# On Iwasawa $\lambda$-invariants for abelian number fields and random matrix
heuristics

Daniel Delbourgo Department of Mathematics and Statistics
University of Waikato
Gate 8
Hillcrest Road
Hamilton
New Zealand
3240<EMAIL_ADDRESS>and  Heiko Knospe Technische Hochschule
Köln
Fakultät für Informations-, Medien- und Elektrotechnik
Institut für Nachrichtentechnik
Betzdorfer Str. 2
50679 Köln
Germany<EMAIL_ADDRESS>
###### Abstract.

Following both Ernvall-Metsänkylä and Ellenberg-Jain-Venkatesh, we study the
density of the number of zeroes (i.e. the cyclotomic $\lambda$-invariant) for
the $p$-adic zeta-function twisted by a Dirichlet character $\chi$ of any
order. We are interested in two cases: (i) the character $\chi$ is fixed and
the prime $p$ varies, and (ii) $\text{\rm ord}(\chi)$ and the prime $p$ are
both fixed but $\chi$ is allowed to vary. We predict distributions for these
$\lambda$-invariants using $p$-adic random matrix theory and provide numerical
evidence for these predictions. We also study the proportion of $\chi$-regular
primes, which depends on how $p$ splits inside $\mathbb{Q}(\chi)$. Finally in
an extensive Appendix A, we tabulate the values of the $\lambda$-invariant for
every character $\chi$ of conductor $\leqslant 1000$ and for odd primes $p$ of
small size.

###### Key words and phrases:

Iwasawa theory, $p$-adic $L$-functions, random matrices

###### 2010 Mathematics Subject Classification:

Primary 11R23; Secondary 11R42, 11S80, 11M41

###### Contents

  1. 1 Introduction and three conjectures
  2. 2 Estimates from $p$-adic random matrix theory
    1. 2.1 The statistics for $\text{\rm GL}(n,\mathcal{O})$ and $\text{\rm GSp}(2n,\mathcal{O})$
    2. 2.2 An application of Euler’s pentagonal theorem
    3. 2.3 Heuristics for the $\chi$-regularity of $p$
    4. 2.4 Heuristics for totally real abelian extensions
  3. 3 The general method and a summary of the data
    1. 3.1 Numerical evidence for Conjecture 1.1
    2. 3.2 Numerical evidence for Conjecture 1.2
    3. 3.3 Numerical evidence for Conjecture 1.3
  4. A Tables of $\lambda$-invariants for small $\chi$ and $p$.

##  1\. Introduction and three conjectures

Let $p$ be an odd prime, and suppose that
$\chi:\mathbb{Z}\rightarrow\mathbb{C}$ denotes a Dirichlet character. The
Kubota-Leopoldt $L$-function $\mathbf{L}_{p}(\chi,s)$ is a meromorphic
function [16] of the $p$-adic variable $s$, which interpolates values of the
$\chi$-twisted Riemann zeta-function along the critical strip of non-positive
integers. More precisely, one has the formula

(1.1)
$\mathbf{L}_{p}(1-n,\chi)\;=\;-\big{(}1-\chi\omega^{-n}(p)p^{n-1}\big{)}\times\frac{B_{n,\chi\omega^{-n}}}{n}\quad\text{at
every $n\in\mathbb{N}$;}$

here $\omega$ indicates the Teichmüller character mod $p$, and
$B_{n,\chi\omega^{-n}}$ equals the $n$-th $\chi\omega^{-n}$-twisted Bernoulli
number. If $\chi$ is non-trivial then the $p$-adic zeta-function is analytic
in $s$, whilst if $\chi=\mathbf{1}$ then it has a simple pole at $s=1$ with
residue $1-\frac{1}{p}$.

Now suppose that $\chi$ is a non-trivial character of the first kind, namely
that $\chi=\theta\omega^{i}$ where $p\nmid\text{\rm cond}(\theta)$ and
$i\in\\{0,\dots,p-2\\}$. We will also set
$\mathcal{O}_{\chi}:=\mathbb{Z}_{p}[\text{\rm Im}(\chi)]$. Iwasawa associated
a unique power series
$\mathcal{G}_{\chi}(T)=\sum_{j=0}^{\infty}c_{j}T^{j}\in\mathcal{O}_{\chi}[\\![T]\\!]$
satisfying

$\mathcal{G}_{\chi}\big{(}(1+p)^{-s}-1\big{)}=\;\mathbf{L}_{p}(s,\chi)\quad\text{for
all $s\in\widehat{\overline{\mathbb{Q}}}_{p}$ with
$\big{|}s\big{|}_{p}\\!<p^{\frac{p-2}{p-1}}$.}$

The $\mu$-invariant of $\mathcal{G}_{\chi}$ is known to vanish by the Ferrero-
Washington theorem [12] in which case this function factorises into a product
of an invertible power series, and a uniquely determined distinguished
polynomial

$T^{\lambda}+a_{\lambda-1}T^{\lambda-1}+\dots+a_{1}T+a_{0}\in\mathcal{O}_{\chi}[T]\quad\text{with
$\;\big{|}a_{\lambda-1}\big{|}_{p},\;\dots\;,\big{|}a_{0}\big{|}_{p}<\;1$.}$

Its degree $\lambda=\lambda_{p}(\chi)$ equals the number of zeroes of
$\mathcal{G}_{\chi}(T)$ on the open unit disk.

######  Conjecture 1.1.

For even characters $\chi$ of the first kind such that $p\nmid\text{\rm
ord}(\chi)$ and $\chi\omega^{-1}(p)\neq 1$, the probability that
$\lambda_{p}(\chi)=r$ is approximately equal to

(1.2) $p^{-fr}\times\;\prod_{t>r}\;\big{(}1-p^{-ft}\big{)}$

where $f\geqslant 1$ denotes the multiplicative order of the prime $p$ modulo
$\text{\rm ord}(\chi)$.

This probability is computed either (i) over primes $p$ and Teichmüller twists
of a fixed character $\chi$, or (ii) over characters $\chi$ of fixed order
$\geqslant 2$ at a fixed prime $p$. The condition $p\nmid\text{\rm ord}(\chi)$
ensures that $\mathcal{O}_{\chi}$ is a finite and unramified extension of
$\mathbb{Z}_{p}$. Moreover we have
$\mathcal{O}_{\chi}=\mathbb{Z}_{p}\big{[}\text{\rm
Im}(\chi)\big{]}=\mathbb{Z}_{p}[\zeta_{\text{\rm ord}(\chi)}]$, hence the
residue class degree of $\mathcal{O}_{\chi}$ over $\mathbb{Z}_{p}$ will be the
smallest positive integer $f$ such that $p^{f}\equiv
1\;\big{(}\\!\\!\\!\\!\mod\text{\rm ord}(\chi)\big{)}$.

######  Warning.

If $\chi\omega^{-1}(p)=1$ then $\mathbf{L}_{p}(s,\chi)$ has a trivial zero at
$s=0$, in which case $\mathcal{G}_{\chi}(T)$ has a trivial zero at $T=0$. This
happens precisely if $\chi=\theta\omega$ where $\theta$ is odd of conductor
prime to $p$ with $\theta(p)=1$. In this situation it is automatically true
that $\lambda_{p}(\chi)\geqslant 1$, so the corrected $\lambda$-invariant is
given by $\lambda_{p}^{\text{corr}}(\chi)=\lambda_{p}(\chi)-1$.

Conjecture 1.1 generalizes a prediction of Ellenberg, Jain and Venkatesh in
[8]. In their work they also provided numerical evidence for $p\in\\{3,5\\}$
and quadratic characters $\chi$ where $p$ does not split, i.e. at which there
is no trivial zero (in the split case they subtracted $1$ from the
$\lambda$-invariant). Furthermore, the first author and Chao [6] studied the
$\lambda$-invariant for cubic characters $\chi$ with the prime
$p\in\\{5,7\\}$, and found distributions for $\lambda_{p}(\chi)$ consistent
with the quantities predicted in (1.2).

As we discuss later in Section 2, following Ellenberg et al. [2, 8] we employ
a $p$-adic random matrix model for the groups $\text{\rm
GL}(n,\mathcal{O}_{\chi})$ to formulate this conjecture. In Section 3 we
provide extensive numerical evidence supporting this conjecture at lots of
primes and characters: for any given prime $p$ and fixed order, we test all
the characters $\chi$ of conductor $<10000$ and their relevant Teichmüller
twists $\chi\omega^{i}$. The probability $\lambda_{p}(\chi)>0$ decreases
sharply when the residue class degree $f$ of $\mathcal{O}_{\chi}$ increases.
For example, if $p\geqslant 11$ and $f\geqslant 2$ then $\text{\rm
Prob}(\lambda_{p}(\chi)>0)\leqslant 0.01$, and if $p\geqslant 101$ and
$f\geqslant 3$ then $\text{\rm Prob}(\lambda_{p}(\chi)>0)\leqslant 10^{-6}$.
It follows that the interesting cases are at small primes $p$ and for small
orders of $\chi$, which is depicted in Figure 1 below.

Figure 1. Predicted probabilities that $\lambda_{p}(\chi)>0$ for primes $p$
and characters $\chi$, under the assumption that $p$ is coprime to $\text{\rm
ord}(\chi)$.

Throughout a prime $p$ is called $\chi$-regular if
$\lambda_{p}(\chi\omega^{i})=0$ at all $i\in\\{0,\dots,p-2\\}$ such that
$\chi\omega^{i}$ is even, unless $\chi\omega^{i}(p)=1$ in which case we use
$\lambda_{p}^{\text{\rm corr}}(\chi\omega^{i})=0$ instead. From the
interpolation rule in Equation (1.1) for even $\chi$, a prime $p$ is
$\chi$-regular if and only if $p$ does not divide the numerators of the
numbers $B_{2,\chi},B_{4,\chi},\dots,B_{p-1,\chi}$. Assuming Conjecture 1.1
holds for a character $\chi$, we derive an asymptotic formula (Theorem 2.8)
that allows us to estimate the exact proportion of $\chi$-regular primes.

######  Conjecture 1.2.

For a fixed character $\chi$, the proportion of $\chi$-regular primes is equal
to $1+\frac{e^{-\frac{1}{2}}-1}{\varphi(\text{\rm ord}(\chi))}$, while the
proportion of $\chi$-irregular primes equals
$\frac{1-e^{-\frac{1}{2}}}{\varphi(\text{\rm ord}(\chi))}$.

Of course, if $\chi$ is trivial then the predicted proportion will be
$e^{-\frac{1}{2}}\approx 0.6065$ which coincides with the standard predicted
proportion of regular primes over $\mathbb{Q}$. However, note that our
derivation is based on Conjecture 1.1, which refines the usual heuristics that
the Bernoulli numbers $\big{\\{}B_{i}\big{\\}}_{i\in 2\mathbb{N}}$ are
uniformly random modulo $p$. The relationship between $\chi$-regularity and
$\text{\rm ord}(\chi)$ is represented in Figure 2 below, and this prediction
is reflected by the numerical data that we collect in Section 3.

Figure 2. The predicted proportion of $\chi$-regular primes versus $\text{\rm
ord}(\chi)$.

