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††footnotetext: Key words and phrases: Factorials; modular mappings; random
sequences; Poisson distribution; fixed points; Stauduhar’s
Conjecture.††footnotetext: 2010 Mathematics Subject Classification: primary
11N69, secondary 49J55, 65C10.

# Factorials $\mkern 4.0mu({\operator@font mod}\mkern 6.0mup)$ and the average
of modular mappings

Cristian Cobeli, Alexandru Zaharescu  CC: Simion Stoilow Institute of
Mathematics of the Romanian Academy, 21 Calea Griviţei Street, P. O. Box
1-764, RO-014700, Bucharest, Romania<EMAIL_ADDRESS>AZ: Department
of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409
W. Green Street, Urbana, IL, 61801, USA and Simion Stoilow Institute of
Mathematics of the Romanian Academy, 21 Calea Griviţei Street, P. O. Box
1-764, RO-014700, Bucharest, Romania<EMAIL_ADDRESS>
###### Abstract.

We have known that most sequences in $\mathcal{M}=\\{1,2,\dots,M\\}$ with
length $n$ will miss $Me^{-\lambda}$ of the total numbers of
$\\{1,2,\dots,M\\}$ as the ratio $n/M$ tends to $\lambda$. Now we consider a
more general case where the numbers in $\\{1,2,\dots,M\\}$ are achieved
exactly k times by a ’random’ sequence $f(1),f(2),\dots,f(n)$. We show that if
$n/M\rightarrow\lambda$, then the limit has a Poisson distribution, that is,
the proportion of sequences for which some number in $\mathcal{M}$ is achieved
exactly $k$ times has the limit $\frac{\lambda^{k}}{k!}e^{-\lambda}$. We
conjecture that this is the behavior of the factorial mapping modulo a prime
and present a few supporting arguments.

##  1\. Introduction

Figure 1. The residue classes $n!\mkern 4.0mu({\operator@font mod}\mkern
6.0mup)$, $1\leq n\leq p-1$, for primes $p<100$, top and for primes $p<1000$,
bottom. Different colors correspond to different residue classes.

In contrast with many of their real siblings, which are continuous, modular
functions are known to be chaotic in nature. This is true for polynomials,
monomials in particular, and this behavior accentuates as the power increases
from $2$ to the value of the modulus minus $2$, that is, from squares to
inverses. The matter was studied extensively in the works of the authors et
al. [CZ’00, CZ’01, CVZ’00a, CVZ’03, CGZ’03], P. Kurlberg et al. [KR’99,
Kur’00, Kur’09], and I. E. Shparlinski [Shp’12].

Since the factorial mapping $x\mapsto x!$ modulo a prime $p$ acts like a
’diagonal of the monomials’, one expects it to induce an even higher degree of
randomness. Many visual representations, such as those in Figure 1, confirm
the expectation. One of the main issues is raised by the following problem.

######  Conjecture 1 (R. Stauduhar).

Let $p$ be a prime and let $h(p)$ be the number of distinct residues of
$1!,2!,\dots,(p-1)!\mkern 4.0mu({\operator@font mod}\mkern 6.0mup)$. Then

$\lim_{p\to\infty}\frac{h(p)}{p}=1-\frac{1}{e}\,.$

Stauduhar’s Conjecture is stated as Problem $7$ in a list of $114$ problems,
which, according to a nice tradition of the time, appeared in the Proceedings
of the 1963 Number Theory Conference [NTC’63]. In several later articles,
various authors referred to the same problem, but in the formulation of R. K.
Guy [Guy’81, Problem F11]. There, Guy draws attention to the phenomenon that
the sequence of factorials $1!,2!,\dots,p!$ misses about $p/e$ residue classes
modulo $p$. The conjecture remains unsolved, but it was proved [CVZ’00b] that
it holds in average for all modular mappings or, in other words, this is a
characteristic of randomly chosen sequences of $p$ classes of residues modulo
$p$. The link with the ’randomness’ was made earlier, as we have found in the
last stage of the preparation of this manuscript. Thus, Brillhart [NTC’63,
Problem 7, page 90] adds a comment to the statement of the conjecture, saying
in parentheses that “from extensive numerical calculation the statement
appears to be true” and ’this is the “random result” for any set of numbers’.

