# A “supernormal” partition statistic

Madeline Locus Dawsey, Matthew Just and Robert Schneider  Department of
Mathematics
University of Texas at Tyler
Tyler, Texas 75799, U.S.A<EMAIL_ADDRESS>Department of Mathematics
University of Georgia
Athens, Georgia 30602, U.S.A<EMAIL_ADDRESS>Department of Mathematics
University of Georgia
Athens, Georgia 30602, U.S.A<EMAIL_ADDRESS>
###### Abstract.

We study a natural bijection from integer partitions to the prime
factorizations of integers that we call the “supernorm” of a partition, in
which the multiplicities of the parts of partitions are mapped to the
multiplicities of prime factors of natural numbers. In addition to being
inherently interesting, the supernorm statistic suggests the potential to
apply techniques from partition theory to identify and prove multiplicative
properties of integers. We prove an analogue of a formula of Kural-McDonald-
Sah: we give arithmetic densities of subsets of $\mathbb{N}$ instead of
natural densities in $\mathbb{P}$ like previous formulas of this type, as an
application of the supernorm, building on works of Alladi, Ono, Wagner and the
first and third authors. We then make an analytic study of related partition
statistics; and suggest speculative applications based on the additive-
multiplicative analogies we prove, to conjecture Abelian-type formulas. In an
appendix, using these same analogies, we give a conjectural partition-
theoretic model of prime gaps.

Keywords: integer partitions; arithmetic density; prime distribution.

##  1\. Introduction and statement of results

Integer partitions are finite multisets of natural numbers, which ripple with
interesting and useful set-theoretic properties. We denote a generic partition
$\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},\dots,\lambda_{r}),\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{r}\geq
1$, with $\emptyset$ denoting the empty partition. There are a number of
natural statistics on partitions that are of interest in number theory and
combinatorics, such as the size $|\lambda|=\sum_{1\leq i\leq r}\lambda_{i}$
(sum of parts) and the related partition function $p(n)$ that counts the
number of size-$n$ partitions, $n\geq 0$; the length $\ell(\lambda)$ (number
of parts); the largest part $\text{lg}(\lambda)=\lambda_{1}$ and smallest part
$\text{sm}(\lambda)=\lambda_{r}$ of the partition; the multiplicity
$m_{i}=m_{i}(\lambda)\geq 0$ (frequency) of the part $i\geq 1$ in $\lambda$;
along with Dyson’s rank, the Andrews-Garvan crank, and other useful maps from
the set of partitions $\mathcal{P}$ to $\mathbb{N}$ (see e.g. [4]).

Beautiful and surprising patterns abound in $\mathcal{P}$ much as in
$\mathbb{N}$; and in the values of these partition statistics, one observes
striking arithmetic properties which often require subtle methods to prove.
Partitions have been connected to algebra since the work of Cauchy [9]; in
recent years, ideas from algebra are used more and more to study partitions
themselves as algebraic and set-theoretic objects on equal footing with
numbers. This shift can perhaps be traced to G. E. Andrews’ far-reaching
theory of partition ideals developed in the 1960s and 70s based on ideas from
lattice theory ([4], Ch. 8), and to a seminal 1977 paper of Alladi and Erdős
[2], which employs a natural bijection between partitions into prime parts and
prime factorizations of integers, to prove analytic properties of arithmetic
functions.

Relatively recently, in the pair of papers [21, 22] and subsequent works, the
third author outlined a multiplicative theory of integer partitions. This
additive-multiplicative theory contains a host of further partition statistics
as direct analogues of classical arithmetic functions, such as an analogue of
the Möbius function $\mu(n)$, viz.

(1) $\mu_{\mathcal{P}}(\lambda)=\left\\{\begin{array}[]{ll}0&\text{if
$\lambda$ has any part repeated,}\\\
(-1)^{\ell(\lambda)}&\text{otherwise},\end{array}\right.$

as well as partition analogues of the Euler phi function and other arithmetic
functions, and a partition generalization of the theory of Dirichlet series.
It is equally closely aligned with the theory of $q$-series (see e.g. [3, 5]).

The focus in these works shifts somewhat from the usual size statistic, to a
multiplicative partition statistic we call the norm $N(\lambda)$ (product of
parts):

(2) $N(\lambda):=\lambda_{1}\lambda_{2}\lambda_{3}\cdots\lambda_{r},$

where $N(\emptyset):=1$ (it is an empty product).111For more in-depth
treatments of the partition norm, see [12, 22, 25]. Partitions of fixed norm
$n$ are called multiplicative partitions of $n$ (or factorizations of $n$) in
the literature; these were apparently first studied by MacMahon [15].
Following Oppenheim [19], the study of these objects is sometimes referred to
as “factorisatio numerorum”.  Furthermore, defined in [22] is a partition
multiplication operation $\lambda\cdot\lambda^{\prime}$ for
$\lambda,\lambda^{\prime}\in\mathcal{P}$, which is concatenation (multiset
union) of their parts, leading also to partition division and a theory of
partition divisors, or subpartitions. The heuristic behind this multiplicative
theory of partitions is: phenomena in classical multiplicative number theory
are special cases of partition-theoretic structures; they can be thought of as
prototypes for more general theorems about partitions.

One potentially promising corollary of this heuristic is to use methods from
partition theory – combinatorial bijections, $q$-series generating functions,
connections with modular forms – to shed new light on problems of interest in
multiplicative number theory, e.g. prime distribution, arithmetic densities,
$L$-functions, etc. Looking in this direction, in recent works [17, 18], Ono,
Wagner and the third author prove partition-theoretic formulas to compute
arithmetic densities of subsets of $\mathbb{N}$ as limiting values of
$q$-series as $q\to 1^{-}$, following up on a 1977 paper of Alladi [1] in
analytic number theory, and work of the first author [6] in algebraic number
theory.

Other authors [14, 27, 28] have since extended Alladi’s techniques from [1],
giving a variety of what we call “Alladi-like” formulas of the shape
$\sum_{n\geq 2}\mu^{*}(n)f(n)/n<\infty$, with $\mu^{*}(n):=-\mu(n)$ and $f(n)$
an arithmetic function, to compute natural densities and other constants, such
as this special case of a result of Kural-McDonald-Sah [14] stated as [27],
equation (4):

(3) $\sum_{\begin{subarray}{c}n\geq 2\\\ p_{\operatorname{min}}(n)\in
A\subseteq\mathbb{P}\end{subarray}}\frac{\mu^{*}(n)}{n}\ =\ d^{*}(A),$

where $A$ is a subset of the primes, $p_{\operatorname{min}}(k)$ denotes the
smallest prime factor of $k\in\mathbb{N}$, and $d^{*}(A)$ denotes the natural
density222Defined in (7). of $A\subseteq\mathbb{P}$. Analogous work [17, 18]
in partition theory gives $q$-series formulas for arithmetic densities of
general subsets of natural numbers. Recall the arithmetic density $d(S)$ of
subset $S\subseteq\mathbb{N}$ is defined by

$d(S):=\lim_{N\to\infty}\frac{\\#\\{n\in S\ :\ n\leq N\\}}{N},$

if the limit exists.