Let $F$ be a totally real abelian extension of $\mathbb{Q}$ such that $p$ is
unramified in $F/\mathbb{Q}$. One says that $p$ is $F$-regular if it is
$\chi$-regular for every character $\chi$ of $\text{Gal}(F/\mathbb{Q})$.
Applying the Iwasawa Main Conjecture [17, 20], $p$ is $F$-regular iff
$\lambda_{p}^{-}(F(\mu_{p}))=0$ i.e. iff the $\lambda_{p}^{-}$ invariant of
the CM field $F(\mu_{p})$ is zero. In this case, the minus-part of the
$p$-component of the class number of $F(\mu_{p^{n}})$ remains bounded as
$n\rightarrow\infty$.

######  Conjecture 1.3.

If $F/\mathbb{Q}$ is a totally real abelian extension of order $m$ and
$\textnormal{Gal}(F/\mathbb{Q})$ is a direct sum of cyclic groups of order
$m_{i}$ with $i\in\\{1,\dots,n\\}$ say, then for those primes $p\gg 0$ with
$p\nmid m\cdot\text{\rm disc}_{F}$, one estimates that

$\text{\rm Prob}\big{(}\text{\rm$p$ is
$F$-regular}\big{)}\;\approx\;\exp\left(\\!-\frac{\prod_{i=1}^{n}\gcd(m_{i},p-1)}{2}\right).$

Recall that a character $\chi$ takes values inside
$\mathbb{Q}(\chi)=\mathbb{Q}\big{(}\text{\rm
Im}(\chi)\big{)}=\mathbb{Q}(\mu_{\text{\rm ord}(\chi)})$. In particular,
$\chi$ contributes to the formula if the residue class degree of $p$ is $1$,
i.e. if $p$ totally splits in $\mathbb{Q}(\mu_{\text{\rm ord}(\chi)})$; in all
other cases, the contribution is close to $1$. Note that $p$ is totally split
in $\mathbb{Q}(\mu_{n})$ if and only if $p-1\equiv 0\;(\\!\\!\\!\\!\mod n)$,
hence exactly $\gcd(m_{i},p-1)$ out of the $m_{i}$ characters of each cyclic
component of the Galois group $\text{Gal}(F/\mathbb{Q})$ play a significant
rôle.

For $F=\mathbb{Q}$, we recover the usual conjecture on the proportion of
regular primes. For real quadratic fields $F$ with non-trivial character
$\chi$, a prime $p$ is $F$-regular iff $\lambda_{p}(\omega^{i})=0$ and
$\lambda_{p}(\chi\omega^{i})=0$ for all $i\in\\{0,2,\dots,p-3\\}$: the
predicted probability is $(e^{-1/2})^{2}=e^{-1}$. For cyclic cubic fields $F$
with non-trivial character $\chi$, a prime $p$ is $F$-regular precisely when
$\lambda_{p}(\chi^{j}\omega^{i})=0$ for all $i\in\\{0,2,\dots,p-3\\}$ and
$j=0,1,2$: if $3\;\big{|}\;p-1$ then one predicts a probability of $e^{-3/2}$,
otherwise one predicts $e^{-1/2}$.

##  2\. Estimates from $p$-adic random matrix theory

To study the behaviour of $\lambda_{p}(\chi)$, we utilise the Iwasawa Main
Conjecture to relate this invariant to the action of a certain operator
‘$\gamma_{0}$’ on an arithmetic module $W$. Let $F$ be a totally real field,
and assume that the prime $p\neq 2$ is unramified in $F$. We write
$F_{\infty}$ for the cyclotomic $\mathbb{Z}_{p}$-extension of $F$, so that
$\Gamma:=\text{\rm Gal}(F_{\infty}/F)\cong\mathbb{Z}_{p}$. Moreover if
$\mu_{p^{\infty}}\\!\subset\overline{F}$ is the group of $p$-power roots, then
$\text{\rm
Gal}\big{(}F(\mu_{p^{\infty}})/F\big{)}\cong\Gamma\times\mathbb{F}_{p}^{\times}$.
Wiles [20] proved for even Dirichlet characters $\chi$ whose restriction to
$\Gamma$ is trivial,

$\mathcal{G}_{\chi}\big{(}\gamma_{0}-1\big{)}\;=\;\text{\rm
char}_{\Lambda_{\mathcal{O}}}\\!\left(W^{(\psi)}\right)\quad\text{\rm up to a
unit, \; where $\psi=\chi^{-1}\omega$.}$

Here $W=\text{\rm Gal}(M_{\infty}/F_{\infty})$ is the Galois group of the
maximal abelian $p$-extension $M_{\infty}$ of $F(\mu_{p^{\infty}})$ unramified
outside $p$, $W^{(\psi)}$ denotes its $\psi$-eigenspace, and
$\Gamma=\gamma_{0}^{\mathbb{Z}_{p}}$. It follows that zeroes of the
$\chi$-twisted $p$-adic zeta-function $\mathcal{G}_{\chi}(T)$ correspond (with
multiplicity) to the zeroes of the characteristic polynomial of
$\gamma_{0}\in\\!\Lambda$ acting on $W^{(\psi)}\\!$. We model the latter
situation using the random matrix approach of [8, Section 4].

###  2.1. The statistics for $\text{\rm GL}(n,\mathcal{O})$ and $\text{\rm
GSp}(2n,\mathcal{O})$

Henceforth we assume $F/\mathbb{Q}$ is an abelian extension. Since
$F\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}$ as $p\nmid\text{\rm disc}(F)$,
one must have

$\text{\rm
Gal}\big{(}F(\mu_{p^{\infty}})/\mathbb{Q}\big{)}\;\cong\;\Gamma\times\mathbb{F}_{p}^{\times}\times\text{\rm
Gal}(F/\mathbb{Q}).$

Note that $W^{(\psi)}:=\big{\\{}w\in\text{\rm
Gal}(M_{\infty}/F_{\infty})\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{\chi}\;\big{|}\;w^{g}=\psi(g)\cdot
w\big{\\}}$ is a compact
$\Lambda\otimes_{\mathbb{Z}_{p}}\\!\mathcal{O}_{\chi}$-torsion module of
finite type, where $\mathcal{O}_{\chi}$ is generated over $\mathbb{Z}_{p}$ by
the values of the character $\chi$. Throughout this section we will suppose
that $\mathcal{O}=\mathcal{O}_{\chi}$ has a chosen uniformiser $\varpi$, with
finite residue field $\mathcal{O}/\varpi\cong\mathbb{F}_{q}$ such that
$q=p^{f}$.

######  Notations.

(i) We write $\text{\rm Mat}_{n\times n}(\mathcal{O})$ for the ring of
$n\times n$ matrices over $\mathcal{O}$, and $\text{\rm
GL}(n,\mathcal{O})\subset\text{\rm Mat}_{n\times n}(\mathcal{O})$ denotes the
multiplicative group of invertible matrices.

(ii) The generalised symplectic group $\text{\rm
GSp}(2n,\mathcal{O})=\bigcup_{\alpha\in\mathcal{O}^{\times}}\text{\rm
GSp}_{\alpha}(2n,\mathcal{O})$ by definition consists of the disjoint matrix
subsets

$\text{\rm GSp}_{\alpha}(2n,\mathcal{O})\;=\;\left\\{A\in\text{\rm
GL}(n,\mathcal{O})\;\Big{|}\;\big{\langle}A\underline{x},A\underline{y}\big{\rangle}=\alpha\cdot\big{\langle}\underline{x},\underline{y}\big{\rangle}\right\\}$

where the pairing $\big{\langle}-,-\big{\rangle}:\mathcal{O}^{\oplus
n}\oplus\mathcal{O}^{\oplus n}\rightarrow\mathcal{O}$ sends
$\big{(}\underline{x},\underline{y}\big{)}\mapsto\sum_{i=1}^{n}x_{i}y_{i+n}-x_{i+n}y_{i}$.

Let $A$ be a matrix selected from either of the groups $\text{\rm
GL}(n,\mathcal{O})$ or $\text{\rm GSp}(2n,\mathcal{O})$. We may consider the
characteristic polynomial of the matrix $A-I_{n}$, namely

$f_{A-I_{n}}(T)\;:=\;\det\big{(}T\cdot
I_{n}-(A-I_{n})\big{)}\;=\;\det\big{(}(1+T)\cdot
I_{n}-A\big{)}\;\in\;\mathcal{O}[T].$

One can view $A$ as acting on the free $\mathcal{O}$-module
$M=\mathcal{O}^{\oplus n}$: there is a Fitting decomposition $M=M^{\text{\rm
uni}}\oplus M^{{\dagger}}$ where $A-I_{n}$ operates on $M^{\text{\rm uni}}$
nilpotently, and on $M^{{\dagger}}$ invertibly. We have a corresponding
factorisation $f_{A-I_{n}}(T)=P_{A}(T)\cdot u(T)$ where $P_{A}(T)$ is a
distinguished polynomial – called the associated polynomial – and $u(T)$
satisfies $u(0)\neq 0\mod\varpi$ (here $P_{A}(T)=1$ iff $A-I_{n}$ is
invertible over $\mathcal{O}$). If one interprets $M^{\text{\rm uni}}$ as the
sum of all generalized eigenspaces $M_{\lambda}$ for eigenvalues
$\lambda\in\overline{\mathbb{Q}}_{p}$ of $A$ satisfying $|\lambda-1|_{p}<1$,
it follows that $P_{A}(T)=\prod_{\lambda}(T+1-\lambda)$ where the product is
taken over all eigenvalues $\lambda$ of $A$ which are $p$-adically close to
one.

Our intended goal is to compute the probability that a uniform random matrix
in either one of the above groups has associated polynomial of a given degree
$r$. For reasons of space only, we limit ourselves to doing the calculation
for $\text{\rm GL}(n,\mathcal{O})$.

######  Definition 2.1.

If $q=p^{f}$ and the integer $r\geqslant 0$, then we define

$\rho(q,r)\;:=\;q^{-r}\times\;\prod_{t>r}\;\big{(}1-q^{-t}\big{)}.$

######  Theorem 2.2.

The proportion of randomly chosen matrices $A\in\text{\rm GL}(n,\mathcal{O})$
whose associated polynomial $P_{A}(T)$ has degree $r$ tends towards
$\rho(q,r)$ as $n\rightarrow\infty$.

###### Proof.

The degree of the associated distinguished polynomial $P_{A}(T)$ only depends
on $A\mod\varpi$, so we reduce the task to a counting problem involving
matrices over $\mathbb{F}_{q}$. For example, if $r=0$ then the proportion of
matrices $A$ in $\text{\rm GL}(n,\mathcal{O})$ such that $A-I_{n}$ is
invertible is the number of $n\times n$ matrices over $\mathbb{F}_{q}$ whose
eigenvalues lie strictly outside $\\{0,1\\}$, divided through by the size of
$\text{\rm GL}(n,\mathbb{F}_{q})$.

Let us choose an $r\in\\{0,1,\dots,n\\}$. Then the number of matrices
$A\in\text{\rm GL}(n,\mathbb{F}_{q})$ such that $\text{\rm deg}(P_{A}(T))=r$
is given by the product of:

  1. (1)

the number of unipotent $r\times r$ matrices over $\mathbb{F}_{q}$;

  2. (2)

the number of $(n-r)\times(n-r)$ matrices $A$ such that $A$ and $A-I_{n-r}$
are invertible; and

  3. (3)

the number of decompositions of $\mathbb{F}_{q}^{n}$ into an $r$-dimensional
subspace and a complementary $(n-r)$-dimensional subspace.

In fact each matrix $A\in\text{\rm GL}(n,\mathbb{F}_{q})$ is uniquely
determined by the associated Fitting decomposition over $\mathbb{F}_{q}$: the
degree of $P_{A}(T)$ is $r$ if and only if the subspace on which $A-I_{n}$
operates nilpotently (or equivalently, $A$ acts unipotently) is
$r$-dimensional.