In this article we show that in average, the expected number of residue
classes that are reached exactly $k\geq 0$ times by a random modular function
defined from $\mathbb{F}_{p}$ with values in $\mathbb{F}_{p}$ equals
$\frac{1}{k!e}$. The limit asymptotic values for the mean and variation are
obtained in Theorems 2 and the size of the error term are evaluated in 3
below. Accordingly, for factorials, this allows us to extend Stauduhar’s
Conjecture, since wide-ranging numerical verifications confirm this trend.

######  Conjecture 2.

For any integer $k\geq 0$, the proportion of elements $y\in\mathbb{F}_{p}$ for
which there are exactly $k$ positive integers $n$ for which $n!\equiv y\mkern
4.0mu({\operator@font mod}\mkern 6.0mup)$ tends to the limit
$\frac{1}{k!}e^{-1}$, as $p\to\infty$.

Notice that Stauduhar’s Conjecture is just the $k=0$ case of Conjecture 2 and
the general statement says that if $p$ is sufficiently large, then the
elements of the sequence $\\{n!\mkern 4.0mu({\operator@font mod}\mkern
6.0mup)\\}_{1\leq n\leq p}$ behave like a Poisson process with mean
$\lambda=1$. More background data are presented in Section 2.

Our main result is a uniform estimation over all modular mappings of the
average number of elements attained exactly a certain fixed number of times.
Let $\mathcal{M}$ be a finite set of $M$ elements and let
$\mathcal{N}\subset\mathcal{M}$ be a subset of $n$ elements. Denote by
$\mathcal{T}$ the set of $n$-tuples with components in $\mathcal{M}$, that is,
$\mathcal{T}=\mathcal{M}^{n}$ or
$\mathcal{T}=\big{\\{}\mathbf{x}=(x_{1},x_{2},\dots,x_{n}):\ \
x_{1},\dots,x_{n}\in\mathcal{M}\big{\\}}$. We say that an element
$y\in\mathcal{N}$ is represented $k$ times in the tuple
$\mathbf{x}\in\mathcal{T}$, if exactly $k$ components of $\mathbf{x}$ coincide
with $y$. Let $m_{k}(\mathbf{x})$ be the number of elements of $\;\mathcal{N}$
that are represented exactly $k$ times by $\mathbf{x}$, that is,

$\begin{split}m_{k}(\mathbf{x}):=\Big{|}\big{\\{}y\in\mathcal{N}:\ \text{ $y$
is represented exactly $k$ times in
$\mathbf{x}$}\big{\\}}\Big{|}\,.\end{split}$ (1)

It turns out that if the sets $\mathcal{N}$ and $\mathcal{M}$ have
sufficiently many elements, than the proportions $m_{k}(\mathbf{x})/n$ cluster
around the values of a Poisson distribution. A controlled bound of the maximum
deviation is obtain in the following theorem.

######  Theorem 1.

Let $\lambda\in(0,1]$, $\gamma\in[0,1)$, $\delta\in(0,(1-\gamma)/2)$ and let
$k\geq 0$ be integer. Suppose the integer variables $M$ and $n$ satisfy the
inequalities $k\leq n\leq M$ and $M=n/\lambda+O(n^{\gamma})$, uniformly on
$\lambda$ and $k$ as $n$ tends to infinity while $\gamma$ is fixed. Then

$\frac{1}{|\mathcal{T}|}\cdot\Big{|}\Big{\\{}\mathbf{x}:\
\Big{|}\frac{m_{k}(\mathbf{x})}{n}-\frac{\lambda^{k}}{k!}e^{-\lambda}\Big{|}<n^{-\delta}\Big{\\}}\Big{|}=1-O\left(\frac{\lambda^{k}}{k!}\cdot\frac{1}{n^{1-2\delta}}\Big{(}1+\frac{\lambda^{k+1}(\lambda+k)}{k!}n^{\gamma}\Big{)}\right),$

and the constant involved in the big Oh term does not dependent on
$\lambda,\gamma,\delta$ and $k$.