Here we prove an Alladi-like formula analogous to (3) above to compute $d(S)$
for general subsets $S\subseteq\mathbb{N}$ (as opposed to subsets of
$\mathbb{P}$) via a map from partition-theoretic density formulas in [17, 18].

Let $\operatorname{idx}(p)$ denote the index of $p\in\mathbb{P}$ in the
sequence of primes, i.e., the $n$th prime has $\operatorname{idx}(p_{n})=n$.
When notations require, we take $p_{0}:=1$ (the “empty prime”) and set
$\operatorname{idx}(1):=0$. Let $\mu^{*}(n):=-\mu(n)$ as above.

######  Theorem 1.

Let $S\subseteq\mathbb{N}$ and let $d(S)$ be the arithmetic density of $S$, if
the density exists. Then

$\sum_{\begin{subarray}{c}n\geq 2\\\
\operatorname{idx}(p_{\operatorname{min}}(n))\in
S\end{subarray}}\frac{\mu^{*}(n)}{n}\ =\ d(S).$

######  Remark.

The sum is taken over integers for which the index of the smallest prime
factor is in $S\subseteq\mathbb{N}$.

We prove this identity via a natural multiplicative partition statistic
resembling the norm $N(\lambda)$ in (2), that we call the supernorm of
$\lambda$:

(4) $\widehat{N}(\lambda)=2^{m_{1}}3^{m_{2}}5^{m_{3}}\cdots
p_{i}^{m_{i}}\cdots$

with $\widehat{N}(\emptyset):=1$, where $m_{i}=m_{i}(\lambda)\geq 0$ is the
multiplicity of part $i$ in partition $\lambda$.333We call $\widehat{N}$ the
“supernorm” because $\widehat{N}(\lambda)\geq N(\lambda)$ for all $\lambda$,
and it is completely multiplicative in the partition sense like the norm,
i.e.,
$\widehat{N}(\lambda\cdot\lambda^{\prime})=\widehat{N}(\lambda)\widehat{N}(\lambda^{\prime})$.
Clearly $\widehat{N}(\lambda)\in\mathbb{N}$ is the unique natural number whose
prime factors are indexed by $\lambda\in\mathcal{P}$. The supernorm statistic
is called the Heinz number of a partition in the OEIS [16], and is studied
analytically by Kumar in recent work [11], with many useful formulas proved.

The supernorm induces a natural isomorphism between partitions and prime
factorizations that can be observed immediately from the dual lattices in
Figure 1. Following Andrews (see [4], Ch. 8), for a partition $\lambda$
written in the classical “multiplicity superscript” notation

$\lambda=\left<1^{m_{1}},2^{m_{2}},3^{m_{3}},\dots,i^{m_{i}},\dots\right>$

where only finitely many $m_{i}\geq 0$ are nonzero, note that the sequence of
multiplicities $(m_{1},m_{2},m_{3},\dots)$ associated to the parts is an
element of the partially ordered set $\bigoplus_{i\geq 1}\mathbb{N}$, the set
of sequences of non-negative integers where only finitely many terms are
nonzero. One can establish a partial ordering on the multiplicities by taking
$(m_{1},m_{2},\ldots)\leq(n_{1},n_{2},\ldots)$ if and only if $m_{i}\leq
n_{i}$ for every $i\geq 1$. Thus we can form a lattice from the set
$\mathcal{P}$ where the ordering is multiset inclusion (partition divisibility
in the sense of [22]); this is a key insight of Andrews’ theory of partition
ideals.

This partition lattice is then isomorphic to the lattice of the positive
integers ordered by divisibility; the supernorm map
$\widehat{N}:\mathcal{P}\to\mathbb{N}$, which is clearly reversible, gives the
isomorphism explicitly.444The second author presented a preliminary overview
of these ideas and Fig. 1 at [8]. Morally speaking, the lattice bijection of
Figure 1 unifies the map of Alladi-Erdős on partitions into prime parts [2],
the lattice-theoretic backdrop of Andrews’ theory of partition ideals [4], and
aspects of the additive-multiplicative theory in [22].555Table 1 lists some of
the isomorphism relations explicitly.

$\emptyset$$\left<1\right>$$\left<2\right>$$\left<3\right>$$\left<1^{2}\right>$$\left<1,2\right>$$\left<2^{2}\right>$$\left<1,3\right>$$\left<2,3\right>$$\left<3^{2}\right>$$\left<1^{3}\right>$$\left<1^{2},2\right>$$\left<1^{2},3\right>$$\left<1,2^{2}\right>$$\left<2^{3}\right>$$\left<2^{2},3\right>$$\left<1,2,3\right>$$\left<1,3^{2}\right>$$\left<2,3^{2}\right>$$\left<3^{3}\right>$$\ldots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$

123546910152581220182745305075125$\ldots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$

Figure 1. Illustration of two lattices: partitions ordered by multiset
inclusion, integers ordered by divisibility

One immediate point comes to mind when considering the isomorphism between
$\mathcal{P}$ and $\mathbb{N}$ in Figure 1: classical bijections between
special subsets of partitions imply analogous bijections between subsets of
natural numbers.

For example, partition conjugation gives a bijection between partitions of
length $k$ and partitions with largest part $k$. Within the set $\mathbb{N}$,
since $\widehat{N}$ maps parts of partitions to indices of prime factors, this
bijection induces a bijection between classes of integers with restricted
prime factorizations.

######  Example 1.

There is a natural bijection between integers with $k\geq 1$ prime factors
including repetition, and integers whose largest prime factor is $p_{k}$ (the
$k$th prime). Furthermore, among the integers whose prime factors’ indices sum
to fixed $n\geq 1$, the number of integers with $k$ prime factors is equal to
the number of integers with largest prime factor $p_{k}$.

######  Remark.

Note that the sum of indices $\sum_{p|n}\operatorname{idx}(p)$ of the prime
factors of integer $n$, including repetition, is analogous to partition size
in this context.

Similarly, the bijection of Euler between partitions into distinct parts and
partitions into odd parts induces another obscure fact about integers.

######  Example 2.

There is a natural bijection between square-free integers and integers with
only odd-indexed prime factors, e.g. $p_{1}=2,p_{3}=5,p_{5}=11,$ etc.
Furthermore, among the integers whose prime factors’ indices sum to fixed
$n\geq 1$, the number of squarefree integers is equal to the number of
integers with only odd-indexed prime factors.

######  Remark.

The general partition result gives the same bijection between $k$th power-free
integers, and integers with no prime factor’s index divisible by $k$.

These bijections were noted in [10] from the perspective of multiplicative
partitions. Such subtle properties of integers would be difficult to prove (or
even notice) without ideas about partitions. Likewise, many theorems about
partitions can be reinterpreted as facts about special subsets of
$\mathbb{N}$.