It is a straightforward exercise to show that $\\#\text{\rm
GL}(r,\mathbb{F}_{q})=q^{r^{2}}\prod_{1\leqslant i\leqslant r}\;(1-q^{-i})$.
We next claim that the cardinalities of the three sets above are respectively:

  1. (1)

$q^{r^{2}-r}$

  2. (2)

$\displaystyle\left(1\;+\;\sum_{i=1}^{n-r}\frac{(-1)^{i}}{(q-1)(q^{2}-1)\dots(q^{i}-1)}\right)\times\\#\text{\rm
GL}(n-r,\mathbb{F}_{q})$

  3. (3)

$\displaystyle\frac{\\#\text{\rm GL}(n,\mathbb{F}_{q})}{\\#\text{\rm
GL}(r,\mathbb{F}_{q})\times\\#\text{\rm GL}(n-r,\mathbb{F}_{q})}$

The formula for (1) is well known, and (3) is quite obvious from the
definition. However for (2), we are looking at the number of invertible
matrices not having $1$ as an eigenvalue. In [13, 19] the proportion of
matrices in $\text{\rm GL}(n,\mathbb{F}_{p})$ having $1$ as an eigenvalue was
computed, and as the proof works over $\mathbb{F}_{q}$, the formula follows.

Multiplying together the above cardinalities, we may thereby conclude that the
proportion of matrices $A$ in $\text{\rm GL}(n,\mathcal{O})$ such that
$\text{\rm deg}(P_{A}(T))=r$ must be equal to

$\frac{q^{-r}}{\prod_{1\leqslant i\leqslant
r}\;(1-q^{-i})}\times\left(1\;+\;\sum_{i=1}^{n-r}\frac{(-1)^{i}}{(q-1)(q^{2}-1)\dots(q^{i}-1)}\right).$

The first factor counts the exact proportion of unipotent matrices in
$\text{\rm GL}(r,\mathcal{O})$, while the initial couple of terms in the
second factor will sum up to equal $1-\frac{1}{q-1}=\frac{q-2}{q-1}$. The rest
of the right-hand series converges very fast, so the proportion mainly depends
on $q$ and $r$, and the dimension of the $\mathcal{O}$-module seems to be less
important. Taking the limit as $n\rightarrow\infty$, the theorem will follow
provided one can show that

$\lim_{n\rightarrow\infty}\left(1+\sum_{i=1}^{n-r}\frac{(-1)^{i}}{(q-1)(q^{2}-1)\dots(q^{i}-1)}\right)=\;\prod_{t=1}^{\infty}\big{(}1-q^{-t}\big{)}\;=\;\rho(q,0).$

This is established in [19] for $q=p$, and the proof extends seamlessly to
$q=p^{f}$. ∎

######  Remark.

(a) If we instead replace the group $\text{\rm GL}(n,\mathcal{O})$ in Theorem
2.2 with $\text{\rm GSp}_{\alpha}(2n,\mathcal{O})$ then the same limiting
value of $\rho(q,r)$ occurs (see [8, §4.2] for $q=p$).

(b) Returning to the equality $\text{\rm
char}_{\Lambda_{\mathcal{O}}}\\!\left(W^{(\psi)}\right)=\mathcal{G}_{\chi}(\gamma_{0}-1)$
from Wiles’ theorem, the probability that $\mathcal{G}_{\chi}(T)$ has exactly
$r$ zeroes on the open unit disk is the same as the probability that the
characteristic power series $\text{\rm
char}_{\Lambda_{\mathcal{O}}}\\!\left(W^{(\psi)}\right)$ has $r$ zeroes.

(c) In the random matrix model, $\mathcal{G}_{\chi}(T)$ represents the
associated distinguished polynomial of a large invertible matrix over
$\mathcal{O}$, and $W^{(\psi)}$ represents the subspace (of a hypothetical
large $\mathcal{O}$-module) on which $\gamma_{0}-1$ acts topologically
nilpotently.

###  2.2. An application of Euler’s pentagonal theorem

If one believes that the statistical behaviour of $\lambda_{p}(\chi)$ is
properly modelled through random matrix theory for either $\text{\rm
GL}(n,\mathcal{O})$ or $\text{\rm GSp}_{\alpha}(2n,\mathcal{O})$ as
$n\rightarrow\infty$, one would then expect

$\text{\rm
Prob}\big{(}\lambda_{p}(\chi)=r\big{)}\;\approx\;\rho\big{(}q,r\big{)}\qquad\text{(see
Theorem \ref{DistThm})}$

at each prime $p$ such that $\mathcal{O}_{\chi}=\mathbb{Z}_{p}[\text{\rm
Im}(\chi)]$ has residue field $\mathbb{F}_{q}\cong\mathcal{O}_{\chi}/\varpi$
with $q=p^{f}$. Indeed this is precisely Conjecture 1.1 that was initially
stated in the Introduction. In this section, we derive some asymptotic
formulae describing the behaviour of $\rho(q,r)$ as a function of
$p\in\mathbb{P}$ (with $q=p^{f}$), which will be needed in our heuristics.

######  Definition 2.3.

For each integer $m\geqslant 1$ and congruence class
$a\in\big{(}\mathbb{Z}/m\mathbb{Z}\big{)}^{\times}\\!$, one partitions the
prime numbers $\mathbb{P}$ into subsets

$\mathbb{P}_{\\!a(m)}\;:=\;\left\\{p\in\mathbb{P}\;\big{|}\;p\equiv
a\;(\\!\\!\\!\\!\\!\\!\mod m)\right\\}$

so in particular, there is a disjoint union
$\mathbb{P}=\coprod_{a(\\!\\!\\!\\!\mod m)}\mathbb{P}_{\\!a(m)}$.

######  Proposition 2.4.

For $X\in\mathbb{R}$, as usual one sets $\pi(X):=\\#\big{\\{}p\in\mathbb{P}\
\big{|}\ p\leqslant X\big{\\}}$.

(i) At every integer $r\geqslant 0$ one has the sequence of bounds

$\qquad 0\;<\;\rho(q,r)\;<\;q^{-r}\;\leqslant\;1,$

thence $\lim_{p\rightarrow\infty}\rho\big{(}p^{f},r\big{)}=0$ if $r>0$ where
$p$ ranges through the prime numbers.

(ii) Given a class $a\in\big{(}\mathbb{Z}/m\mathbb{Z}\big{)}^{\times}\\!$
and a real number $X>0$, we have the estimate

$\qquad\qquad\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant
X}\rho(p,r)\;\approx\;\begin{cases}\frac{1}{\varphi(m)}\big{(}\pi(X)-\log\log
X\big{)}+O(1)&\text{if $r=0$}\\\ \frac{\log\log
X}{\varphi(m)}\times\big{(}1+o(1)\big{)}&\text{if $r=1$}\\\ \;O(1)&\text{if
$r\geqslant 2$.}\end{cases}$ $\quad\text{(iii) If the degree $f\geqslant 2$
then}\;\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant
X}\rho\big{(}p^{f},r\big{)}\;\approx\;\begin{cases}\frac{\pi(X)}{\varphi(m)}+O(1)&\text{if
$r=0$}\\\ \;O(1)&\text{if $r\geqslant 1$.}\end{cases}$

###### Proof.

(i) is completely obvious. To establish parts (ii) and (iii), we recall
Euler’s Pentagonal Formula from [9]: for every $y\in\mathbb{R}$ with $|y|<1$,

$\prod_{t\geqslant
1}\;(1-y^{t})\;=\;1\;+\;\sum_{k=1}^{\infty}\;(-1)^{k}\Big{(}y^{\frac{3k^{2}-k}{2}}+y^{\frac{3k^{2}+k}{2}}\Big{)}.$

Now substituting in $y=q^{-1}=p^{-f}$, one immediately deduces that

$\rho\big{(}p^{f},0\big{)}=\;\prod_{t\geqslant
1}\;(1-p^{-ft})\;=\;1-p^{-f}-p^{-2f}+\sum_{k=2}^{\infty}(-1)^{k}\Big{(}p^{-\frac{(3k^{2}-k)f}{2}}+p^{-\frac{(3k^{2}+k)f}{2}}\Big{)}.$

In particular, we have $\lim_{p\rightarrow\infty}\rho\big{(}p^{f},0\big{)}=1$.
Summing over the primes $p$ congruent to $a\;(\\!\\!\\!\\!\mod m)$ and less
than or equal to $X$:

$\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant
X}\rho\big{(}p^{f},0\big{)}\;\;\approx\;\;\text{\rm
dens}(\mathbb{P}_{\\!a(m)})\cdot\pi(X)\;-\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant
X}p^{-f}\;+\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant X}a(p^{f})$

where the coefficients
$a(p^{f})=-p^{-2f}+\sum_{k=2}^{\infty}(-1)^{k}\Big{(}p^{-\frac{(3k^{2}-k)f}{2}}+p^{-\frac{(3k^{2}+k)f}{2}}\Big{)}\in\mathbb{R}$,
and the notation $\text{\rm dens}(\mathcal{S})$ indicates the Dirichlet
density of the set $\mathcal{S}$ inside $\mathbb{P}$.

######  Remarks.

(a) If $f=1$ then $\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant
X}\;p^{-f}\approx\text{\rm dens}(\mathbb{P}_{\\!a(m)})\cdot\log\log X+O(1)$
from [1], and if $f>1$ this sum is bounded above independently of the choice
of $X$.

(b) The infinite series $\sum_{p\in\mathbb{P}_{\\!a(m)}}p^{-2f}$ will always
converge; in fact $\sum_{p\in\mathbb{P}_{\\!a(m)}}p^{-2f}$ is bounded above by
$\wp(2f)$ where $\wp(s)=\sum_{p\in\mathbb{P}}\;p^{-s}$ is the prime zeta-
function.

(c) If $\wp_{a(m)}(s)=\sum_{p\in\mathbb{P}_{\\!a(m)}}\;p^{-s}$ denotes a
Hurwitz-type prime zeta-function, the convergence of

$\sum_{p\in\mathbb{P}_{\\!a(m)}}\;\sum_{k=2}^{\infty}\;(-1)^{k}\Big{(}p^{-\frac{(3k^{2}-k)f}{2}}+p^{-\frac{(3k^{2}+k)f}{2}}\Big{)}$

to the explicit value
$\sum_{k=2}^{\infty}\;(-1)^{k}\Big{(}\wp_{a(m)}\Big{(}\frac{(3k^{2}-k)f}{2}\Big{)}+\wp_{a(m)}\Big{(}\frac{(3k^{2}+k)f}{2}\Big{)}\Big{)}$
inside $\mathbb{R}$ follows from the absolute convergence of the double sum.