Let us notice that both summands in the error term have their particular type
of contribution. The distinction can be seen whether or not $\lambda$ is very
small, or whether $k=0$, or else, whether $k$ is small or it becomes large
enough to make one or the other of the terms of the sum dominate on one side
or the other of the balance point, which is attained if
$n\approx\left(k!/(\lambda^{k+1}(\lambda+k)\right)^{1/\gamma}$.

##  2\. Ramification points and factorials $\mkern 4.0mu({\operator@font
mod}\mkern 6.0mup)$

Referring to the distribution of the sequence of factorials modulo a prime
$p$, an earlier simpler problem was proposed by P. Erdős [Guy’81, problem
F11]. He asks whether there exists $p>5$ for which the numbers
$2!,3!,\dots,(p-1)!$ are all distinct modulo $p$. If there were such a prime,
then T. Trudgian [Tru’14] verified that it must be greater than $10^{9}$. The
problem of Erdős is still unsolved, although B. Rokowska, A. Schinzel [RS’60]
and Trudgian [Tru’14] showed that for large classes of primes the sequence of
factorials modulo $p$ can not be so close to a permutation, while Klurman and
Munsch [KM’17] obtained non-trivial bounds for the average deviation.

In reality, for any given $p$, one can check that there are many residue
classes modulo $p$ that are hit more than once by the sequence of factorials.
Iterating the factorial function modulo $p$, two types of points distinguish
in the complex created tree. They are the fixed points, or the roots of the
tree, and the ramification points, which are the residue classes reached
several times by the factorials.

In precise terms, we say that $1\leq x\leq p-1$ is a fixed point or, shortly,
$x$ is an FP of the factorial function modulo $p$, if $x!\equiv x\mkern
4.0mu({\operator@font mod}\mkern 6.0mup)$. The residue classes modulo $p$,
viewed as stacks that are reached a different number of times by the sequence
$1!,2!,3!,\dots,(p-1)!\mkern 4.0mu({\operator@font mod}\mkern 6.0mup)$, are
also called ramification points of the factorial function modulo $p$ or,
shortly, RPs. Thus, rigorously, we say that $y\in\\{1,\dots,p-1\\}$ is a
$k$-RP, if it is hit exactly $k$ times by the sequence
$1!,2!,3!,\dots,(p-1)!$, that is, there exist exactly $k$ distinct integers
$x_{1},\dots,x_{k}\in\\{1,2,\dots,p-1\\}$ such that $x_{1}!\equiv\cdots\equiv
x_{k}!\equiv y\mkern 4.0mu({\operator@font mod}\mkern 6.0mup)$. For example,
in Table 1 one sees that for $p=23$, the points $a=1,2,5,9,12,22$ are FPs;
$10,15,16,17,19,20$ are $0$-RPs (they are missed by the factorial function
modulo $23$); $2,3,4,\dots$, $9$, $11$, $12$, $13$, $14$, $18$, $21$ are
$1$-RPs (they are hit exactly once by the factorial function); $22$ is a
$3$-RP (it is attained three times) and $1$ is a $5$-RP (it appears five times
on the second row of the table).

Table 1. The values of the modular factorial function $x\mapsto x!\pmod{p}$,
for $p=23$.

$x$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$ | $16$ | $17$ | $18$ | $19$ | $20$ | $21$ | $22$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$x!\pmod{p}$ | $1$ | $2$ | $6$ | $1$ | $5$ | $7$ | $3$ | $1$ | $9$ | $21$ | $1$ | $12$ | $18$ | $22$ | $8$ | $13$ | $14$ | $22$ | $4$ | $11$ | $1$ | $22$

Remark that $1,2$ and $p-1$ are always FPs (by Wilson’s Theorem), but
congruences such as

$\begin{split}4244208!\equiv 4244208\mkern 4.0mu({\operator@font mod}\mkern
6.0mu9991769),\quad\qquad 5112195!\equiv 5112195\mkern 4.0mu({\operator@font
mod}\mkern 6.0mu5444407)\end{split}$

are sporadic. Quite often, the median $m=(p+1)/2$ is an almost trivial fixed
point, too, since $m!\equiv m\mkern 4.0mu({\operator@font mod}\mkern 6.0mup)$
or $m!\equiv m-1\mkern 4.0mu({\operator@font mod}\mkern 6.0mup)$ for any prime
$p\equiv 3\mkern 4.0mu({\operator@font mod}\mkern 6.0mu4)$.