Then a natural question to ask is: what can we say about the arithmetic
densities of subsets of integers arising in the image of the supernorm?

In Section 2, we follow up on this question of arithmetic densities, a thread
that leads to our proof of Theorem 1. In Section 3.1 we make a brief analytic
study of connections between various partition statistics, particularly the
norm, size and length; this continues in Section 3.2 where we look at analysis
related to the supernorm. In Section 4, we use the heuristic discussed above,
together with a theorem from [23] and a table of additive-multiplicative
correspondences, to conjecture a generalization of Theorem 1. In an Appendix,
we conjecture a surprisingly accurate partition-theoretic model of prime gaps
based on properties of the supernorm.

######  Remark.

The supernorm can be viewed as a composite map involving the norm. For a
partition $\lambda$, let the “superpartition” of $\lambda$ be the prime
partition
$\widehat{\lambda}:=\left<2^{m_{1}},3^{m_{2}},5^{m_{3}},\cdots,p_{i}^{m_{i}},\cdots\right>$,
with $m_{i}=m_{i}(\lambda)$. Then $\widehat{N}(\lambda)=N(\widehat{\lambda})$,
and other arithmetic functions translate similarly such as
$\mu(\widehat{N}\left(\lambda)\right)=\mu_{\mathcal{P}}(\widehat{\lambda})$.

##  2\. Proof of Theorem 1: Partition bijections
and arithmetic densities

An interesting feature of Example 1 in the preceding section, is that the
arithmetic densities of these two bijective subsets are equal: a moment’s
thought will convince the reader that almost all natural numbers do not have
exactly $k$ prime factors; it is just as intuitively obvious that almost all
natural numbers do not have the largest prime factor $p_{k}$ for fixed $k$.
Both of the subsets of $\mathbb{N}$ in Example 1 provably have arithmetic
density zero.

Is it always the case that bijective subsets arising from the supernorm map
also have matching arithmetic densities? Let us look at Example 2 above: the
density of squarefree numbers is well known to be $6/\pi^{2}$. Is this also
the density of the set $S_{k}$ of integers with no prime factor’s index
divisible by $k$?

######  Proposition 1.

Let $S_{k}\subset\mathbb{N}$ be the natural numbers with no prime factor’s
index divisible by fixed $k\geq 2$. Then

$d(S_{k})=0.$

###### Proof of Proposition 1.

We use a straightforward sieving argument. Let $\mathbb{P}_{k}$ be the set of
all primes that are indexed by a multiple of $k$, i.e.,
$p_{j}\in\mathbb{P}_{k}$ if and only if $k\ |\ j$. Then if we choose a
parameter $T=T(x)$ depending on $x\in\mathbb{R}^{+}$ and define

$\mathbb{P}_{k}(T):=\mathbb{P}_{k}\cap[2,T]$

and

$S_{k}(x):=\\#\left\\{n\in S_{k}:n\leq x\right\\},$

we can write

$0\leq S_{k}(x)\leq x-\sum_{j\geq
1}\sum_{\begin{subarray}{c}A\subset\mathbb{P}_{k}(T)\\\
|A|=j\end{subarray}}(-1)^{j}\left\lfloor\frac{x}{\prod_{p\in
A}p}\right\rfloor,$

which is an application of the principle of inclusion-exclusion, with
$\left\lfloor X\right\rfloor$ the floor function of $X\in\mathbb{R}$. We can
break up the floor function using $\left\lfloor X\right\rfloor=X-\\{X\\}$ with
$\\{X\\}$ the fractional part of $X\in\mathbb{R}$, to see that

$S_{k}(x)\leq
x\prod_{p\in\mathbb{P}_{k}(T)}\left(1-\frac{1}{p}\right)+O\left(2^{\pi(T)}\right)$

where $\pi(X)$ is the total number of primes not exceeding $X\geq 2$; we
recall that $\pi(X)\sim X/\log X$ as $X\to\infty$ by the prime number theorem.
Setting $T=\log x$ gives

$S_{k}(x)\leq x\prod_{p\in\mathbb{P}_{k}(\log
x)}\left(1-\frac{1}{p}\right)+o(x).$

This gives the upper bound on the arithmetic density:

(5) $d(S_{k})\ =\ \lim_{x\to\infty}\frac{S_{k}(x)}{x}\ \leq\
\prod_{p\in\mathbb{P}_{k}}\left(1-\frac{1}{p}\right).$

That the product diverges to zero is equivalent to the divergence of the sum

$\sum_{p\in\mathbb{P}_{k}}\frac{1}{p},$

which is clear since in the tail of the series, $p_{mk}\sim k\cdot m\log m$ as
$m\rightarrow\infty$ by the prime number theorem. Thus $d(S_{k})$ is zero. ∎

So the arithmetic densities of bijective subsets do not necessarily match up
in the image of the supernorm map, as Example 2 demonstrates. As we prove in
Section 3.2, the supernorm scatters partitions of size $n$ over a very large
interval in $\mathbb{N}$ as $n\to\infty$, namely

(6) $p_{n}\ \leq\ \widehat{N}(\lambda)\ \leq\ 2^{n}$

for $|\lambda|=n$. We note that $p_{n}\sim n\operatorname{log}n$ by the prime
number theorem, while $p(n)=o(2^{n})$ by the Hardy-Ramanujan asymptotic.
Seeking explicit transformations between densities of resulting bijective
subsets on this large interval would be a difficult (but potentially
worthwhile) undertaking.

An easier route to computing arithmetic densities via “supernormal”
bijections, is to map from natural densities; we take this path to prove
Theorem 1. Recall the natural density $d^{*}(A)$ of a subset
$A\subseteq\mathbb{P}$ is defined by

(7) $d^{*}(A)=\lim_{x\rightarrow\infty}\frac{\\#\\{p\in A\ :\ p\leq
x\\}}{\pi(x)},$

if the limit exists, where $\pi(x)$ is the number of primes not exceeding
$x\geq 2$.

We use the observation that the supernorm maps the partition consisting of the
single part $k\geq 1$ to the $k$th prime number. Therefore, conversely,
special subsets of $\mathbb{P}$ are in bijection with special subsets of
$\mathbb{N}$, viz. the prime indices. We record this intuitive observation as
a lemma.

######  Lemma 1.

Suppose $p\in A\subseteq\mathbb{P}$ if and only if $\operatorname{idx}(p)\in
S\subseteq\mathbb{N}$; then $d^{*}(A)=d(S)$, assuming one of the two limits
exists.

