}\sum_{\ell\mid n\atop a\equiv
1\bmod(d,2n)}+\frac{1}{2}\sum_{\ell\nmid n\atop a\equiv
1\bmod(d,2n)}\Bigg{)}\frac{\mu(n)}{\omega_{d}(n)}.$

If $a\equiv 1\bmod 4$, then

$\Sigma_{1}=\frac{1}{2}\Bigg{(}\sum_{n\geq 1\atop a\equiv
1\bmod(d,n)}+\sum_{\ell\mid n\atop a\equiv
1\bmod(d,n)}\Bigg{)}\frac{\mu(n)}{\omega_{d}(n)}=\frac{1}{2}(S(1)+S(\ell))=\frac{A}{2}\left(1-\frac{1}{\ell^{2}-\ell-1}\right)R(d,a),$

and hence $c^{-}(d,a)=\frac{1}{2}(1-\frac{1}{\ell^{2}-\ell-1})$.

Next suppose $a\equiv 3\bmod 4$. In this case in the density sums we can
restrict to odd $n$.
_Subcase 2.1: $\ell\equiv 3\bmod 4$._ Now

$\alpha^{-}_{d,a}=\Sigma_{1}-\Sigma_{3}=\frac{1}{2}\sum_{\ell\nmid n,\ 2\nmid
n\atop a\equiv 1\bmod(d,n)}\frac{\mu(n)}{\omega_{d}(n)},$

which is easily seen to equal

$\frac{1}{2}(S(1)-S_{2}(1)-S(\ell)+S_{2}(\ell))=S(1)-S(\ell)=A\left(1+\frac{1}{\ell^{2}-\ell-1}\right)R(d,a).$

_Subcase 2.2: $\ell\equiv 1\bmod 4$._ Now $\Sigma_{3}=0$ and

$\alpha^{-}_{d,a}=\Sigma_{1}=\Bigg{(}\sum_{\ell\mid n,\ 2\nmid n\atop a\equiv
1\bmod(d,n)}+\frac{1}{2}\sum_{\ell\nmid n,\ 2\nmid n\atop a\equiv
1\bmod(d,n)}\Bigg{)}\frac{\mu(n)}{\omega_{d}(n)},$

which equals

$(S(\ell)-S_{2}(\ell))+(S(1)-S(\ell))=S(1)+S(\ell)=A\left(1-\frac{1}{\ell^{2}-\ell-1}\right)R(d,a).$

We conclude that
$c^{-}(d,a)=1-{\bigl{(}\frac{-1}{\ell}\bigr{)}}\frac{1}{\ell^{2}-\ell-1}$.

This completes the proof in the case $4\mid d$. Suppose now that $4\nmid d$.
We may argue as in Sect. 6.1.2 and, keeping the notation $d,a_{1},a_{3}$ as
there, we obtain

$\alpha_{d,a}^{-}=\frac{\alpha_{d_{1},a_{1}}^{-}+\alpha_{d_{1},a_{3}}^{-}}{2}=\frac{A}{2}(c^{-}(d_{1},a_{1})+c^{-}(d_{1},a_{3}))R(d,a),$

and hence

$c^{-}(d,a)=\frac{c^{-}(d_{1},a_{1})+c^{-}(d_{1},a_{3})}{2}.$

Since
${\bigl{(}\frac{a}{\ell}\bigr{)}}={\bigl{(}\frac{a_{1}}{\ell}\bigr{)}}={\bigl{(}\frac{a_{3}}{\ell}\bigr{)}}$,
the idea is to rewrite (42) using quadratic reciprocity in terms of
${\bigl{(}\frac{a}{\ell}\bigr{)}}$. This gives rise to more cases, but makes
it easy to determine $c^{-}(d,a)$. For example, if $\ell\mid
d,\,{\bigl{(}\frac{\ell}{a}\bigr{)}}=1$, and $\ell\equiv 1\bmod{4}$, then
$c^{-}(d_{1},a_{1})=1$ and $c^{-}(d_{1},a_{3})=2$, and thus
$c^{-}(d,a)=\frac{3}{2}$ if $\ell\mid d,{\bigl{(}\frac{a}{\ell}\bigr{)}}=1$,
and $\ell\equiv 1\bmod{4}$. ∎