Also, notice this intrinsic connection between FPs and the particular
ramification point $1$:
for any prime $p\geq 3$, the number of FPs is $r$ if and only if $1$ is an
$(r-1)$-RP.

Let $m(p)$ be the number of $0$-RPs, that is, the number of residual classes
$1,2,\dots,p-1$ that are missed by the sequence of factorials modulo $p$.
Table 2 shows evidences in favor of Stauduhar’s Conjecture 1.

With so many residue classes missed by the factorials, we expect that the
sequence of factorials is always very far from a permutation, but a proof of
this fact is still needed. A comparison approach is taken by Lev [Lev’06] who
tests the size of the partial sums of the elements of permutations in abelian
groups. Upper and lower bounds for the number of distinct residue classes of
$n!\mkern 4.0mu({\operator@font mod}\mkern 6.0mup)$ under different conditions
were obtained by Banks et al. [BLSS’05], Garaev et al. [GLS’04, GH’17],
Klurman and Munsch [KM’17]. The interesting properties of $n!\mkern
4.0mu({\operator@font mod}\mkern 6.0mup)$ have been studied from different
perspectives by us [CVZ’00b], Shub and Smale [SS’95], Markström [Mar’15], Luca
et al. [LS’03, LS’05a, LS’05b], Cheng [Che’04], Broughan et al. [BB’09],
Garaev et al. [GLS’04, GLS’05], Banks et al. [BLSS’05], Dai et al. [CD’06,
Dai’08], García [Gar’07], [Gar’08].

Table 2. The proportion of residue classes missed by factorials for different
primes. Compare the values from the third column with $1/e\approx 0.3678794$.

$n\qquad\qquad$ | $p_{n}$ | $\quad m(p_{n})/p_{n}\quad$ | $1/e-m(p_{n})/p_{n}$
---|---|---|---
$5$ | $11$ | $0.5454545$ | $-0.1775751$
$7$ | $17$ | $0.3529412$ | $\phantom{-}0.0149382$
$8$ | $19$ | $0.4210526$ | $-0.0531731$
$26$ | $101$ | $0.3663366$ | $\phantom{-}0.0015428$
$100$ | $541$ | $0.3789279$ | $-0.0110484$
$1000$ | $7919$ | $0.3725218$ | $-0.0046423$
$10000$ | $104729$ | $0.3681884$ | $-0.0003089$
$100000$ | $1299709$ | $0.3679662$ | $-0.0000867$
$1000000$ | $15485863$ | $0.3676930$ | $\phantom{-}0.0001864$

Together with our colleague M. Vâjâitu [CVZ’00b], we showed that when $p$ is
sufficiently large the number of $0$-RPs is about $p/e$ for almost all
sequences. Since most sequences have no particular simple defining rule, we
may say that the non-representation of $p/e$ residue classes modulo $p$ is a
general feature that characterizes randomness.

Figure 2. The proportions of $k$-RPs of the $30$th prime (the figures on the
left) and of the $10\,000$th prime (the figures on the right). In the figures
at the bottom, only the RPs obtained from factorials from the first half, that
is, only the frequencies of $1!$, $2!$, $\dots$,
$\big{(}\frac{p-1}{2}\big{)}!\mkern 4.0mu({\operator@font mod}\mkern 6.0mup)$
are counted. The proportions are compared with the Poisson distribution with
means $\lambda=1$ and $\lambda=1/2$, respectively.

The more general statement [CVZ’00b, Theorem 1] says that if $\lambda\in(0,1]$
is fixed, $p$ is a large prime number, and $n\sim\lambda p$, then almost all
sequences of length $n$ chosen from a subset
$\mathcal{N}\subset\mathbb{F}_{p}$ having $n$ elements omit about
$n/e^{\lambda}$ classes of $\mathcal{N}$.