###### Proof.

Assume that $d^{*}(A)$ exists. Set $S(x):=\\#\\{n\in S\ :\ n\leq x\\}$. Then
for any $\varepsilon>0$ there exists an $N\geq 1$ such that for all $x>N$ we
have

$d^{*}(A)-\varepsilon\ <\ \frac{\pi_{A}(x)}{\pi(x)}\ <\ d^{*}(A)+\varepsilon,$

where the we count primes $p\leq x$ with

$\displaystyle\pi_{A}(x)\ :=\ \\#\\{p\leq x:p\in A\\}\ =\
\\#\\{n\leq\pi(x):n\in S\\}\ =\ S(\pi(x)).$

Thus if we choose $M=\pi(N)$, we have for all $y>M$

$d^{*}(A)-\varepsilon\ <\ \frac{S(y)}{y}\ <\ d^{*}(A)+\varepsilon,$

which shows that $d(S)$ exists and is equal to $d^{*}(A)$. By symmetry, a
similar argument proves that if $d(S)$ exists, then $d^{*}(A)$ exists and is
equal to $d(S)$. ∎

######  Remark.

Similar arguments might be applied to compare the density of a subset
$\\{c_{n_{1}},c_{n_{2}},c_{n_{3}},\dots\\}\subseteq\\{c_{1},c_{2},c_{3},\dots\\}=:C$
within any discrete set $C$, with the arithmetic density of the set of indices
$\\{n_{1},n_{2},n_{3},\dots\\}\subseteq\mathbb{N}$.

Applying Lemma 1 together with equation (3), Theorem 1 follows easily.

###### Proof of Theorem 1.

We start from equation (3) above. Now let $S$ be any set of positive integers
such that $d(S)$ exists, and define

(8) $A=A(S):=\left\\{p_{a}\in\mathbb{P}:a\in S\right\\}.$

Then by Lemma 1, $d^{*}(A)$ exists and is equal to $d(S)$. Therefore, using
(3):

$\displaystyle d^{*}(A)=\sum_{\begin{subarray}{c}n\geq 2\\\
p_{\operatorname{min}}(n)\in
A\end{subarray}}\frac{\mu^{*}(n)}{n}=\sum_{\begin{subarray}{c}n\geq 2\\\
\operatorname{idx}(p_{\operatorname{min}}(n))\in
S\end{subarray}}\frac{\mu^{*}(n)}{n}\ =\ d(S).$

The final equality above is the statement of the theorem. ∎

##  3\. Analytic comparisons of partition statistics

Since $p_{i}\sim i\log i$ by the prime number theorem, equation (4) suggests

(9) $\widehat{N}(\lambda)\ \approx\ 2^{m_{1}(\lambda)}\prod_{i\geq 2}(i\log
i)^{m_{i}(\lambda)}\ \approx\ N(\lambda)\cdot 2^{m_{1}(\lambda)}\prod_{i\geq
2}(\log i)^{m_{i}(\lambda)},$

where we pulled out the $i=1$ approximating factor to avoid getting zeroes. As
we discuss in Section 3.2, this approximation has considerable error at
smaller indices $i\geq 2$, but it does accurately display the following:
partitions with greater lengths have supernorm a lot larger than their norm –
and therefore much larger than their length and size, since

(10) $\ell(\lambda)\ \leq\ |\lambda|\ \leq\ N(\lambda)\
\leq\widehat{N}(\lambda).$

Conversely, _partitions with fewer parts have supernorm closer to the norm_.

In this section we look into some background analysis to aid our intuition in
comparing the relative magnitudes of these partition statistics. Excellent
studies of these connections can be found in works of Kumar-Rana [12, 11].

###  3.1. Comparing $\ell(\lambda),|\lambda|$ and $N(\lambda)$

####  3.1.1. Explicit formula for $\log N(\lambda)$

In this subsection and the next, we examine interrelations and asymptotics
between the length, size, and norm. First we prove a formula for the natural
logarithm of the norm, written in terms of the length and largest part
statistics.

For a partition $\lambda=\left\langle
1^{m_{1}},2^{m_{2}},\dots,r^{m_{r}}\right\rangle$ with
$\operatorname{lg}(\lambda)=r$, define $\lambda(m,n)$ to be the subpartition
of $\lambda$ containing only the parts of $\lambda$ with sizes
$m,m+1,m+2,\dots,n-1,n$.

######  Theorem 2.

Let $r>1$ be an integer, and define $\varepsilon_{r}$ to be 1 if $r$ is odd or
0 if $r$ is even. For a partition $\lambda=\left\langle
1^{m_{1}},2^{m_{2}},\dots,r^{m_{r}}\right\rangle$, we have that

$\displaystyle\log N(\lambda)=\ell(\lambda)\log
r+\varepsilon_{r}m_{\left\lfloor\frac{r}{2}\right\rfloor}\log\left(\left\lfloor\frac{r}{2}\right\rfloor\right)$
$\displaystyle\hskip
28.45274pt+\sum_{i=2}^{\left\lfloor\frac{r}{2}\right\rfloor}\big{[}\ell(\lambda(i,r-i+1))-\ell(\lambda(1,r-i))\big{]}\log(r-i+1).$

###### Proof.

Applying Abel summation to $\log N(\lambda)=\sum\limits_{i=1}^{r}m_{i}\log i$
yields

$\displaystyle\sum_{i=1}^{r}m_{i}\log i$
$\displaystyle=\sum_{i=1}^{r}m_{i}\log
r-\int_{1}^{r}\left(\frac{1}{u}\sum_{i=1}^{u}m_{i}\right)du$
$\displaystyle=\sum_{i=1}^{r}m_{i}\log
r-\int_{1}^{r}\frac{\ell(\lambda(1,u))}{u}\,du.$

In other words, $\ell(\lambda(1,u))$ is the number of parts in $\lambda$ with
size up to $u$. Since $\ell(\lambda(1,u))$ is a stair-step function which only
changes value at each integer value of $u$, we can break up the integral into
$r-1$ separate integrals:

$\displaystyle\int_{1}^{r}\frac{\ell(\lambda(1,u))}{u}\,du$
$\displaystyle=\int_{1}^{2}\frac{\ell(\lambda(1,1))}{u}\,du+\int_{2}^{3}\frac{\ell(\lambda(1,2))}{u}\,du+\cdots+\int_{r-1}^{r}\frac{\ell(\lambda(1,r-1))}{u}\,du.$

The lengths of the subpartitions are now constant on each interval, so we can
evaluate each integral separately, viz.