###  7.2. The $H^{+}$-sequence

Let $\ell$ be an odd prime number and $1\leq a<d$ be coprime integers. In this
section we consider the set of rational (odd) primes

$\mathcal{P}_{d,a}^{+}:=\\{p\equiv a\bmod
d:\operatorname{ord}_{p}(\ell)=p-1\text{\leavevmode\nobreak\
or\leavevmode\nobreak\ }\operatorname{ord}_{p}(\ell)\text{\leavevmode\nobreak\
is\leavevmode\nobreak\ odd}\\}.$

This set can be written as a disjoint union of two sets:

$\mathcal{P}_{d,a}^{+}=\mathcal{A}_{d,a}\cup\\{p\equiv a\bmod
d:\operatorname{ord}_{p}(\ell)\text{\leavevmode\nobreak\
is\leavevmode\nobreak\ odd}\\}.$

Under GRH the density of $\mathcal{A}_{d,a}$ is given in Theorem 5. The
density of the second set has been unconditionally determined in case $\ell$
is a positive rational number by Moree and Sury [26]. Unfortunately, even in
the case where $\ell$ is an odd prime, there are many cases and for this
reason we will not write the details out here. However, under this restriction
on $\ell$ the density is always positive (this follows from [26, Theorem 5]).
If the discriminant of $\mathbb{Q}(\sqrt{\ell})$ divides $d$ and
${\bigl{(}\frac{\ell}{a}\bigr{)}}=1$, then the set $\mathcal{A}_{d,a}$ is
empty (see Remark 2) and we obtain an unconditional asymptotic for
$\mathcal{P}_{d,a}^{+}(x)$. A particular easy case arises for $\ell=3,a=11$
and $d=12$. Then we have $\mathcal{P}^{+}_{12,11}=\\{p:p\equiv
11\bmod{12}\\}$. This basically is a claim Fermat made in 1641! He made some
similar, but unfortunately wrong ones, for the details see [26].

###  7.3. Counting prime divisors of the $H$-, $H^{-}$\- and $H^{+}$-sequences

Denote the sets of prime divisors in the section header by, respectively,
$\mathcal{Q}_{H}$, ${\mathcal{Q}}_{H^{-}}$ and ${\mathcal{Q}}_{H^{+}}$. By
Proposition 3 and Lemma 4 these sets are very closely related to
$\mathcal{P}_{G}$, $\mathcal{P}_{H^{-}}$ and $\mathcal{P}_{H^{+}}$,
respectively. Assuming that the associated Wieferich sets are $o(x/\log x)$
(see Sect. 3.4), we arrive at the following conjecture.

######  Conjecture 5.

Asymptotically we have ${\mathcal{Q}}_{H}(x)\sim\mathcal{P}_{G}(x)$,
${\mathcal{Q}}_{H^{-}}(x)\sim\mathcal{P}_{H^{-}}(x)$ and
${\mathcal{Q}}_{H^{+}}(x)\sim\mathcal{P}_{H^{+}}(x)$, where the asymptotic
behavior of $\mathcal{P}_{G}(x),\mathcal{P}_{H^{-}}(x),\mathcal{P}_{H^{+}}(x)$
is given by (27), (25), respectively (26).

##  8\. Earlier work on the Genocchi $\ell$-integers

We recapitulate some earlier work on $\ell$-Genocchi integers. None of it is
directly relevant for the proofs presented in this paper, and so it can be
regarded as background reading.

###  8.1. The case $\ell=2$

The original Genocchi numbers are obtained on taking $\ell=2$. These numbers
have received considerable attention in the literature. It follows from (12)
and $H_{n}=G_{n}/2n$ that

$\frac{2t}{e^{t}+1}=\sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!}.$

It is well-known that $G_{1}=1$, $G_{2n+1}=0$ for $n\geq 1$, and that
$(-1)^{n}G_{2n}$ is an odd positive integer.

Dumont [8] showed that $|G_{2n}|$ equals the number of permutations $p$ of
$\\{1,2,\ldots,2n-1\\}$ such that $p(i)<p(i+1)$ for $p(i)$ odd and
$p(i)>p(i+1)$ for $p(i)$ even $(1\leq i\leq 2n-2)$. For a survey of related
material see Stanley [33].