We remark that the proportions of $k$-RPs of the factorials are close to their
conjectured limits even for small primes $p$, as can be seen from the two
example shown in Figure 2 for $p=113$ and $p=104729$ (the $30$th and the
$10\,000$th primes). For $p=113$, there are $40$ points that are missed by
factorials, $45$ points that are hit exactly once, $19$ points that are hit
exactly twice and so on. The record values for the two primes are the residue
classes $57$, which is hit exactly $5$ times, since

$20!\equiv 40!\equiv 50!\equiv 89!\equiv 101!\equiv 57\mkern
4.0mu({\operator@font mod}\mkern 6.0mu113)\,,$

and $78919$, which is hit exactly $9$ times, since

$\begin{split}2470!&\equiv 2742!\equiv 20986!\equiv 29734!\equiv 36188!\\\
&\equiv 39370!\equiv 39865!\equiv 40787!\equiv 65457!\equiv 78919\mkern
4.0mu({\operator@font mod}\mkern 6.0mu104729)\,.\end{split}$

The fast decay of the number of $k$-RPs as $k$ increases is a widespread
phenomenon in different contexts and it can be investigated under the
following generic query.

######  Problem.

Let $\mathcal{M}$ be a set of positive integers and suppose
$\mathcal{N}\subset\mathcal{M}$ is a ’large enough’ subset. What is the
expected proportion of numbers that are represented exactly $k$ times by the
sequence $f(1)$,$f(2)$, $\dots$, $f(n)$, where
$\mathcal{N}:=\big{\\{}f(1),f(2),\dots,f(n)\big{\\}}$?

The most expected answer to the question in this problem is given in the
following section.

##  3\. The limits of the average and of the variation

Denote by $A(k;M,n)$ the average of the proportions of elements of
$\mathcal{N}\subset\mathcal{M}$ that are represented $k$ times by vectors
$\mathbf{x}\in\mathcal{T}$, that is,

$A(k;M,n):=\frac{1}{|\mathcal{T}|}\sum_{\mathbf{x}\in\mathcal{T}}\frac{m_{k}(\mathbf{x})}{n}\,,$
(2)

where the counter $m_{k}(\mathbf{x})$ is defined by (1). For any pair of
elements $x,y\in\mathcal{M}$, we define

$\delta(x,y)=\left\\{\begin{array}[]{ll}1,&\mbox{if }x=y\\\ 0,&\mbox{if }x\neq
y\end{array}\right.\,.$

Similarly, for any vector $\mathbf{x}=(x_{1},x_{2},\dots,x_{n})\in\mathcal{T}$
and any $y\in\mathcal{M}$, we denote

$\delta_{k}(\mathbf{x},y)=\begin{cases}1,&\text{if exactly $k$ components of
$\mathbf{x}$ coincide with $y$,}\\\ 0,&\text{else.}\end{cases}$

Then, by the inclusion-exclusion principle, the counting function
$m_{k}(\mathbf{x})$ can be expressed as

$\begin{split}m_{k}(\mathbf{x})&=\Big{|}\\{y\in\mathcal{N}:\
\delta_{k}(\mathbf{x},y)=1\\}\Big{|}\\\
&=\sum_{y\in\mathcal{N}}\sum_{\begin{subarray}{c}\mathcal{L}\subset\\{1,2,\dots,n\\}\\\
|\mathcal{L}|=k\end{subarray}}\prod_{\begin{subarray}{c}j\in\mathcal{L}\\\
i\in\\{1,2,\dots,n\\}\setminus\mathcal{L}\end{subarray}}\delta(x_{j},y)\Big{(}1-\delta(x_{i},y)\Big{)}\,.\end{split}$
(3)

Since for each $y\in\mathcal{N}$ there are ${n\choose k}$ choices for the
positions of the components of $\mathbf{x}$ occupied by $y$, while the
remaining $M-k$ positions can take any of the remaining $M-1$ values, by (3)
we obtain the following closed form expression of the average introduced by
(2):