$\displaystyle\int_{1}^{2}\frac{\ell(\lambda(1,1))}{u}\,du$
$\displaystyle=\ell(\lambda(1,1))\log 2,$
$\displaystyle\int_{2}^{3}\frac{\ell(\lambda(1,2))}{u}\,du$
$\displaystyle=\ell(\lambda(1,2))\big{(}\log 3-\log 2\big{)},$
$\displaystyle\vdots$
$\displaystyle\int_{r-1}^{r}\frac{\ell(\lambda(1,r-1))}{u}\,du$
$\displaystyle=\ell(\lambda(1,r-1))\big{(}\log r-\log(r-1)\big{)}.$

The sum of these integrals simplifies when we rewrite
$\ell(\lambda(1,n))=m_{1}+m_{2}+\cdots+m_{n}$ for each $n$:

$\displaystyle\int_{1}^{2}\frac{\ell(\lambda(1,1))}{u}\,du$
$\displaystyle+\cdots+\int_{r-1}^{r}\frac{\ell(\lambda(1,r-1))}{u}\,du$
$\displaystyle=-m_{2}\log 2-m_{3}\log 3-\cdots-m_{r-1}\log(r-1)$
$\displaystyle\hskip 28.45274pt+m_{1}\log r+m_{2}\log r+\cdots+m_{r-1}\log r$
$\displaystyle=\ell(\lambda(1,r-1))\log r-\sum_{i=2}^{r-1}m_{i}\log i.$

Now, we can write $\log N(\lambda)$ as follows:

$\sum_{i=1}^{r}m_{i}\log i=\ell(\lambda)\log r-\ell(\lambda(1,r-1))\log
r+\sum_{i=2}^{r-1}m_{i}\log i.$

The remaining sum is $\log N(\lambda(2,r-1))$, which we can rewrite by using
Abel summation again:

$\sum_{i=2}^{r-1}m_{i}\log
i=\big{[}\ell(\lambda(2,r-1))-\ell(\lambda(1,r-2))\big{]}\log(r-1)+\sum_{i=3}^{r-2}m_{i}\log
i.$

More generally, the $j$th remaining sum, $1\leq
j\leq\left\lfloor\frac{r}{2}\right\rfloor$, can be written as follows:

$\sum_{i=j}^{r-j+1}m_{i}\log
i=\big{[}\ell(\lambda(j,r-j+1))-\ell(\lambda,1,r-j))\big{]}\log(r-j+1)+\sum_{i=j+1}^{r-j}m_{i}\log
i.$

Repeating Abel summation on the remaining sum each time until the bounds meet
at $\left\lfloor\frac{r}{2}\right\rfloor$ yields the desired formula for $\log
N(\lambda)$. ∎

####  3.1.2. Explicit formula for $|\lambda|$

In this subsection we record an explicit formula for the partition size,
written in terms of the length and largest part.

######  Theorem 3.

For a partition $\lambda=\left\langle
1^{m_{1}},2^{m_{2}},\dots,r^{m_{r}}\right\rangle$, we have that

$|\lambda|=\ell(\lambda)r+\sum_{i=3}^{r}\big{(}r-i\big{)}\ell(\lambda(1,i-1))-rm_{1}.$

###### Proof.

Applying Abel summation to $|\lambda|=\sum\limits_{i=1}^{r}m_{i}\cdot i$
yields

$\displaystyle\sum_{i=1}^{r}m_{i}\cdot i$
$\displaystyle=\sum_{i=1}^{r}m_{i}\cdot
r-m_{1}-\int_{1}^{r}\left(\sum_{i=1}^{u}m_{i}\right)du$
$\displaystyle=\ell(\lambda)r-m_{1}-\int_{1}^{r}\ell(\lambda(1,u))\,du.$

Again using the fact that $\ell(\lambda(1,u))$ is a stair-step function which
only changes value at the integers, we can break up the integral into $r-1$
integrals and evaluate each integral separately:

$\displaystyle\int_{1}^{r}\ell(\lambda(1,u))\,du$
$\displaystyle=\int_{1}^{2}\ell(\lambda(1,1))\,du+\int_{2}^{3}\ell(\lambda(1,2))\,du+\cdots+\int_{r-1}^{r}\ell(\lambda(1,r-1))\,du$
$\displaystyle=\ell(\lambda(1,1))+\ell(\lambda(1,2))+\cdots+\ell(\lambda(1,k-1))=\sum_{i=1}^{r-1}\ell(\lambda(1,i)).$

Rewriting each subpartition length as a sum of multiplicities, we obtain the
following:

$\displaystyle\sum_{i=1}^{r-1}\ell(\lambda(1,i))$
$\displaystyle=\sum_{i=1}^{r-1}\sum_{j=1}^{i}m_{j}$
$\displaystyle=(r-1)m_{1}+(r-2)m_{2}+\cdots+2m_{r-2}+m_{r-1}$
$\displaystyle=\sum_{i=1}^{r-1}(r-i)m_{i}$
$\displaystyle=\sum_{i=1}^{r-1}rm_{i}-\sum_{i=1}^{r-1}m_{i}\cdot i$
$\displaystyle=\ell(\lambda(1,r-1))r-\sum_{i=1}^{r-1}m_{i}\cdot i.$

Then we obtain the following formula for $|\lambda|$:

$\sum_{i=1}^{r}m_{i}\cdot
i=\ell(\lambda)r-m_{1}-\ell(\lambda(1,r-1))r+\sum_{i=1}^{r-1}m_{i}\cdot i.$

Repeating Abel summation on the remaining sum, we have that

$\sum_{i=1}^{r-1}m_{i}\cdot
i=\ell(\lambda(1,r-1))r-m_{1}-(r-1)\ell(\lambda(1,r-2))+\sum_{i=1}^{r-2}m_{i}\cdot
i.$

In general, applying Abel summation to the remaining sum up to $j$ yields the
following formula:

$\sum_{i=1}^{j}m_{i}\cdot
i=\ell(\lambda(1,j))r-m_{1}-j\ell(\lambda(1,j-1))+\sum_{i=1}^{j-1}m_{i}\cdot
i.$

Therefore, the formula for $|\lambda|$ simplifies to the following:

$\displaystyle\sum_{i=1}^{r}m_{i}\cdot i=\ell(\lambda)r-\ell(\lambda(1,r-1))r$
$\displaystyle\hskip
56.9055pt+\ell(\lambda(1,r-1))r-(r-1)\ell(\lambda(1,r-2))$
$\displaystyle\hskip
56.9055pt+\cdots+\ell(\lambda(1,2))r-2\ell(\lambda(1,1))+m_{1}-(r-1)m_{1}$
$\displaystyle=\sum_{i=1}^{r-1}\big{[}\ell(\lambda(1,r-i+1))r-(r-i+1)\ell(\lambda(1,r-i))\big{]}-(r-2)m_{1}$
$\displaystyle=r\sum_{i=2}^{r}\ell(\lambda(1,i))-\sum_{i=1}^{r-1}(i+1)\ell(\lambda(1,i))-(r-2)m_{1}.$

This gives the desired result. ∎

###  3.2. Analysis of the supernorm

####  3.2.1. Explicit bounds

In this subsection and the next, we look at estimating the magnitude of the
supernorm, and finish with a nice combinatorial connection between the length,
size, norm and supernorm statistics.