Setting $\ell=2$ in (13), in case $p-1\nmid 2n$ we obtain

$H_{2n}\equiv-2^{2n-1}\sum_{j=(p+1)/2}^{p-1}j^{2n-1}\equiv
2^{2n-1}\sum_{j=1}^{(p-1)/2}j^{2n-1}\mkern 4.0mu({\operator@font mod}\mkern
6.0mup).$ (43)

A related result is due to Emma Lehmer [17], who showed in case $2n\not\equiv
2\bmod{(p-1)}$ that

$H_{2n}\equiv-\sum_{j=1}^{(p-1)/2}(p-2j)^{2n-1}\bmod{p^{2}}.$

Under the same assumption on $n$ and in the same spirit she showed that

$\sum_{j=1}^{[p/4]}(p-4j)^{2n-1}\equiv-H_{2n}(2^{2n-1}+1)\bmod{p^{2}},\quad
p>3.$

In case $2n\equiv 2\bmod{(p-1)}$ both congruences still hold true modulo $p$.

For $|t|<\pi/2$ we have (see, e.g., [2, Prop. 1.17])

$\tan
t=\sum_{n=1}^{\infty}T_{n}\frac{t^{2n-1}}{(2n-1)!},\,\text{\leavevmode\nobreak\
with\leavevmode\nobreak\ }\,T_{n}=(-4)^{n}H_{2n}.$

The numbers $T_{n}$ are called _tangent numbers_ and count the number of all
alternating permutations of length $2n-1$ (see Entringer [9] or Knuth and
Buckholtz [15]).

Let $p>3$ be a prime satisfying $p\equiv 3\mkern 4.0mu({\operator@font
mod}\mkern 6.0mu4)$ and put $m=(p-1)/2$. Then the class number $h(-p)$ of the
quadratic field $\mathbb{Q}(\sqrt{-p})$ satisfies

$h(-p)=\frac{1}{2-{\bigl{(}\frac{2}{p}\bigr{)}}}\sum_{j=1}^{m}{\bigl{(}\frac{j}{p}\bigr{)}}\equiv\frac{1}{2-2^{m}}\sum_{j=1}^{m}j^{m}\equiv\frac{2^{-m}}{2-2^{m}}H_{\frac{p+1}{2}}\equiv-2B_{\frac{p+1}{2}}\,\mkern
4.0mu({\operator@font mod}\mkern 6.0mup),$

a congruence due to Cauchy. The first identity is an easy consequence of
Dirichlet’s class number formula (see, for example, [2, p. 99]), the second
congruence follows from (43) on setting $n=(p+1)/4$. The first identity
implies that $h(-p)\leq(p-1)/2$ and thus the congruence uniquely determines
the value of $h(-p)$. For some related results we refer to the recent preprint
by Mináč et al. [20].

###  8.2. The case $\ell$ is odd

Despite the enormous literature on variations of Bernoulli numbers, we found
only very little earlier work on $\ell$-Genocchi numbers for odd $\ell$. For
example, in case $\ell=3$ and $p>3$, Emma Lehmer [17] showed that

$H_{2n}\equiv-2\sum_{j=1}^{[p/3]}(p-3j)^{2n-1}\bmod{p^{2}}.$

The deepest result we found gives a connection with functions related to
polylogarithms. Namely, given any integer $k$ consider the formal series

$l_{k}(s)=\sum_{n=1}^{\infty}\frac{s^{n}}{n^{k}}.$

Wójcik [35] found an explicit formula for $l_{k}$ for $k\leq 0$, namely

$l_{k}(s)=-\frac{sR_{n}(s)}{(s-1)^{n+1}},$

where $n=-k$ and the $R_{n}\in\mathbb{Z}[s]$ are the classical Euler-Frobenius
polynomials defined by the formula

$\frac{1-s}{e^{t}-s}=\sum_{n=0}^{\infty}\frac{R_{n}(s)}{(1-s)^{n}}\frac{t^{n}}{n!}.$

The individual terms in the first sum of (12) can be connected with the series
$l_{k}(s)$, namely it can be shown (cf. Urbanowicz and Williams [34, p. 132])
that

$\frac{z}{e^{t}-z}=\sum_{n=1}^{\infty}(-1)^{n}l_{-n}(z)\frac{t^{n}}{n!}.$

From (12) we then infer on equating Taylor coefficients

$H_{n}=(-1)^{n}\sum_{a=1}^{\ell-1}l_{1-n}(\zeta_{\ell}^{a}).$

##  9\. Experimental data

In this section we provide some numerical examples for the densities of $G$-,
$H^{+}$\- and $H^{-}$-irregular primes, and $G$-irregular primes in arithmetic
progressions and compare them with our conjectural predictions. As customary,
$\pi(x)$ is the number of primes $p\leq x$. All data have been produced with
SageMath [30].