$\begin{split}A(k;M,n)&=\frac{1}{|\mathcal{T}|}\sum_{\mathbf{x}\in\mathcal{T}}\frac{m_{k}(\mathbf{x})}{n}\\\
&=\frac{1}{nM^{n}}\sum_{\mathbf{x}\in\mathcal{T}}\sum_{y\in\mathcal{N}}\sum_{\begin{subarray}{c}\mathcal{L}\subset\\{1,2,\dots,n\\}\\\
|\mathcal{L}|=k\end{subarray}}\prod_{\begin{subarray}{c}j\in\mathcal{L}\\\
i\in\\{1,2,\dots,n\\}\setminus\mathcal{L}\end{subarray}}\delta(x_{j},y)\Big{(}1-\delta(x_{i},y)\Big{)}\\\
&=\frac{1}{M^{n}}{n\choose k}\big{(}M-1\big{)}^{n-k}\,.\end{split}$ (4)

This implies that

$\begin{split}A(k;M,n)=\frac{1}{k!}\cdot\frac{n(n-1)\cdots(n-k+1)}{M^{k}}\left(1-\frac{1}{M}\right)^{n-k}\to\frac{\lambda^{k}}{k!}e^{-\lambda},\end{split}$
(5)

provided that $k$ is fixed and both $M,n\to\infty$ such that the ratio $n/M$
tends to $\lambda$.

Next we find the limit of the variance of $A(k;M,n)$ about its limit mean
determined by the assymptotic estimate (5) Let

$M_{2}(k;M,n):=\frac{1}{|\mathcal{T}|}\sum_{\mathbf{x}\in\mathcal{T}}\left(\frac{m_{k}(\mathbf{x})}{n}-\frac{\lambda^{k}}{k!}e^{-\lambda}\right)^{2}$
(6)

be the second moment about the mean. Expanding the binomial, we find that it
can be written as

$\begin{split}M_{2}(k;M,n)&=\frac{\lambda^{2k}}{k!^{2}}e^{-2\lambda}-\frac{2\lambda^{k}}{k!}e^{-\lambda}A(k;M,n)+S_{2}(k;M,n)\,,\end{split}$
(7)

where we denoted

$S_{2}(k;M,n)=\frac{1}{|\mathcal{T}|}\sum_{\mathbf{x}\in\mathcal{T}}\left(\frac{m_{k}(\mathbf{x})}{n}\right)^{2}=\frac{1}{M^{n}n^{2}}\sum_{\mathbf{x}\in\mathcal{T}}m^{2}_{k}(\mathbf{x}).$

By (3) and the same argument used to derive the expression (4), we see that

$\begin{split}S_{2}(k;M,n)&=\frac{1}{M^{n}n^{2}}\sum_{\mathbf{x}\in\mathcal{T}}\Bigg{(}\sum_{y\in\mathcal{N}}\delta_{k}(\mathbf{x},y)\Bigg{)}^{2}\\\
&=\frac{1}{M^{n}n^{2}}\sum_{\mathbf{x}\in\mathcal{T}}\sum_{y,y^{\prime}\in\mathcal{N}}\delta_{k}(\mathbf{x},y)\delta_{k}(\mathbf{x},y^{\prime})\\\
&=\frac{1}{M^{n}n^{2}}\Bigg{(}\sum_{\mathbf{x}\in\mathcal{T}}\sum_{y\not=y^{\prime}\in\mathcal{N}}\delta_{k}(\mathbf{x},y)\delta_{k}(\mathbf{x},y^{\prime})+\sum_{\mathbf{x}\in\mathcal{T}}\sum_{y=y^{\prime}\in\mathcal{N}}\delta_{k}(\mathbf{x},y)\delta_{k}(\mathbf{x},y^{\prime})\Bigg{)}\\\
&=\frac{1}{M^{n}n^{2}}\left(n(n-1){n\choose k}{n-k\choose
k}(M-2)^{n-2k}+n{n\choose k}(M-1)^{n-k}\right).\end{split}$ (8)