######  Theorem 4.

If $\lambda$ is a partition with $N(\lambda)=N$ then

$\widehat{N}(\lambda)\leq N^{\frac{\log 3}{\log 2}}.$

###### Proof.

If $\lambda=\langle 1^{m_{1}},2^{m_{2}},\ldots\rangle$ and we define
$N_{d}=d^{m_{d}}$, then a simple computation shows

$\widehat{N}(\lambda)=\prod_{d}N_{d}^{\frac{\log p_{d}}{\log d}}$

where $p_{d}$ is the $d$th prime. We now focus on the exponents

$a(d):=\frac{\log p_{d}}{\log d}$

and note it suffices to show $a(d)\leq\log 3/\log 2$ for all $d$. This can be
shown quickly using the prime number theorem, but we choose to give a more
elementary proof. One can easily obtain the upper bound for $p_{d}$:

$p_{d}\leq 2^{e/2}d\log d,$

and can use this to obtain an upper bound for $a(d)$:

$a(d)\leq 1+\frac{e\log 2}{2\log d}+\frac{\log_{2}d}{\log d}.$

Now we see the inequality $a(d)\leq\log 3/\log 2$ certainly holds for $d>52$;
the remaining cases can quickly be checked. ∎

######  Theorem 5.

If $\lambda$ is a partition with $|\lambda|=n$ and $p_{n}$ is the $n$th prime
then

$p_{n}\leq\widehat{N}(\lambda)\leq 2^{n}.$

###### Proof.

We start with the upper bound. Suppose among all partitions of size $n$ that
$\lambda$ maximizes $\widehat{N}(\lambda)$. Then it suffices to show that
$\ell(\lambda)=n$, since in this case $\lambda=\left\langle
1^{n}\right\rangle$ and $\widehat{N}(\lambda)=2^{n}$. Suppose then that
$\ell(\lambda)=k<n$. There must be a part of $\lambda$ that is larger than
one, so let $r$ denote a largest such part. We can then inductively assume
that the partition is of the form $\lambda=\left<1^{n-r},r\right>$ (for $n=1$,
$n=2$, etc. the hypothesis is easily verified). Now we have that
$\widehat{N}(\left<1^{n-r},r\right>)=2^{n-r}p_{r}$, but a simple application
of Bertrand’s postulate shows that $p_{r}<2p_{r-1}$. This shows that as long
as $k<n$ we can increase $k$ while only increasing the integer our partition
is mapped to by $\widehat{N}$; in other words, we must have $k=n$.

For the lower bound, we argue similarly except now assuming that $\lambda$
minimizes $\widehat{N}(\lambda)$ among all partitions of size $n$. It suffices
to show that $\ell(\lambda)=1$, and we notice that for $n=1$, $n=2$, $n=3$,
etc. this is the case. Thus we can inductively assume that
$\widehat{N}(\lambda)=p_{r}p_{n-r}$. It now suffices to show that
$p_{a}p_{b}>p_{a+b}$ for all positive integers $a$ and $b$. Notice that if
$a=1$ this is simply Bertrand’s postulate, so assume without loss of
generality that $1<a\leq b$. Thus $p_{a+b}\leq p_{2b}$ and $p_{a}p_{b}\geq
3p_{b}$. We now seek to show $p_{2b}\leq 3p_{b}$ for all $b$. By a result of
Rosser and Shoenfeld [20] we have for $n\geq 2$

$p_{n}>n(\log n+\log\log n-3/2)$

and for $n\geq 20$

$p_{n}<n(\log n+\log\log n-1/2).$

A simple computation shows that $p_{2b}\leq 3p_{b}$ for all $b\geq 48$. The
remaining cases can be quickly verified with a computer. ∎

####  3.2.2. Other considerations

Since the supernorm of a partition $\lambda$ shares multiplicative
similarities to the norm, we might hope to develop formulas like those
described in Theorems 2 and 3. Let $\lambda=\langle
1^{m_{1}},2^{m_{2}},\ldots\rangle$. Then

$\log\widehat{N}(\lambda)=\sum_{i\geq 1}m_{i}\log p_{i},$

and a reasonable thing to do is try and apply the prime number theorem to
estimate $\log p_{i}$. The trouble is most numbers will have small prime
factors where the prime number theorem is not accurate. For example,
$p_{3}=5$, while $3\log 3\approx 3.30$. If the part 3 appears in the partition
$\lambda$ multiple times, the error in this approximation compounds.

However, it is possible to explicitly relate the length, size, norm and
supernorm for special classes of partitions. Here we give an example of such a
class, noting that other related classes could be studied as well.

Let $\lambda_{[n]}:=\left<1,2,3,\dots,n-1,n\right>\in\mathcal{P}$ denote the
partition into distinct parts consisting of the first $n$ natural numbers.
Then we see immediately666For $p,p^{\prime}\in\mathbb{P}$, recall
$p$-primorial is defined by $p\\#:=\prod_{p^{\prime}\leq p}p^{\prime}$
(product of primes $\leq p$).:

$\displaystyle\ell\left(\lambda_{[n]}\right)\ =\ n\ =\ \exp(\log n)\text{ as
$n\rightarrow\infty$},$ $\displaystyle\left|\lambda_{[n]}\right|\ =\
{n+1\choose 2}=\ \exp(2\log n+O(1))\text{ as $n\rightarrow\infty$},$
$\displaystyle N\left(\lambda_{[n]}\right)\ =\ n!\ =\ \exp(n\log
n-n+o(n))\text{ as $n\rightarrow\infty$},$
$\displaystyle\widehat{N}\left(\lambda_{[n]}\right)\ =\ p_{n}\\#\ =\
\exp(n\log n(1+o(1)))\text{ as $n\rightarrow\infty$}.$

Interestingly, the left-hand sides give natural partition-theoretic
interpretations for the classical combinatorial statistics and asymptotics on
the right.

##  4\. “Supernormal” analogies and Abelian-type conjectures

In light of the ideas presented in this paper, Table 1 gives some
“supernormal” analogies – i.e., consistent with the supernorm map
$\widehat{N}$ in (4) – between partition-theoretic objects and their
arithmetic counterparts. We note similar but distinct additive-multiplicative
analogies exist in other settings; for instance, there is the web of partition
zeta function and arithmetic function analogies proved in [21, 22] related to
the norm instead of supernorm, and tables in [17, 18, 24] give analogies
arising in the contexts of those works.