Combining together (a)-(c) above and also noting that $\text{\rm
dens}(\mathbb{P}_{\\!a(m)})=1/\varphi(m)$, the stated estimate for $\rho(q,r)$
follows immediately in the situation where $r=0$. Let us instead suppose that
$r=1$. Then for any choice of $\epsilon>0$,

$p^{-f}\cdot\rho\big{(}p^{f},0\big{)}\;<\;\rho\big{(}p^{f},1\big{)}=\frac{p^{-f}}{1-p^{-f}}\cdot\rho\big{(}p^{f},0\big{)}\;<\;(1+\epsilon)\times
p^{-f}\cdot\rho\big{(}p^{f},0\big{)}$

provided that the prime
$p>X_{\epsilon}:=\big{(}1+\frac{1}{\epsilon}\big{)}^{1/f}$ say. From the
previous estimates,

$\sum_{p\in\mathbb{P}_{\\!a(m)},\;X_{\epsilon}<p\leqslant
X}\rho\big{(}p^{f},1\big{)}\;\;\lesssim\;\;\left(1+\epsilon\right)\times\sum_{p\in\mathbb{P}_{\\!a(m)},\;X_{\epsilon}<p\leqslant
X}p^{-f}.$

If $f=1$ then $\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant
X}\;p^{-f}\approx\frac{\log\log X}{\varphi(m)}+O(1)$, otherwise the sum is
bounded. As a direct consequence, one has
$\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant
X}\rho\big{(}p,1\big{)}\approx\Big{(}\frac{1+o(1)}{\varphi(m)}\Big{)}\cdot\log\log
X+O(1)$ whilst if $f\geqslant 2$ then
$\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant X}\;\rho\big{(}p^{f},1\big{)}$ is
bounded above independently of $X$.

Finally, let us suppose the integer $r\geqslant 2$. Then one has the strict
upper bounds

$\rho\big{(}p^{f},r\big{)}=\prod_{1\leqslant t\leqslant
r}\frac{1}{1-p^{-ft}}\times
p^{-fr}\cdot\rho\big{(}p^{f},0\big{)}\;\;<\;\;2^{r}\times
p^{-fr}\cdot\rho\big{(}p^{f},0\big{)}\;\;<\;\;\left(\frac{2}{p^{f}}\right)^{r}$

because $\rho\big{(}p^{f},0\big{)}<1$. Clearly
$\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant
X}\left(\frac{2}{p^{f}}\right)^{r}$ is bounded independently of $X$ hence the
sum of these $\rho\big{(}p^{f},r\big{)}$ is too, which completes all cases in
(ii) and (iii). ∎

######  Definition 2.5.

Given any integer $r\geqslant 0$, a real number $X>0$ and a Dirichlet
character $\chi$, let us define

$\theta_{\leqslant
X}^{(r)}(\chi)\;:=\;\\#\left\\{p\in\mathbb{P}\;\Big{|}\;p\leqslant
X\;\text{and}\;\lambda_{p}^{\text{corr}}(\chi)=r\right\\}$

where (as before) one sets
$\lambda_{p}^{\text{corr}}(\chi):=\lambda_{p}(\chi)$ in the situation for
which $\chi\omega^{-1}(p)\neq 1$, and otherwise sets
$\lambda_{p}^{\text{corr}}(\chi):=\lambda_{p}(\chi)-1$ if
$\chi\omega^{-1}(p)=1$ (the trivial zero situation).

Equivalently, $\theta_{\leqslant X}^{(r)}(\chi)$ counts up precisely those
primes less than or equal to $X$ whose $\chi$-twisted $\lambda$-invariant is
equal to $r$. Putting $m=\text{\rm ord}(\chi)$ we may partition the primes
into $\mathbb{P}=\coprod_{a(\\!\\!\\!\\!\mod m)}\mathbb{P}_{\\!a(m)}$, which
yields the decomposition

$\theta_{\leqslant
X}^{(r)}(\chi)\;\;=\;\sum_{a\in(\mathbb{Z}/m\mathbb{Z})^{\times}}\;\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant
X}\delta\big{(}\lambda_{p}^{\text{corr}}(\chi)=r\big{)}$

where $\delta(\text{\rm TRUE})=1$ and $\delta(\text{\rm FALSE})=0$.

For example, if $a=1$ then $\mathbb{P}_{\\!1(m)}$ are the primes $p$ congruent
to $1$ modulo $\text{\rm ord}(\chi)$ i.e. exactly those prime numbers $p$ such
that $\mathcal{O}_{\chi}=\mathbb{Z}_{p}[\text{\rm Im}(\chi)]$ has residue
field $\mathbb{F}_{p}$. Given the random matrix model is accurate, the above
sum is approximated by

$\theta_{\leqslant
X}^{(r)}(\chi)\;\;\approx\;\sum_{p\in\mathbb{P}_{\\!1(m)},\;p\leqslant
X}\rho\big{(}p,r\big{)}\;\;+\\!\\!\sum_{\tiny\begin{array}[]{c}a=2,\\\
\gcd(a,m)=1\end{array}}^{m}\\!\sum_{p\in\mathbb{P}_{\\!a(m)},\;p\leqslant
X}\rho\big{(}\\#(\mathcal{O}_{\chi}/\varpi),r\big{)}.$

As we shall shortly see, as an isolated function of $r$ (with $X$ fixed) the
summation attains its largest values when $r=0$, and tends towards zero as $r$
becomes large. The terms on the right-hand side of this approximation can be
estimated using Proposition 2.4, in the following manner:

  * •

if $r=0$ then the first sum behaves like $\frac{\pi(X)-\log\log
X}{\varphi(m)}+O(1)$ while the second sum behaves like
$\big{(}1-\frac{1}{\varphi(m)}\big{)}\cdot\pi(X)+O(1)$, as the value
$X\rightarrow\infty$;

  * •

if $r=1$, the first sum is $\frac{\log\log
X}{\varphi(m)}\cdot\big{(}1+o(1)\big{)}$ and the second sum is $O(1)$;

  * •

if $r\geqslant 2$ then both of these sums will be bounded independently of
$X>0$.

Noting the various predictions arising from this model, one should therefore
expect

$\theta_{\leqslant
X}^{(r)}(\chi)\;\;\approx\;\;\begin{cases}\pi(X)-\frac{\log\log
X}{\varphi(m)}+O(1)&\text{if $r=0$}\\\ \frac{\log\log
X}{\varphi(m)}+o\big{(}\log\log X\big{)}&\text{if $r=1$}\\\ \;O(1)&\text{if
$r\geqslant 2$.}\end{cases}$

Lastly we recall that the prime number theorem implies
$\pi(X)\approx\frac{X}{\log(X)}$ for $X\gg 0$.

######  Conjecture 2.6.

For increasing $X$, one predicts the asymptotic behaviour:

$\displaystyle\;\text{(i)}\;\;\;\lim_{X\rightarrow\infty}\frac{\theta_{\leqslant
X}^{(0)}(\chi)}{X/\log(X)}\;=\;1$
$\displaystyle\text{(ii)}\;\;\;\lim_{X\rightarrow\infty}\frac{\theta_{\leqslant
X}^{(1)}(\chi)}{\log\log(X)}\;=\;\frac{1}{\varphi\big{(}\text{\rm
ord}(\chi)\big{)}}$ $\displaystyle\text{(iii)}\;\;\;\theta_{\leqslant
X}^{(r)}(\chi)\;\;\text{is bounded above if $r\geqslant
2$.}\qquad\qquad\qquad\qquad\qquad\qquad$

###  2.3. Heuristics for the $\chi$-regularity of $p$

Recall that a prime number $p$ is called regular if it does not divide the
class number of the cyclotomic field $\mathbb{Q}(\mu_{p})$. Indeed it is
essentially a consequence of Kummer’s criterion that a prime $p$ is regular if
and only if each power series
$\mathcal{G}_{\\!\omega^{2j}}(T)\in\mathbb{Z}_{p}[\\![T]\\!]$ is a unit of the
Iwasawa algebra.

######  Definition 2.7.

(i) For any Dirichlet character $\chi$, we shall define the total
$\lambda$-invariant associated to $\chi$ by

$\qquad\lambda_{p}^{\text{\rm tot}}(\chi)\;:=\sum_{\begin{subarray}{c}j=0\\\
\chi\omega^{j}\text{even}\end{subarray}}^{p-2}\lambda_{p}^{\text{corr}}(\chi\omega^{j})\;\in\;\mathbb{Z}_{\geqslant
0}.$

(ii) A prime $p$ is called $\chi$-regular111This differs in general from
Ernvall’s definition of $\chi$-regularity for a prime $p$, as he requires that
$\lambda_{p}(\chi^{\sigma}\omega^{j})=0$ for all even branches and elements
$\sigma\in\text{\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, which is a
stronger condition. if $\lambda_{p}^{\text{corr}}(\chi\omega^{j})=0$ for every
$j\in\\{0,\dots,p-2\\}$ such that $\chi\omega^{j}$ is even; otherwise it is
called a $\chi$-irregular prime, not surprisingly.

Applying the interpolation formula (1.1) for the $\chi$-twisted $p$-adic zeta-
function, if $\chi$ is even then it follows that a prime $p$ is $\chi$-regular
if and only if $p$ does not divide the numerator of any of the $\chi$-twisted
Bernoulli numbers $B_{2,\chi},\dots,B_{p-1,\chi}$. Alternatively, if $\chi$ is
an odd Dirichlet character then $p$ will be $\chi$-regular if and only if $p$
does not divide the numerator of any of the $\chi$-twisted numbers
$B_{1,\chi},\dots,B_{p-2,\chi}$. Now assuming our $p$-adic random matrix
models are accurate, the probability that $\lambda_{p}(\chi\omega^{j})=0$
should be independent of the branch $\omega^{j}$. This independence implies

$\text{\rm Prob}\big{(}\lambda_{p}^{\text{\rm
tot}}(\chi)=0\big{)}\;\;\approx\;\;\prod_{\begin{subarray}{c}j=0\\\
\chi\omega^{j}\text{ even}\end{subarray}}^{p-2}\;\text{\rm
Prob}\big{(}\lambda^{\text{corr}}_{p}(\chi\omega^{j})=0\big{)}\;\;\approx\;\;\rho(q,0)^{\frac{p-1}{2}}$

where the residue field of $\mathbb{Z}_{p}[\text{\rm Im}(\chi)]$ has $q=p^{f}$
elements, thus one should expect

$\pi(X)^{-1}\;\times\sum_{p\in\mathbb{P},\;p\leqslant
X}\;\rho\big{(}\\#(\mathcal{O}_{\chi}/\varpi),0\big{)}^{\frac{p-1}{2}}$

to tend towards the true proportion of $\chi$-regular primes as the bound $X$
gets larger.