We denote and rewrite the two terms on the last line of (8) as

$\begin{split}S_{I}(k;M,n)=&\left(1-\frac{1}{n}\right)\cdot\frac{1}{k!^{2}}\cdot\frac{n(n-1)\cdots(n-k+1)}{M^{k}}\\\
\phantom{S_{I}(k;M,n)=}&\phantom{\left(1-\frac{1}{n}\right)\cdot\frac{1}{k!^{2}}}\times\frac{(n-k)(n-k-1)\cdots(n-2k+1)}{M^{k}}\cdot\left(1-\frac{2}{M}\right)^{n-2k},\\\
S_{II}(k;M,n)=&\frac{1}{n}\cdot\frac{1}{k!}\cdot\frac{n(n-1)\cdots(n-k+1)}{M^{k}}\cdot\left(1-\frac{1}{M}\right)^{n-k}.\end{split}$
(9)

Provided that $k$ is fixed and both $M,n\to\infty$ such that the ratio $n/M$
tends to some fixed constant $\lambda\in(0,1]$, it follows that

$\begin{split}S_{I}(k;M,n)\to\frac{\lambda^{2k}}{k!^{2}}e^{-2\lambda}\quad\text{
and }\quad S_{II}(k;M,n)\to 0\,,\end{split}$

which means that

$\begin{split}\lim_{\begin{subarray}{c}n,M\rightarrow\infty\\\
\frac{n}{M}\rightarrow\lambda\end{subarray}}S_{2}(k;M,n)=\left(\frac{\lambda^{k}}{k!}e^{-\lambda}\right)^{2}.\end{split}$

On inserting this limit and the asymptotic estimate (5) on the right side of
relation (7), we find that the limit of the second moment about the mean is
zero.

######  Theorem 2.

Let $\lambda\in(0,1]$ and let $k$ be a fixed integer. Suppose that $n$ and $M$
are integer variables and both increase tending to infinity while their ratio
$n/M$ tends to $\lambda$. Then, the average defined by relation (2) and the
second square moment defined by (6) have the following limits:

$\lim_{\begin{subarray}{c}n,M\rightarrow\infty\\\
n/M\to\lambda\end{subarray}}A(k;M,n)=\frac{\lambda^{k}}{k!}e^{-\lambda}\quad\text{and}\quad\lim_{\begin{subarray}{c}n,M\rightarrow\infty\\\
n/M\to\lambda\end{subarray}}M_{2}(k;M,n)=0\,.$

##  4\. Sharp uniform estimate of the average and of the second moment

######  Theorem 3.

Let $\lambda\in(0,1]$, $\gamma\in[0,1)$ and let $k\geq 0$ be integer. Suppose
the integer variables $M$ and $n$ satisfy the inequalities $k\leq n\leq M$ and
$M=n/\lambda+O(n^{\gamma})$, uniformly on $\lambda$ and $k$ as $n$ tends to
infinity while $\gamma$ is fixed. Then

$\displaystyle A(k;M,n)$
$\displaystyle=\frac{\lambda^{k}}{k!}e^{-\lambda}\left(1+O\big{(}\lambda(\lambda+k)n^{\gamma-1}\big{)}\right)$
(10) and $\displaystyle M_{2}(k;M,n)$
$\displaystyle=O\left(\frac{\lambda^{k}}{k!}\cdot\frac{1}{n}\Big{(}1+\frac{\lambda^{k+1}(\lambda+k)}{k!}n^{\gamma}\Big{)}\right),$
(11)

where the constants implied in the estimates are independent of
$\lambda,\gamma$ and $k$.

###### Proof.

By the hypothesis of the theorem, we see that
$n/M=1/(\lambda^{-1}+O(n^{\gamma-1}))$, so that $1/M=\lambda(n^{-1}+O(\lambda
n^{\gamma-2}))$ and $n/M=\lambda(1+O(\lambda n^{\gamma-1}))$. Then the
asymptotic approximations of the main exp-log functions involved are:

$\begin{split}\left(\frac{n}{M}\right)^{k}=\lambda^{k}\Big{(}1+O\big{(}\lambda
n^{\gamma-1}\big{)}\Big{)}^{k}=\lambda^{k}\Big{(}1+O\big{(}\lambda
kn^{\gamma-1}\big{)}\Big{)}\end{split}$ (12)

and, if $a=1$ or $2$, then

$\begin{split}\left(1-\frac{a}{M}\right)^{n-k}=e^{(n-k)\log(1-a/M)}=e^{\frac{-an}{M}+O\left(\frac{k}{M}\right)}&=e^{-a\lambda}\left(1+O\big{(}\lambda^{2}n^{\gamma-1}+\lambda
kn^{-1}\big{)}\right).\end{split}$ (13)