| Partition side of analogy | Arithmetic side of analogy
---|---|---
1 | $\mathcal{P}$ | $\mathbb{N}$
2 | $\lambda$ |  $n$ (supernorm)
3 | $\lambda\cdot\lambda^{\prime}$ | $n\cdot n^{\prime}$
4 | Subpartitions of $\lambda$ | Divisors of $n$
5 | Parts of $\lambda$ | Prime factors of $n$ (indices)
6 | $\left<1^{m_{1}},2^{m_{2}},\dots,i^{m_{i}},\dots\right>$ | $p_{1}^{m_{1}}p_{2}^{m_{2}}\cdots p_{i}^{m_{i}}\cdots$
7 | $\ell(\lambda)$ |  $\Omega(n)$, number of prime factors
8 | $\rm{sm}(\lambda)$ |  $p_{\rm{min}}(n)$ (index)
9 | $\rm{lg}(\lambda)$ |  $p_{\rm{max}}(n)$ (index)
10 | Partitions with no $1$’s | Odd integers
11 | Partitions containing $1$’s | Even integers
12 | Partitions into distinct parts | Square-free integers
13 | $\mu_{\mathcal{P}}(\lambda)$ | $\mu(n)$
14 |  $q$-series | Dirichlet series
15 |  $q^{|\lambda|}$ in generating functions |  $n^{-s}$ in Dirichlet series
16 | $\prod_{i=1}^{\infty}(1-q^{i})$ | ${\zeta(s)^{-1}}$
17 | $q^{i}$ | $p_{i}^{-s}$
18 | $q\rightarrow 1^{-}$ | $s\to 1^{+}$
19 | $\sum_{i}a_{i}q^{i}$ | $\sum_{i}a_{i}p_{i}^{-s}$
20 | $(1-q)^{-1}$ | $1+\sum_{p\in\mathbb{P}}p^{-s}$
Table 1. “Supernormal” analogies between partition theory and arithmetic
consistent with the mapping $\widehat{N}:\mathcal{P}\to\mathbb{N}$.

In the citations noted above, the third author and his coauthors used
analogies like those in Table 1 to guess, and subsequently prove, new theorems
on both the additive and multiplicative sides of number theory. Indeed, the
present authors used the analogies of Table 1, plus the result (3) of Kural-
McDonald-Sah and analogies with [17, 18], to intuit and prove Theorem 1. As
long as one takes care to use the appropriate analogies depending on context,
such identities seem to arise naturally from the background combinatorics and
convergence properties – and display an interesting, useful fusion of features
from both sides of number theory. Here, as examples of this method of
guesswork, we formulate a couple of conjectures that are speculative, but do
specialize to known cases.

We note that the right-hand column of Table 1 reflects direct mappings from
the left column consistent with the supernorm
$\widehat{N}:\mathcal{P}\to\mathbb{N}$. The first thirteen rows of the table
are obvious from the supernorm map; e.g. it is clear that
$\mu_{\mathcal{P}}(\lambda)$ in equation (1) transforms to the definition of
the classical $\mu(n)$ simply by substituting right-hand-column symbols and
words, for the corresponding terms from the left column, and vice versa. This
theme continues in the subsequent rows, but the analogies are more analytic in
nature. Even so, it is just as clear that, for instance, replacing $q^{i}$
with $p_{i}^{-s}$ transforms $\prod_{i=1}^{\infty}(1-q^{i})$ into
$\prod_{i=1}^{\infty}(1-p_{i}^{-s})=\zeta(s)^{-1}$. Expanding these products
yields the analogy between
$\sum_{\lambda\in\mathcal{P}}\mu_{\mathcal{P}}(\lambda)q^{|\lambda|}=\sum_{\lambda\in\mathcal{P}}\mu_{\mathcal{P}}(\lambda)q^{\lambda_{1}}q^{\lambda_{2}}\cdots
q^{\lambda_{r}}$ and
$\sum_{n\in\mathbb{N}}\mu(n)n^{-s}=\sum_{\lambda\in\mathcal{P}}\mu_{\mathcal{P}}(\lambda)p_{\lambda_{1}}^{-s}p_{\lambda_{2}}^{-s}\cdots
p_{\lambda_{r}}^{-s}$; we note from the product forms that both cases have
limiting value zero as $q\to 1^{-}$ and $s\to 1^{+}$,
respectively.777Conformal mapping between the complex regions $|q|<1$ and
$\operatorname{Re}(s)>1$ gives $s=2/(1+q)\to 1^{+}$ as $q\to 1^{-}$. The last
two rows give the analogues of power series and geometric series consistent
with this scheme.

Now, let us apply these analogies. In a recent paper [23] generalizing [17,
18], the third author proves families of Abelian theorems using partition
theory and $q$-series. Motivated by this work, we will conjecture Dirichlet
series and Alladi-like Abelian theorems using the analogies in Table 1.

Let us start with a theorem of Frobenius [7] that is a central component of
the proofs of [23]. For an arithmetic function $f:\mathbb{N}\to\mathbb{C}$
(for which we take $f(0):=0$ when needed), if the limit of the average value
$L_{f}:=\lim_{N\to\infty}\ \frac{1}{N}\sum_{k=1}^{N}f(k)$, then for
$|q|<1,q\to 1^{-}$ radially,888In fact, the limit (11) holds if $q\to 1$
through any path in a Stolz sector of the unit disk, i.e., a region with
vertex at $z=1$ such that $\frac{|1-q|}{1-|q|}\leq M$ for some $M>0$. Do Stolz
sectors map to regions in $\operatorname{Re}(s)>1$ that give analogous
convergence under $s=2/(1+q)$? Frobenius gives

(11) $\lim_{q\to 1^{-}}\ (1-q)\sum_{n\geq 1}f(n)q^{n}\ =\ L_{f}.$

If the sequence $\\{f(n)\\}$ converges, then (11) is the statement of Abel’s
convergence theorem. If $\\{f(n)\\}$ diverges, we assign the limiting value
$L_{f}$ to the sequence so long as the average value converges.

Under the analogies of Table 1, the limit of (11) transforms like $\lim_{q\to
1^{-}}\mapsto\ \lim_{s\to 1^{+}}$, and the summands of the power series
transform like $f(n)q^{n}\mapsto
f(n)p_{n}^{-s}=f(\operatorname{idx}(p_{n}))p_{n}^{-s}$. Does the
multiplicative factor $1-q=1-q^{1}$ also transform to
$1-p_{1}^{-s}=1-2^{-s}\to 1/2$, by the $q^{i}\mapsto p_{i}^{-s}$
correspondence, to yield as the appropriate analogy

(12) $\lim_{q\to 1^{-}}\ (1-q)\sum_{n\geq 1}f(n)q^{n}\ \
\stackrel{{\scriptstyle?}}{{\longmapsto}}\ \ \lim_{s\to 1^{+}}\
\frac{1}{2}\sum_{n\geq 1}f(n)p_{n}^{-s}\ \ ?$

This is a good example of how the analysis, as well as the exact structural
analogues, should inform our intuition. Ideally, we want our analogies to have
analogous limiting behaviors. Note that if the limit in $q$ on the left of
(12) is $0<L_{f}<\infty$, then the power series must diverge to infinity as
$q\to 1^{-}$ to achieve the finite nonzero limiting value. Morally, equation
(11) is an advanced instance of L’Hospital’s rule since, in fact (recalling
$f(0):=0$),

$\lim_{q\to 1^{-}}\ (1-q)\sum_{n\geq 1}f(n)q^{n}\ =\ \lim_{q\to 1^{-}}\
\frac{\sum_{n\geq 0}f(n)q^{n}}{\sum_{n\geq 0}q^{n}}\ =\
\frac{\infty}{\infty}.$

An appropriate analogy to (11) would mimic the implicit use of L’Hospital’s
rule as well as the structural analogies. Now, the Dirichlet series on the
right of (12) also diverges to infinity; but with no divergent denominator
term to serve as a counter-balance, it will not yield analogous limiting
behavior. To achieve the correct analogy, we want to transform
$1-q=1/(1-q)^{-1}$ via the geometric series analogy in row 20 of Table 1,
instead of using $q^{i}\mapsto p_{i}^{-s}$.