######  Theorem 2.8.

Given a real number $X\gg 0$, one has the approximation

$\sum_{p\in\mathbb{P},\;p\leqslant
X}\rho\big{(}\\#(\mathcal{O}_{\chi}/\varpi),0\big{)}^{\frac{p-1}{2}}\approx\;\pi(X)\times\left(\frac{\varphi(\text{\rm
ord}(\chi))+e^{-1/2}-1}{\varphi(\text{\rm
ord}(\chi))}\right)+\;O\big{(}1\big{)}.$

###### Proof.

We begin with some simple estimates. If one has $y\in(0,1)$ and $p>3$ then
$1-\left(\frac{p-1}{2}\right)y<(1-y)^{\frac{p-1}{2}}<1-y$, in which case

$1-p^{-(t-1)f}\;<\;1-\left(\frac{p-1}{2}\right)p^{-tf}\;<\;\big{(}1-p^{-tf}\big{)}^{\frac{p-1}{2}}\;<\;1-p^{-tf}.$

Taking a product over $t$: $\prod_{t\geqslant
2}\big{(}1-p^{-tf}\big{)}<\prod_{t\geqslant
3}\big{(}1-p^{-tf}\big{)}^{\frac{p-1}{2}}<\prod_{t\geqslant
3}\big{(}1-p^{-tf}\big{)}.$ If we write $\beta(y)=(1-y^{-1})(1-y^{-2})$, then
by Definition 2.1 one has the identity

$\prod_{t\geqslant
3}\big{(}1-p^{-tf}\big{)}^{\frac{p-1}{2}}=\left(\frac{\rho(p^{f},0)}{(1-p^{-f})(1-p^{-2f})}\right)^{\frac{p-1}{2}}\\!=\;\beta\big{(}p^{f}\big{)}^{\frac{1-p}{2}}\\!\cdot\rho\big{(}p^{f},0\big{)}^{\frac{p-1}{2}}.$

Plugging this expression into our inequalities, one obtains the strict bounds

(2.1) $\;\beta\big{(}p^{f}\big{)}^{\frac{p-1}{2}}\\!\cdot\prod_{t\geqslant
2}\big{(}1-p^{-tf}\big{)}\;\;<\;\;\rho\big{(}p^{f},0\big{)}^{\frac{p-1}{2}}\;\;<\;\;\beta\big{(}p^{f}\big{)}^{\frac{p-1}{2}}\\!\cdot\prod_{t\geqslant
3}\big{(}1-p^{-tf}\big{)}.$

It is well known that
$\lim_{n\rightarrow\infty}\big{(}1+x/n\big{)}^{n}=\exp(x)$, so for $x=-1$ we
deduce that

$\lim_{p\rightarrow\infty}\beta(p)^{\frac{p-1}{2}}=\;\sqrt{\lim_{p\rightarrow\infty}\\!\left(1-\frac{1}{p}\right)^{p}\times\lim_{p\rightarrow\infty}\\!\frac{\big{(}1-\frac{1}{p^{2}}\big{)}^{p-1}}{\big{(}1-\frac{1}{p}\big{)}}}\;=\;\sqrt{\exp\left(-1\right)\\!\times
1}\;=\;e^{-1/2}.$

Exploiting the algebra of limits, if the exponent $f\geqslant 2$ then
$\lim_{p\rightarrow\infty}\beta\big{(}p^{f}\big{)}^{\frac{p-1}{2}}=1$. Now fix
any $\epsilon>0$ and write $m=\text{\rm ord}(\chi)$. Then there exists
$N_{\epsilon}\in\mathbb{N}$ such that

  * •

$\Big{|}\beta\big{(}\\#(\mathcal{O}_{\chi}/\varpi)\big{)}^{\frac{p-1}{2}}-e^{-1/2}\Big{|}<\epsilon$
for all $p\in\mathbb{P}_{\\!1(m)}$ with $p\geqslant N_{\epsilon}$, and

  * •

$\Big{|}\beta\big{(}\\#(\mathcal{O}_{\chi}/\varpi)\big{)}^{\frac{p-1}{2}}-1\Big{|}<\epsilon$
for all $p\in\mathbb{P}-\mathbb{P}_{\\!1(m)}$ with $p\geqslant N_{\epsilon}$.

Firstly, the bounds in (2.1) and our computation of
$\lim_{p\rightarrow\infty}\beta(p)^{\frac{p-1}{2}}$ imply that

$\displaystyle\big{(}e^{-1/2}-\epsilon\big{)}\times\\!\\!\\!\sum_{\tiny\begin{array}[]{c}p\in\mathbb{P}_{\\!1(m)},\\\
N_{\epsilon}\leqslant p\leqslant X\end{array}}\prod_{t\geqslant
2}\big{(}1-p^{-t}\big{)}$
$\displaystyle\;\;<\\!\sum_{\tiny\begin{array}[]{c}p\in\mathbb{P}_{\\!1(m)},\\\
N_{\epsilon}\leqslant p\leqslant
X\end{array}}\rho\big{(}p,0\big{)}^{\frac{p-1}{2}}\;\;<\;\;\big{(}e^{-1/2}+\epsilon\big{)}\times\\!\\!\\!\sum_{\tiny\begin{array}[]{c}p\in\mathbb{P}_{\\!1(m)},\\\
N_{\epsilon}\leqslant p\leqslant X\end{array}}\prod_{t\geqslant
3}\big{(}1-p^{-t}\big{)}.$

Then it is an exercise involving Euler’s pentagonal theorem to show

$\sum_{\tiny\begin{array}[]{c}p\in\mathbb{P}_{\\!1(m)},\\\
N_{\epsilon}\leqslant p\leqslant X\end{array}}\prod_{t\geqslant
d}\big{(}1-p^{-t}\big{)}\;\approx\;\;\frac{\pi(X)}{\varphi(m)}\;+\;O(1)\quad\text{with
$d=2$ or $3$;}$

consequently $\sum_{p\in\mathbb{P}_{\\!1(m)},\;p\leqslant
X}\;\;\rho\big{(}p,0\big{)}^{\frac{p-1}{2}}\approx\;e^{-1/2}\cdot\frac{\pi(X)}{\varphi(m)}+O(1)$
for $X\gg 0$.

Secondly, if $q=p^{f}=\\#(\mathcal{O}_{\chi}/\varpi)$ with $f\geqslant 2$ as
before, then again the bounds in (2.1) and our computation of
$\lim_{p\rightarrow\infty}\beta(p^{f})^{\frac{p-1}{2}}$ together imply

$\displaystyle\big{(}1-\epsilon\big{)}\times\\!\\!\\!\sum_{\tiny\begin{array}[]{c}p\in\mathbb{P}-\mathbb{P}_{\\!1(m)},\\\
N_{\epsilon}\leqslant p\leqslant X\end{array}}\prod_{t\geqslant
2}\big{(}1-p^{-tf}\big{)}$
$\displaystyle\;\;<\\!\sum_{\tiny\begin{array}[]{c}p\in\mathbb{P}-\mathbb{P}_{\\!1(m)},\\\
N_{\epsilon}\leqslant p\leqslant
X\end{array}}\rho\big{(}p^{f},0\big{)}^{\frac{p-1}{2}}\;\;<\;\;\big{(}1+\epsilon\big{)}\times\\!\\!\\!\sum_{\tiny\begin{array}[]{c}p\in\mathbb{P}-\mathbb{P}_{\\!1(m)},\\\
N_{\epsilon}\leqslant p\leqslant X\end{array}}\prod_{t\geqslant
3}\big{(}1-p^{-tf}\big{)}.$

Another exercise involving Euler’s pentagonal theorem shows that for $d=2$ or
$3$,

$\sum_{\tiny\begin{array}[]{c}p\in\mathbb{P}-\mathbb{P}_{\\!1(m)},\\\
N_{\epsilon}\leqslant p\leqslant X\end{array}}\prod_{t\geqslant
d}\big{(}1-p^{-tf}\big{)}\;\approx\;\;\frac{(\varphi(m)-1)\cdot\pi(X)}{\varphi(m)}\;+\;O(1);$

it follows that $\sum_{p\in\mathbb{P}-\mathbb{P}_{\\!1(m)},\;p\leqslant
X}\;\rho\big{(}p^{f},0\big{)}^{\frac{p-1}{2}}\approx\;\left(1-\frac{1}{\varphi(m)}\right)\cdot\pi(X)+O(1)$.

Finally, one can always decompose

$\sum_{p\in\mathbb{P},\;p\leqslant
X}\rho\big{(}\\#(\mathcal{O}_{\chi}/\varpi),0\big{)}^{\frac{p-1}{2}}\;=\;\sum_{p\in\mathbb{P}_{\\!1(m)},\;p\leqslant
X}\rho\big{(}p,0\big{)}^{\frac{p-1}{2}}\;+\\!\\!\sum_{p\in\mathbb{P}-\mathbb{P}_{\\!1(m)},\;p\leqslant
X}\rho\big{(}p^{f},0\big{)}^{\frac{p-1}{2}}$

and the first summation behaves like
$e^{-1/2}\cdot\frac{\pi(X)}{\varphi(m)}+O(1)$ as we just found, whilst the
second sum behaves like $\left(1-\frac{1}{\varphi(m)}\right)\cdot\pi(X)+O(1)$
with $X\gg 0$. One thereby concludes that

$\sum_{p\in\mathbb{P},\;p\leqslant
X}\rho\big{(}\\#(\mathcal{O}_{\chi}/\varpi),0\big{)}^{\frac{p-1}{2}}\approx\;e^{-1/2}\cdot\frac{\pi(X)}{\varphi(m)}+\left(1-\frac{1}{\varphi(m)}\right)\cdot\pi(X)+O(1)$

and the result is now fully established. ∎

######  Remarks.

(i) A straightforward consequence of Theorem 2.8 is that

$\lim_{X\rightarrow\infty}\left(\frac{\sum_{p\in\mathbb{P},\;p\leqslant
X}\;\rho\big{(}\\#(\mathcal{O}_{\chi}/\varpi),0\big{)}^{\frac{p-1}{2}}}{\pi(X)}\right)\;=\;\;1+\frac{e^{-\frac{1}{2}}-1}{\varphi(\text{\rm
ord}(\chi))}$

which was the principal motivation for making Conjecture 1.2 in the first
place.

(ii) If $\chi$ is trivial or quadratic then $\varphi(\text{\rm ord}(\chi))=1$,
hence $1+\frac{e^{-\frac{1}{2}}-1}{\varphi(\text{\rm
ord}(\chi))}=e^{-\frac{1}{2}}$ which is consistent with the proportion of
regular primes observed in nature.

(iii) As Im$(\chi)$ grows larger so does $\varphi(\text{\rm ord}(\chi))$, thus
one expects the proportion of $\chi$-regular primes to approach $1$ and that
the $\chi$-irregular primes become sparser.

###  2.4. Heuristics for totally real abelian extensions

Let $F$ be an arbitrary totally real number field (not necessarily an abelian
extension of $\mathbb{Q}$), and choose a prime $p\neq 2$ that is unramified in
$F$. Deligne and Ribet [7] associated a $p$-adic $L$-function,
$\zeta_{p,F}(s)$, whose $\omega^{2j}$-branches interpolate special values of
the complex Dedekind zeta-function $\zeta_{F}(s,\omega^{2j})$. Let
$\mathcal{D}_{F,\omega^{2j}}(T)$ be the associated power series.

######  Definition 2.9.

(i) We say that a prime $p$ is $F$-regular if $p$ is unramified in $F$ and
secondly $\lambda\big{(}\mathcal{D}_{F,\omega^{2j}}(T)\big{)}=0$ for all
$j\in\\{1,\dots,\frac{p-1}{2}\\}$; otherwise $p$ is $F$-irregular.

(ii) In particular, if one defines $\lambda_{p}^{\text{\rm
tot}}\big{(}F\big{)}:=\sum_{j=1}^{\frac{p-1}{2}}\;\lambda\big{(}\mathcal{D}_{F,\omega^{2j}}(T)\big{)}\in\mathbb{Z}_{\geqslant
0}$ then the prime $p$ is $F$-regular if and only if both $p\nmid\text{\rm
disc}_{F}$ and $\lambda_{p}^{\text{\rm tot}}(F)$ is equal to zero.

We now study totally real abelian extensions $F/\mathbb{Q}$ of order $m$.
Suppose that $\text{Gal}(F/\mathbb{Q})$ is a direct sum of cyclic groups of
order $m_{i}$ where each $i\in\\{1,\dots,n\\}$. Recall from earlier that
Conjecture 1.3 predicted for $p\gg 0$ with $p\nmid m\cdot\text{\rm disc}_{F}$
that

$\text{\rm Prob}\big{(}\text{\rm$p$ is
$F$-regular}\big{)}\;\approx\;\exp\left(\\!-\frac{\prod_{i=1}^{n}\gcd(m_{i},p-1)}{2}\right)$

where the probability is computed over a random field extension (of fixed
order), and a randomly chosen prime number.