On combining (4), (12) and (13), we find that

$\begin{split}A(k;M,n)&=\frac{1}{k!}\cdot\lambda^{k}\Big{(}1+O\big{(}\lambda
kn^{\gamma-1}\big{)}\Big{)}e^{-\lambda}\left(1+O\big{(}\lambda^{2}n^{\gamma-1}+\lambda
kn^{-1}\big{)}\right)\\\
&=\frac{\lambda^{k}}{k!}e^{-\lambda}\left(1+O\big{(}\lambda(\lambda+k)n^{\gamma-1}\big{)}\right)\,.\end{split}$

The estimation of $M_{2}(k;M,n)$ is obtained similarly on combining (7), (8),
(9),(12) and (13), which completes the proof of the theorem,

∎

Now we can prove the uniform result in Theorem 1. For any $\eta>0$, let us
split $\mathcal{T}$ in two disjoint parts, that is,
$\mathcal{T}=\mathcal{T}^{<}(\eta)\cup\mathcal{T}^{\geq}(\eta)$ and
$\mathcal{T}^{<}(\eta)\cap\mathcal{T}^{\geq}(\eta)=\emptyset$, where

$\begin{split}\mathcal{T}^{<}(\eta)=\Big{\\{}\mathbf{x}\in\mathcal{T}:\
\Big{|}\frac{m_{k}(\mathbf{x})}{n}-\frac{\lambda^{k}}{k!}e^{-\lambda}\Big{|}<\eta\Big{\\}}\quad\text{
and }\quad\mathcal{T}^{\geq}(\eta)=\Big{\\{}\mathbf{x}\in\mathcal{T}:\
\Big{|}\frac{m_{k}(\mathbf{x})}{n}-\frac{\lambda^{k}}{k!}e^{-\lambda}\Big{|}\geq\eta\Big{\\}}\,.\
\end{split}$

Then, accordingly,

$\begin{split}M_{2}(k;M,n)&=\frac{1}{|\mathcal{T}|}\sum_{\mathbf{x}\in\mathcal{T}^{<}(\eta)}\left(\frac{m_{k}(\mathbf{x})}{n}-\frac{\lambda^{k}}{k!}e^{-\lambda}\right)^{2}+\frac{1}{|\mathcal{T}|}\sum_{\mathbf{x}\in\mathcal{T}^{\geq}(\eta)}\left(\frac{m_{k}(\mathbf{x})}{n}-\frac{\lambda^{k}}{k!}e^{-\lambda}\right)^{2}.\\\
\end{split}$

Ignoring the contribution of the first sum, we obtain for $M_{2}(k;M,n)$ a
bound from below:

$\begin{split}M_{2}(k;M,n)\geq\frac{1}{|\mathcal{T}|}\sum_{\mathbf{x}\in\mathcal{T}^{\geq}(\eta)}\eta^{2}=\frac{\eta^{2}}{|\mathcal{T}|}\big{(}|\mathcal{T}|-|\mathcal{T}^{<}(\eta)|\big{)}\,.\end{split}$

This can be rewritten conveniently as the following inequality:

$\begin{split}\frac{|\mathcal{T}^{<}(\eta)|}{|\mathcal{T}|}\geq
1-\eta^{-2}M_{2}(k;M,n)\,.\end{split}$ (14)

On using the estimate (11) on the right side of (14) and choosing
$\eta=n^{-\delta}$, we find that

$\begin{split}\frac{|\mathcal{T}^{<}(n^{-\delta})|}{|\mathcal{T}|}\geq
1-O\left(\frac{\lambda^{k}}{k!}\cdot\frac{1}{n^{1-2\delta}}\Big{(}1+\frac{\lambda^{k+1}(\lambda+k)}{k!}n^{\gamma}\Big{)}\right),\end{split}$

which concludes the proof of the theorem.

Acknowledgements:  Calculations and plots created using the free open-source
mathematics software system SAGE: http://www.sagemath.org.

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