Table 1. The ratio $\mathcal{P}_{G}(x)/\pi(x)$ for $x=10^{5}$ $\ell$ | experimental | theoretical
---|---|---
2 | 0.661593 | 0.659776
3 | 0.635113 | 0.637095
5 | 0.657214 | 0.653807
7 | 0.660863 | 0.657010
11 | 0.660133 | 0.658736
13 | 0.659612 | 0.659045
17 | 0.662948 | 0.659358
19 | 0.657110 | 0.659444
Table 2. The ratios $\mathcal{P}_{H^{+}}(x)/\pi(x)$ and $\mathcal{P}_{H^{-}}(x)/\pi(x)$ for $x=10^{5}$ | $\mathcal{P}_{H^{+}}(x)/\pi(x)$ |  | $\mathcal{P}_{H^{-}}(x)/\pi(x)$ |
---|---|---|---|---
$\ell$ | experimental | theoretical | experimental | theoretical
2 | 0.599145 | 0.596279 | 0.603315 | 0.603072
3 | 0.568390 | 0.571007 | 0.588198 | 0.591731
5 | 0.563699 | 0.559070 | 0.599458 | 0.600088
7 | 0.575271 | 0.571007 | 0.604671 | 0.601689
11 | 0.571726 | 0.571007 | 0.604462 | 0.602552
13 | 0.571518 | 0.569544 | 0.600292 | 0.602706
17 | 0.573499 | 0.570170 | 0.607173 | 0.602863
19 | 0.569537 | 0.571007 | 0.599875 | 0.602906
Table 3. The ratio $\mathcal{P}_{G}(d,a)(x)/\pi(x;d,a)$ for $x=10^{5}$

$\ell$ | $d$ | $a$ | experimental | theoretical
---|---|---|---|---
3 | 3 | 1 | 0.816097 | 0.818547
| 3 | 2 | 0.454337 | 0.455642
| 4 | 1 | 0.717473 | 0.727821
| 4 | 3 | 0.552752 | 0.546368
5 | 3 | 1 | 0.725396 | 0.723046
| 3 | 2 | 0.589241 | 0.584569
| 4 | 1 | 0.763970 | 0.761246
| 4 | 3 | 0.550459 | 0.546368
7 | 3 | 1 | 0.723311 | 0.725608
| 3 | 2 | 0.598624 | 0.588412
| 4 | 1 | 0.764178 | 0.767652
| 4 | 3 | 0.557548 | 0.546368

$\ell$ | $d$ | $a$ | experimental | theoretical
---|---|---|---|---
11 | 7 | 1 | 0.695580 | 0.700353
| 7 | 2 | 0.661802 | 0.650412
| 7 | 3 | 0.649291 | 0.650412
| 7 | 4 | 0.649917 | 0.650412
| 7 | 5 | 0.670559 | 0.650412
| 7 | 6 | 0.634279 | 0.650412
| 15 | 1 | 0.753962 | 0.770096
| 15 | 2 | 0.576314 | 0.568929
| 15 | 4 | 0.706422 | 0.712620
| 15 | 7 | 0.724771 | 0.712620
| 15 | 8 | 0.573812 | 0.568929
| 15 | 11 | 0.649708 | 0.655144
| 15 | 13 | 0.712260 | 0.712620
| 15 | 14 | 0.583820 | 0.568929

## Acknowledgement

The authors thank Karl Dilcher, Bakir Farhi, Masanobu Kaneko and Bernd Kellner
for kindly answering some Bernoulli number related questions. A very helpful
e-mail exchange with Efthymios Sofos led to a simplified proof of Proposition
7. Thanks to Antonella Perucca for her constant support and encouragement.

## References

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