In the sequel, we provide some theoretical evidence for the conjecture above.
Firstly a prime $p$ satisfying $p\nmid m\cdot\text{disc}_{F}$ is $F$-regular
if and only if $\lambda_{p}(\chi\omega^{2j})=0$ for all characters $\chi$ of
$\text{Gal}(F/\mathbb{Q})$ and $j\in\\{1,\dots,\frac{p-1}{2}\\}$. If the
random matrix model is accurate for a fixed prime $p$, then the probability of
$\lambda_{p}(\chi\omega^{2j})=0$ is given by $\rho(q,0)$ where
$q=p^{f}=\\#(\mathcal{O}_{\chi}/\varpi)$. Assuming that the distribution of
the $\lambda$-invariants for characters $\chi$ of $\text{Gal}(F/\mathbb{Q})$
in the same Galois orbit is independent222This is probably true for totally
split primes in $\mathbb{Q}(\chi)$, as the decomposition group is trivial.
However this is not true for inert primes, but luckily
Prob$\big{(}\lambda_{p}(\chi\omega^{2j})=0\big{)}$ is close to $1$ if $f>1$.
we should then have an approximation

(2.6) $\text{\rm Prob}\big{(}\text{$p$ is
$F$-regular}\big{)}\;\approx\\!\prod_{\begin{subarray}{c}\chi:\text{Gal}(F/\mathbb{Q})\rightarrow\overline{\mathbb{Q}}^{\times}\\!\\!,\\\
\\#(\mathcal{O}_{\chi}/\varpi)=q\end{subarray}}\\!\rho(q,0)^{\frac{p-1}{2}}.$

We already saw from Equation (2.1) that (as in the proof of Theorem 2.8):

$\;\beta\big{(}p^{f}\big{)}^{\frac{p-1}{2}}\\!\cdot\prod_{t\geqslant
2}\big{(}1-p^{-tf}\big{)}\;\;<\;\;\rho\big{(}p^{f},0\big{)}^{\frac{p-1}{2}}\;\;<\;\;\beta\big{(}p^{f}\big{)}^{\frac{p-1}{2}}\\!\cdot\prod_{t\geqslant
3}\big{(}1-p^{-tf}\big{)}.$

Furthermore, one has
$\lim_{p\rightarrow\infty}\beta\big{(}p^{f}\big{)}^{\frac{p-1}{2}}=1$ if $f>1$
and $\lim_{p\rightarrow\infty}\beta(p)^{\frac{p-1}{2}}=e^{-1/2}$, in which
case $\lim_{p\rightarrow\infty}\rho(p,0)^{\frac{p-1}{2}}=e^{-1/2}$ while
$\lim_{p\rightarrow\infty}\rho(p^{f},0)^{\frac{p-1}{2}}=1$ when $f>1$. Thus we
obtain a factor of $e^{-1/2}$ for each character $\chi$ of
$\text{Gal}(F/\mathbb{Q})$ such that $\\#(\mathcal{O}_{\chi}/\varpi)=p$, and
the other characters yield a factor of $1$ for the product in (2.6).
Consequently, it is enough to count up the characters $\chi$ such that
$\mathcal{O}_{\chi}$ has residue class degree $f=1$, i.e. those $\chi$ at
which $p$ is totally split inside $\mathbb{Q}(\chi)=\mathbb{Q}(\mu_{\text{\rm
ord}(\chi)})$.

######  Lemma 2.10.

Let $p$ be a prime number, and let $G$ be a cyclic group of order $m$ such
that $p\nmid m$. Then there exist precisely $\gcd(m,p-1)$ distinct characters
$\chi$ of $G$ having the property that $p$ is totally split inside the
cyclotomic extension $\mathbb{Q}(\chi)/\mathbb{Q}$. More generally, if $G$ is
a direct product of cyclic groups of order $m_{i}$ where each
$i\in\\{1,\dots,n\\}$, then the number of such characters is
$\prod_{i=1}^{n}\gcd(m_{i},p-1)$.

###### Proof.

The prime $p$ totally splits in $\mathbb{Q}(\mu_{N})/\mathbb{Q}$ if and only
if $p-1\equiv 0\;(\\!\\!\\!\\!\mod N)$. For each character $\chi$ of a cyclic
group of order $m$ clearly one has $\text{\rm ord}(\chi)\big{|}m$, and the
largest possible order of a character at which $p$ is totally split is
$\gcd(m,p-1)$. Indeed such a character generates the (cyclic subgroup of)
characters $\chi$ at which $p$ totally splits in $\mathbb{Q}(\chi)$. We can
decompose a character $\chi$ of a finite abelian group $G$ into a product
$\chi_{1}\cdots\chi_{n}$, where the $\chi_{i}$’s are characters on the cyclic
components. We have $\text{\rm ord}(\chi)=\text{lcm}(\text{\rm
ord}(\chi_{1}),\dots,\text{\rm ord}(\chi_{n}))=N$ say: by the Chinese
remainder theorem and its generalisation to non-coprime moduli,

$p-1\equiv 0\;(\\!\\!\\!\\!\\!\mod N)\;\Longleftrightarrow\;p-1\equiv
0\;\big{(}\\!\\!\\!\\!\\!\mod\text{\rm ord}(\chi_{i})\big{)}\;\text{ for all
}i\in\\{1,\dots,n\\}.$

There are $\gcd(m_{i},p-1)$ distinct characters $\chi_{i}$ such that
$p-1\equiv 0\;\big{(}\\!\\!\\!\\!\mod\text{\rm ord}(\chi_{i})\big{)}$, which
completes the proof of the lemma. ∎

######  Example.

Suppose that $F/\mathbb{Q}$ denotes a totally real cyclic extension of order
$l^{n}$ where $l\neq p$ is a prime. If $n(p):=\text{\rm ord}_{l}(p-1)$ then
$p$ splits completely in $\mathbb{Q}(\mu_{l^{n(p)}})$ i.e.
$(p)=\mathfrak{p}_{1}\cdots\mathfrak{p}_{\varphi(l^{n(p)})}$, and moreover the
prime ideals $\mathfrak{p}_{j}$ are inert in the extension
$\mathbb{Q}(\mu_{l^{n}})\big{/}\mathbb{Q}(\mu_{l^{n(p)}})$ for each $j$. We
must then have $\gcd(l^{n},p-1)=l^{n(p)}$, and hence

$\text{\rm Prob}\big{(}\text{$p$ is
$F$-regular}\big{)}\;\approx\;e^{-\frac{l^{n(p)}}{2}}.$

It follows directly that the probability $p$ is $F$-regular stabilises in any
tower of extensions of $l$-power order. For instance if $l\nmid p-1$ then this
probability is $e^{-1/2}$, the same as for the classical case $F=\mathbb{Q}$.

##  3\. The general method and a summary of the data

We compile some evidence supporting Conjectures 1.1-1.3 by computing
$\lambda_{p}(\chi)$ for a large number of characters $\chi$ of a given order,
as well as their Teichmüller twists. Let us make a few general comments.
Firstly, the SageMath class DirichletGroup was used to find the distinct
Dirichlet characters $\chi$ of a given conductor and order. In our numerical
experiments, we usually looked at characters $\theta$ with $\text{\rm
cond}(\theta)<10^{5}$ and $\gcd\big{(}p,\text{\rm cond}(\theta)\big{)}=1$, and
at all Teichmüller twists $\chi=\theta\omega^{i}$ so that $\chi$ was even. To
compute $\lambda_{p}(\chi)$ itself we employed two different methods, as we
will now outline.

Method I - Interpolation of Bernoulli numbers. One commences by choosing $15$
separate values $n=i+k(p-1)$ of the power series
$\mathcal{G}_{\theta\omega^{i}}(T)\in\mathcal{O}_{\chi}[\\![T]\\!]$. Applying
Equation (1.1), we immediately discover that

$\mathcal{G}_{\theta\omega^{i}}\big{(}(1+p)^{n-1}-1\big{)}=\;-\big{(}1-\chi(p)p^{n-1}\big{)}\times\frac{B_{n,\theta}}{n}\quad\text{for
positive $n\equiv i\;(\\!\\!\\!\\!\\!\mod p-1)$. }$

An interpolation of these 15 values produces a Lagrange polynomial in
$\mathbb{Q}_{p}(\chi)\big{[}T\big{]}$, and to calculate $\lambda_{p}(\chi)$ we
need only look at its first few coefficients modulo $\varpi$.

As is described at length in [8, Section 5.2] for the ring
$\mathcal{O}_{\psi}=\mathbb{Z}_{p}$, the difference between the $t$-th
coefficient of the power series and the interpolated polynomial lies in
$p^{C-t}\mathbb{Z}_{p}$ where $C$ denotes the number of values used for the
interpolation. The statement easily extends to the valuation ring
$\mathcal{O}_{\chi}$, and so we can compute $\lambda$-invariants $<C=15$ if we
work at the required $p$-adic precision (see also [10]).

The numbers $B_{n,\theta}$ themselves are obtained by calling the SageMath
function bernoulli on a DirichletGroup object. The generalised Bernoulli
numbers are elements of the global field $\mathbb{Q}(\zeta_{\text{\rm
ord}(\chi)})$, which we need to embed into a $p$-adic field. To this end, we
exploit SageMath’s implementation Qq of field extensions of $\mathbb{Q}_{p}$:
we compute the extension $K=\mathbb{Q}_{p}(\chi)$, so that $B_{n,\theta}$ is
sent to an element in $K$.

Method II - $p$-adic Dirichlet series expansions. Over the last fifteen years
or so, it was found by various authors that the Kubota-Leopoldt zeta-function
exhibits a nice expansion as a non-archimedean Dirichlet series [4, 5, 6, 14,
21]. For $x\in\mathbb{Z}_{p}^{\times}$ we write $\langle
x\rangle=x\cdot\omega(x)^{-1}\in 1+p\mathbb{Z}_{p}$ for its image in the
principal units. If
$\mathcal{G}_{\chi,\beta}^{\natural}(T)\in\mathcal{O}_{\chi}[\\![T]\\!]$ is
the power series corresponding to the Iwasawa function

$\big{(}2\omega^{\beta}(2)\langle
2\rangle^{-s}-1\big{)}\cdot\mathbf{L}_{p}\big{(}s,\chi\omega^{1+\beta}\big{)}$

and assuming the prime $p\nmid\text{\rm cond}(\chi)$, then by [6, Theorem 1]
there are congruences

$\mathcal{G}_{\chi,\beta}^{\natural}(T)\;\equiv\;\;\sum_{j=0}^{p^{N}}\;c_{j}^{(N)}\\!\big{(}\mathcal{G}_{\chi,\beta}^{\natural}\big{)}\cdot
T^{j}\mod\mathcal{J}_{N}\quad\text{for all integers $N\geqslant 1$,}$

where $\mathcal{J}_{N}$ denotes the $\mathcal{O}_{\chi}[\\![T]\\!]$-ideal
$\prod_{j=1}^{N}\big{(}T^{p^{j-1}}\\!,p\big{)}=\big{(}T,p\big{)}\cdot\big{(}T^{p},p\big{)}\dots\big{(}T^{p^{N-1}},p\big{)}$.
These $p$-adic coefficients $c_{j}^{(N)}$ can be easily calculated via
Equation (2) of op. cit. and individually require roughly $N\cdot\log(p)\cdot
p^{N}$ operations to work out each time. Also, the ideals
$\\{\mathcal{J}_{N}\\}$ form a decreasing set of neighborhoods of zero as
$N\rightarrow\infty$. Put $N=2$: by Proposition 1 of op. cit., if $\text{\rm
ord}_{p}\big{(}c_{j}^{(2)}\\!(\mathcal{G}_{\chi,\beta}^{\natural})\big{)}=0$
for some $j\leqslant p$ then

$\lambda_{p}(\chi\omega^{1+\beta})\;=\;\min\Big{\\{}j\geqslant
0\;\Big{|}\;\text{\rm
ord}_{p}\big{(}c_{j}^{(2)}\\!(\mathcal{G}_{\chi,\beta}^{\natural})\big{)}=0\Big{\\}}-\begin{cases}1&\text{if
$\beta\equiv-1\mod p-1$}\\\ 0&\text{if $\beta\not\equiv-1\mod
p-1$.}\end{cases}$

Provided the $\lambda$-invariant does not exceed $p$ (which is most often the
case anyway), the above formula provides a simple and efficient method of
determining $\lambda_{p}(\chi)$ using approximately $O\big{(}\log(p)\cdot
p^{2})$ arithmetic operations.

Alternatively, one may instead use a complementary expansion derived by the
second author and Washington in [14], namely

$-\big{(}1-\theta\omega^{i}(2)\langle
2\rangle^{1-s}\big{)}\cdot\mathbf{L}_{p}(s,\theta\omega^{i})\;=\;\;\sum^{F}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\frac{(-1)^{a}}{2}\,\theta\omega^{i-1}(a)\langle
a\rangle^{-s}+\delta(F/p^{2})$

where $s\in\mathbb{C}_{p}$ with $|s|<p^{(p-2)/(p-1)}$, $F$ is any chosen
multiple of $p\cdot\text{\rm cond}(\theta)$, and one has
$|\delta(F/p^{2})|\leqslant|F/p^{2}|$. The regularisation at $c=2$ works best
for odd characters $\theta$, but other choices for $c$ are also possible. If
we set $T=(1+pd)^{s}-1$ with $d=\text{\rm cond}(\theta)$ then we have
$s=\frac{\log_{p}(1+T)}{\log_{p}(1+pd)}$, and thereby define the quantity
$l(x)=-\frac{\log_{p}\langle x\rangle}{\log_{p}(1+pd)}$ for all
$x\in\mathbb{Z}_{p}^{\times}$. The above Dirichlet series expansion yields for
$T\in\mathbb{C}_{p}$ with $|T|<p^{-1/(p-1)}$:

$\displaystyle-\big{(}1-\theta\omega^{i}(2)\langle
2\rangle(1+T)^{l(2)}\big{)}\cdot\mathcal{G}_{\theta\omega^{i}}(T)\;=\;\sum^{F}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\frac{(-1)^{a}}{2}\,\theta\omega^{i-1}(a)(1+T)^{l(a)}+\delta(F/p^{2})\;$
$\displaystyle=\;\sum_{j=0}^{\infty}\left(\sum^{F}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\frac{(-1)^{a}}{2}\,\theta\omega^{i-1}(a)\binom{l(a)}{j}\right)T^{j}+\delta(F/p^{2}).$

If we put $F=d\cdot p^{N}$ with $N$ large enough, the inner bracket gives an
easily computable formula for the initial coefficients of the power series
$\mathcal{G}_{\theta\omega^{i}}(T)$(multiplied with the regularisation
factor), from which we readily obtain the value of
$\lambda_{p}(\theta\omega^{i})$.

######  Remarks.

(i) In practice we found that Method II worked perfectly well if both $p$ and
$\text{\rm cond}(\chi)$ were not too big, but Method I was quicker in the long
term.

(ii) To apply Method I for a fixed $\chi$, one simply generates a vast array
of Bernoulli numbers (once and for all), and then varying the prime $p$ is not
an issue.

(iii) In work in progress of the authors and Luochen Zhao, using [18, 21] we
treat certain totally real fields $F$ which are non-abelian extensions of
$\mathbb{Q}$, and study the proportion of $F$-regular primes (N.B. this is not
covered by Conjecture 1.3).

(iv) The SageMath code which implements Methods I and II is freely available
from the website https://github.com/knospe/iwasawa

###  3.1. Numerical evidence for Conjecture 1.1

We have computed the distribution of $\lambda$-invariants for all primitive
even characters $\chi=\theta\omega^{i}$ satisfying $\gcd\\!\big{(}\text{\rm
cond}(\theta),p\big{)}\\!=1$ and $\text{\rm cond}(\theta)<10000$, where we fix
both $p$ and the order of $\chi$. The tables contain the predicted and the
computed proportion for $N$ characters and all relevant Teichmüller twists $i$
(we omit the case of a trivial zero where $\lambda\geqslant 1$). These
computed numbers agree with the prediction, except for some mysterious larger
deviation for the combination $p=3$ and $\text{\rm ord}(\chi)=2$. The
proportion with $\lambda=0$ is systematically slightly underestimated. For
quadratic characters with large conductors, the computations in [8, 15]
indicate convergence towards this value.

$p=3$, $\text{\rm ord}(\chi)=2$ (even and odd quadratic characters), $f=1$

$i$ | $N$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---|---
| pred. | $0.5601$ | $0.2801$ | $0.1050$ | $0.0364$ | $0.0123$ | $0.0041$ | $0.0014$ | $0.0005$
$0$ | $2280$ | $0.6460$ | $0.2439$ | $0.0798$ | $0.0219$ | $0.0066$ | $0.0013$ | $0.0004$ | $0.0000$
$1$ | $1140$ | $0.6544$ | $0.2412$ | $0.0702$ | $0.0228$ | $0.0079$ | $0.0026$ | $0.0009$ | $0.0000$

$p=5$, $\text{\rm ord}(\chi)=2$ (even and odd quadratic characters), $f=1$

$i$ | $N$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---|---
| pred. | $0.7603$ | $0.1901$ | $0.0396$ | $0.0080$ | $0.0016$ | $0.0003$ | $0.0001$ | $0.0000$
$0$ | $2535$ | $0.7854$ | $0.1732$ | $0.0316$ | $0.0087$ | $0.0012$ | $0.0000$ | $0.0000$ | $0.0000$
$1$ | $1267$ | $0.8082$ | $0.1650$ | $0.0205$ | $0.0063$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$2$ | $2535$ | $0.7953$ | $0.1692$ | $0.0292$ | $0.0051$ | $0.0012$ | $0.0000$ | $0.0000$ | $0.0000$
$3$ | $2534$ | $0.8062$ | $0.1539$ | $0.0328$ | $0.0059$ | $0.0012$ | $0.0000$ | $0.0000$ | $0.0000$

$p=5$, $\text{\rm ord}(\chi)=3$ (even cubic characters), $f=2$

$i$ | $N$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---|---
| pred. | $0.9584$ | $0.0399$ | $0.0016$ | $0.0001$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$0$ | $3184$ | $0.9579$ | $0.0408$ | $0.0013$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$2$ | $3184$ | $0.9579$ | $0.0415$ | $0.0006$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$

$p=7$, $\text{\rm ord}(\chi)=3$ (even cubic characters), $f=1$

$i$ | $N$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---|---
| pred. | $0.8368$ | $0.1395$ | $0.0203$ | $0.0029$ | $0.0004$ | $0.0001$ | $0.0000$ | $0.0000$
$0$ | $2470$ | $0.8336$ | $0.1368$ | $0.0263$ | $0.0028$ | $0.0004$ | $0.0000$ | $0.0000$ | $0.0000$
$2$ | $2470$ | $0.8490$ | $0.1283$ | $0.0190$ | $0.0032$ | $0.0004$ | $0.0000$ | $0.0000$ | $0.0000$
$4$ | $2470$ | $0.8559$ | $0.1219$ | $0.0202$ | $0.0020$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$

$p=11$, $\text{\rm ord}(\chi)=3$ (even cubic characters), $f=2$

$i$ | $N$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---|---
| pred. | $0.9917$ | $0.0083$ | $0.0001$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$0$ | $3184$ | $0.9900$ | $0.0101$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$2$ | $3184$ | $0.9956$ | $0.0044$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$4$ | $3184$ | $0.9912$ | $0.0088$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$6$ | $3184$ | $0.9931$ | $0.0069$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$8$ | $3184$ | $0.9962$ | $0.0038$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$

$p=13$, $\text{\rm ord}(\chi)=3$ (even cubic characters), $f=1$

$i$ | $N$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---|---
| pred. | $0.9172$ | $0.0764$ | $0.0059$ | $0.0005$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$0$ | $2742$ | $0.9168$ | $0.0762$ | $0.0069$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$2$ | $2742$ | $0.9172$ | $0.0766$ | $0.0055$ | $0.0007$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$4$ | $2742$ | $0.9209$ | $0.0715$ | $0.0069$ | $0.0007$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$6$ | $2742$ | $0.9263$ | $0.0686$ | $0.0047$ | $0.0004$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$8$ | $2742$ | $0.9238$ | $0.0697$ | $0.0058$ | $0.0007$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$10$ | $2742$ | $0.9154$ | $0.0806$ | $0.0029$ | $0.0011$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$

$p=5$, $\text{\rm ord}(\chi)=4$ (even and odd quartic characters), $f=1$

$i$ | $N$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---|---
| pred. | $0.7603$ | $0.1901$ | $0.0396$ | $0.0080$ | $0.0016$ | $0.0003$ | $0.0001$ | $0.0000$
$0$ | $5038$ | $0.7848$ | $0.1757$ | $0.0308$ | $0.0073$ | $0.0014$ | $0.0000$ | $0.0000$ | $0.0000$
$1$ | $4018$ | $0.7867$ | $0.1737$ | $0.0299$ | $0.0075$ | $0.0017$ | $0.0005$ | $0.0000$ | $0.0000$
$2$ | $5038$ | $0.7823$ | $0.1739$ | $0.0357$ | $0.0058$ | $0.0022$ | $0.0000$ | $0.0000$ | $0.0002$
$3$ | $5056$ | $0.7777$ | $0.1719$ | $0.0396$ | $0.0089$ | $0.0020$ | $0.0000$ | $0.0000$ | $0.0000$

$p=11$, $\text{\rm ord}(\chi)=5$ (even quintic characters), $f=1$

$i$ | $N$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---|---
| pred. | $0.9008$ | $0.0901$ | $0.0083$ | $0.0008$ | $0.0001$ | $0.0000$ | $0.0000$ | $0.0000$
$0$ | $1944$ | $0.9059$ | $0.0875$ | $0.0062$ | $0.0005$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$2$ | $1944$ | $0.8935$ | $0.0972$ | $0.0077$ | $0.0010$ | $0.0005$ | $0.0000$ | $0.0000$ | $0.0000$
$4$ | $1944$ | $0.8997$ | $0.0911$ | $0.0082$ | $0.0005$ | $0.0005$ | $0.0000$ | $0.0000$ | $0.0000$
$6$ | $1944$ | $0.9012$ | $0.0844$ | $0.0129$ | $0.0015$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$
$8$ | $1944$ | $0.9105$ | $0.0813$ | $0.0072$ | $0.0010$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$

$p=3$, $\text{\rm ord}(\chi)=8$ (even and odd octal characters), $f=2$

$i$ | $N$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---|---
| pred. | $0.8766$ | $0.1096$ | $0.0123$ | $0.0014$ | $0.0002$ | $0.0000$ | $0.0000$ | $0.0000$
$0$ | $7264$ | $0.8945$ | $0.0950$ | $0.0080$ | $0.0017$ | $0.0006$ | $0.0000$ | $0.0003$ | $0.0000$
$1$ | $6656$ | $0.8846$ | $0.1019$ | $0.0126$ | $0.0009$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$

###  3.2. Numerical evidence for Conjecture 1.2

We computed the Bernoulli numbers $B_{n,\chi}$ for all primitive $\chi$ with
$1<\text{\rm cond}(\chi)<1000$ and fixed order $\text{\rm ord}(\chi)$. We
checked if $p$ is $\chi$-regular for the first $100$ odd primes with
$p^{f}\equiv 1\;\big{(}\\!\\!\\!\mod\text{\rm ord}(\chi)\big{)}$ where $f$ is
fixed; the conjecture states that the proportion of regular primes is
$e^{-1/2}\approx 0.6065$ if $f=1$, while for $f>1$ the proportion should be
very close to $1$. The Kummer congruences imply that modulo $p$,

$\mathbf{L}_{p}(0,\chi\omega^{n})=-\big{(}1-\chi\omega^{n-1}(p)\big{)}B_{1,\chi\omega^{n-1}}\;\equiv\;-\big{(}1-\chi(p)p^{n-1}\big{)}\frac{B_{n,\chi}}{n}=\mathbf{L}_{p}(1-n,\chi\omega^{n})$

for all integers $n\geqslant 1$. In order to compute
$\mathbf{L}_{p}(0,\chi\omega^{n})\mod p$ for all Teichmüller twists such that
$\chi\omega^{n}$ is even, it suffices to check the untwisted Bernoulli numbers
$B_{n,\chi}$ mod $p$ where $n\in\\{2,\dots,p-1\\}$ if $\chi$ is even, and
$n\in\\{1,\dots,p-2\\}$ if $\chi$ is odd.

######  Remark.

We observe that $\big{(}1-\chi(p)p^{n-1}\big{)}\not\equiv 0\mod\varpi$ for all
integers $n\geqslant 2$, however for $n=1$ we always obtain a trivial zero if
$\chi(p)=1$. Hence we test regularity using Bernoulli numbers
$B_{n,\chi}\mod\varpi$ for all cases (including $n=1$), which is in line with
the definition of generalized (irr-)regular primes given in [11].

The tables below contain the number of evaluated characters, $N$ say, and the
average proportion of regular primes we found experimentally. The first line
of each table indicates the expected proportion. They also include the
standard deviation, $\sigma$, as well as the minimum/maximum proportion for
the corresponding data set.

$\text{\rm ord}(\chi)=2$ (even and odd quadratic characters), $f=1$

$N$ | proportion | standard dev. | min. value | max. value
---|---|---|---|---
pred. | $0.6065$ | - | - | -
$607$ | $0.6086$ | $0.0486$ | $0.48$ | $0.75$

$\text{\rm ord}(\chi)=3$ (even cubic characters), primes with $f=1$

$N$ | proportion | standard dev. | min. value | max. value
---|---|---|---|---
pred. | $0.6065$ | - | - | -
$162$ | $0.6107$ | $0.0477$ | $0.48$ | $0.77$

$\text{\rm ord}(\chi)=3$ (even cubic characters), primes with $f=2$

$N$ | proportion | standard dev. | min. value | max. value
---|---|---|---|---
pred. | $1$ | - | - | -
$162$ | $0.9970$ | $0.0058$ | $0.97$ | $1.00$

$\text{\rm ord}(\chi)=4$ (even and odd quartic characters), primes with $f=1$

$N$ | proportion | standard dev. | min. value | max. value
---|---|---|---|---
pred. | $0.6065$ | - | - | -
$1144$ | $0.6066$ | $0.0481$ | $0.46$ | $0.78$

$\text{\rm ord}(\chi)=4$ (even and odd quartic characters), primes with $f=2$

$N$ | proportion | standard dev. | min. value | max. value
---|---|---|---|---
pred. | $1$ | - | - | -
$1242$ | $0.9961$ | $0.0062$ | $0.97$ | $1.00$

###  3.3. Numerical evidence for Conjecture 1.3

Lastly for totally real cyclic extensions $F/\mathbb{Q}$ of conductor
$<10^{5}$, we computed the $\lambda_{p}(\chi)$-invariants for
$\chi=\theta\omega^{i}$ where $\theta$ is a character factoring through
Gal$(F/\mathbb{Q})$, and $\omega^{i}$ is an even power of the Teichmüller
character. We next determined the invariant $\lambda_{p}^{\text{\rm
tot}}(F)=\sum_{\theta,i}\lambda_{p}(\theta\omega^{i})$ by summing over even
characters $\theta\omega^{i}$ factoring through the group
Gal$\big{(}F(\mu_{p})/\mathbb{Q}\big{)}$.

For small primes $p<1000$, it is known [3, 10] that the $\lambda$-invariants
of $\mathbb{Q}(\mu_{p})$ are zero more often than generally predicted. In fact
prime numbers $p<37$ are always regular, and so here the modified prediction
for $F$-regular primes should be

$\text{\rm Prob}\big{(}\text{\rm$p$ is
$F$-regular}\big{)}\approx\;\exp\left(\\!-\frac{\gcd(m,p-1)-1}{2}\right)\quad\\!\text{if
the prime $p$ is $\mathbb{Q}$-regular.}$

For instance, if $m=[F:\mathbb{Q}]=2$ we should expect the proportion
$e^{-1/2}\approx 0.6065$, and if $m=[F:\mathbb{Q}]=3$ and $p\equiv
1\;(\\!\\!\\!\\!\mod 3)$ we should instead expect $e^{-1}\approx 0.3679$. If
$m=[F:\mathbb{Q}]=3$ and $p\equiv 2\;(\\!\\!\\!\\!\mod 3)$ we expect a
proportion close to $1$.

The tables shown below give the number $N$ of totally real fields $F$ that we
checked, as well as the distribution of the $\lambda_{p}^{\text{\rm
tot}}(F)$-invariant averaged over these sample fields.

$p=5$, $[F:\mathbb{Q}]=2$, $f=1$
$N$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $2535$ $0.6252$ $0.2694$ $0.0781$ $0.0209$
$0.0051$ $0.0012$ $0.0000$ $0.0000$

$p=7$, $[F:\mathbb{Q}]=2$, $f=1$
$N$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $2662$ $0.6078$ $0.2960$ $0.0770$ $0.0162$
$0.0023$ $0.0004$ $0.0000$ $0.0000$

$p=11$, $[F:\mathbb{Q}]=2$, $f=1$
$N$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $2786$ $0.6027$ $0.2893$ $0.0876$ $0.0183$
$0.0022$ $0.0000$ $0.0000$ $0.0000$

$p=13$, $[F:\mathbb{Q}]=2$, $f=1$
$N$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $2825$ $0.6000$ $0.2988$ $0.0835$ $0.0149$
$0.0028$ $0.0000$ $0.0000$ $0.0000$

$p=5$, $[F:\mathbb{Q}]=3$, $f=2$
$N$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $1592$ $0.9171$ $0.0000$ $0.0798$ $0.0000$
$0.0031$ $0.0000$ $0.0000$ $0.0000$

$p=7$, $[F:\mathbb{Q}]=3$, $f=1$
$N$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $1235$ $0.3595$ $0.3482$ $0.1870$ $0.0696$
$0.0219$ $0.0097$ $0.0040$ $0.0000$

$p=11$, $[F:\mathbb{Q}]=3$, $f=2$
$N$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $1592$ $0.9667$ $0.0000$ $0.0327$ $0.0000$
$0.0006$ $0.0000$ $0.0000$ $0.0000$

$p=13$, $[F:\mathbb{Q}]=3$, $f=1$
$N$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $1371$ $0.3669$ $0.3538$ $0.1875$ $0.0657$
$0.0190$ $0.0058$ $0.0015$ $0.0000$

Acknowledgements. The authors warmly thank Larry Washington, Qin Chao and
Luochen Zhao for their helpful comments and suggestions related to this paper.
The numerical experiments were done using SageMath, developed by William
Stein.

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##  Appendix A Tables of $\lambda$-invariants for small $\chi$ and $p$.

We write $\zeta_{m}$ to denote the $m$-th root of unity $\exp\big{(}\frac{2\pi
i}{m}\big{)}$. The tables below list for primes $p=3,5,7$ all primitive
Dirichlet characters of the form $\chi=\theta\omega^{i}$ satisfying:

  * •

$\text{\rm cond}(\theta)<1000$

  * •

$p\nmid\text{\rm cond}(\theta)$

  * •

$p\nmid\text{\rm ord}(\chi)$

  * •

$f<10$

such that $\lambda_{p}(\chi)>0$ if there is no trivial $p$-adic zero, and for
$\lambda_{p}(\chi)>1$ if there is. If a character is not listed then
$\lambda_{p}(\chi)=0$ (resp. $\lambda_{p}(\chi)=1$ in the trivial zero case).

N.B. The probability of finding a primitive character $\chi=\theta\omega^{i}$
with residue class degree $f\geqslant 10$ and non-trivial
$\lambda_{p}(\chi)$-invariant is small according to Conjecture 1.1.

I. $p=3$

[longtable=—p8mm—p8cm—p9mm—p9mm—p6mm—p6mm—, separator=semicolon, column
count=7, no head, table head=Dirichlet characters $\chi=\theta\omega^{i}$ with
nontrivial $\lambda_{3}(\chi)$ invariant.
$\mathbf{\lambda}_{3}(\chi)$ Dirichlet character $\theta$ twist $i$ $\text{\rm
ord}(\chi)$ $f$ triv. zero
$\mathbf{\lambda}_{3}(\chi)$ Dirichlet character $\theta$ twist $i$ $\text{\rm
ord}(\chi)$ $f$ triv. zero
]lambda3-1000.csv $\csvcoli$ $\csvcoliii$ $\csvcoliv$ $\csvcolv$ $\csvcolvi$

II. $p=5$

[longtable=—p8mm—p8cm—p9mm—p9mm—p6mm—p6mm—, separator=semicolon, column
count=7, no head, table head=Dirichlet characters $\chi=\theta\omega^{i}$ with
nontrivial $\lambda_{5}(\chi)$ invariant.
$\mathbf{\lambda}_{5}(\chi)$ Dirichlet character $\theta$ twist $i$ $\text{\rm
ord}(\chi)$ $f$ triv. zero
$\mathbf{\lambda}_{5}(\chi)$ Dirichlet character $\theta$ twist $i$ $\text{\rm
ord}(\chi)$ $f$ triv. zero
]lambda5-1000.csv $\csvcoli$ $\csvcoliii$ $\csvcoliv$ $\csvcolv$ $\csvcolvi$

III. $p=7$

[longtable=—p8mm—p8cm—p9mm—p9mm—p6mm—p6mm—, separator=semicolon, column
count=7, no head, table head=Dirichlet characters $\chi=\theta\omega^{i}$ with
nontrivial $\lambda_{7}(\chi)$ invariant.
$\mathbf{\lambda}_{7}(\chi)$ Dirichlet character $\theta$ twist $i$ $\text{\rm
ord}(\chi)$ $f$ triv. zero
$\mathbf{\lambda}_{7}(\chi)$ Dirichlet character $\theta$ twist $i$ $\text{\rm
ord}(\chi)$ $f$ triv. zero
]lambda7-1000.csv $\csvcoli$ $\csvcoliii$ $\csvcoliv$ $\csvcolv$ $\csvcolvi$