Based on these analogies,999And noting $1+\sum_{p}p^{-s}\sim\sum_{p}p^{-s}$ as
$s\to 1^{+}$. we make a somewhat speculative conjecture.

######  Conjecture 1.

For $f(n)$ an arithmetic function, if
$\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}f(k)=L_{f}$ exists, then

$\lim_{s\to 1^{+}}\
\frac{\sum_{p\in\mathbb{P}}f\left(\operatorname{idx}(p)\right)p^{-s}}{\sum_{p\in\mathbb{P}}p^{-s}}\
\ =\ \ L_{f}.$

We are encouraged to note that if $f(n)$ is the indicator function of the set
$S\subseteq\mathbb{N}$, and $A=A(S)\subseteq\mathbb{P}$ is as defined in (8),
then $L_{f}=d(S)=d^{*}(A)$. In this case, Conjecture 1 reduces to the
definition of the Dirichlet density of $A\subseteq\mathbb{P}$, which is well
known to equal the natural density $d^{*}(A)$:

$\lim_{s\to 1^{+}}\ \frac{\sum_{p\in A}p^{-s}}{\sum_{p\in\mathbb{P}}p^{-s}}\
=\ d^{*}(A).$

######  Remark.

A related conjecture is that $\sum_{n\geq 1}a_{n}q^{n}$ $\sim\sum_{n\geq
1}b_{n}q^{n}$ as $q\to 1^{-}$ if and only if
$\sum_{p}a_{\operatorname{idx}(p)}p^{-s}$
$\sim\sum_{p}b_{\operatorname{idx}(p)}p^{-s}$ as $s\to 1^{+}$, based on Table
1.

Next we conjecture an Alladi-like formula, in analogy with the following
partition-theoretic $q$-series result from the third author’s paper [23]. For
$f(n)$ an arithmetic function, setting
$\mu_{\mathcal{P}}^{*}(\lambda):=-\mu_{\mathcal{P}}(\lambda)$, it is proved in
[23] that

(13) $\displaystyle\lim_{q\to 1^{-}}\
\sum_{\lambda\in\mathcal{P}}\mu_{\mathcal{P}}^{*}(\lambda)f\left(\operatorname{sm}(\lambda)\right)q^{|\lambda|}\
\ =\ \ L_{f}.$

Using Table 1 together with (13), we swap the limits in $q$ and $s$ as before,
replace partitions with natural numbers in the indices of the series, and swap
$\mu_{\mathcal{P}}(\lambda)\mapsto\mu(n)$ and $q^{|\lambda|}\mapsto n^{-s}$.
The only subtlety to arriving at another “supernormal” conjecture, is to note
that by Table 1, $\operatorname{sm}(\lambda)$ maps to the index of
$p_{\operatorname{min}}(n)$, as opposed to the smallest prime factor of $n$
itself.

Taking the limit as $s\to 1^{+}$, the analogies lead us to guess an Alladi-
like identity; note $\mu^{*}(n):=-\mu(n)$, $f(\operatorname{idx}(1)):=0$,
$(f\circ\operatorname{idx})(p):=f(\operatorname{idx}(p))$.

######  Conjecture 2.

For $f(n)$ an arithmetic function, if
$\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^{N}f(k)=L_{f}$ exists, then

$\displaystyle\sum_{n\geq
2}\frac{\mu^{*}(n)\left(f\circ\operatorname{idx}\right)\left(p_{\operatorname{min}}(n)\right)}{n}\
\ =\ \ L_{f}.$

Again, we are encouraged by the observation that if $f(n)$ is the indicator
function of $S\subseteq\mathbb{N}$, then $L_{f}=d(S)$ and Conjecture 2 reduces
to Theorem 1.

It is very interesting that combinatorial and set-theoretic aspects of
partition theory are expressed in the prime factorizations of integers via the
indices of prime numbers – inducing observable (if obscure) patterns and
bijections within the set of natural numbers.101010We note converting
partition parts to indices of variables is a key insight of Sills [26].

## Appendix: Conjectural partition model of prime gaps

In this appendix, we use “supernormal” analogies from Table 1 to conjecture a
partition-theoretic model of prime number distribution111111We direct readers
interested in this direction to work of Kumar [11] and Kumar-Rana [13], who
give interpretations of Goldbach’s conjecture using arguments about
partitions. – continuing the spirit of Section 4, viz. using the supernorm to
formulate conjectures.

While the set $\mathbb{P}$ is highly enigmatic in number theory and known for
its seemingly stochastic growth, from the perspective of the supernorm map,
prime numbers are as easy and orderly as a set can be: the $n$th prime $p_{n}$
represents the supernorm $\widehat{N}(\lambda)$ of the partition $\lambda=(n)$
with a single part $n\in\mathbb{N}$ (recalling $\emptyset$ maps to $1$). The
sequence of primes is essentially the image of the sequence of natural
numbers. The complexities from this perspective come from the also enigmatic,
seemingly stochastic proliferation of partitions with many parts, which are
mapped by $\widehat{N}$ to gaps between primes.

Thus prime gaps have a combinatorial interpretation, suggesting the following
heuristic: understanding prime gaps is equivalent to counting the partitions
that map into respective gaps under the supernorm $\widehat{N}$.

A clear goal based on this heuristic is to identify partitions that map into
each prime gap so we can count them (at least approximately), which is to say,
to identify $\lambda\in\mathcal{P}$ such that for $n\geq 2$,

(14) $p_{n}\ \ \leq\ \ \widehat{N}(\lambda)\ \ <\ \ p_{n+1}.$

Ideally, one wants to do this without generating and checking partitions
computationally. We will use the above heuristic together with the analogies
of Table 1 to reason toward an educated guess of such a method.

The statement of the above heuristic is a leap of faith in the usefulness of
“supernormal” analogies, but is not in itself conjectural. Let us refine the
heuristic from elementary considerations, to conjecture a certainly-over-
simplified partition-theoretic approach to prime gaps that agrees with the
facts surprisingly well.121212The second author discussed a preliminary
version of our conjectural model at [8].

We will employ a few simplifying assumptions, captured in three postulates
below. The end goal is to compute (at least approximately) the length of the
interval $[p_{n},p_{n+1})$ for each $n\geq 1$, recalling the prime number
theorem gives the average prime gap $p_{n+1}-p_{n}\sim\log n$.

As a matter of principle, we assume $n$ somehow identifies partitions mapping
into the prime gap immediately above $p_{n}$. Right away one observes that
partitions of size $|\lambda|>n$ have $\widehat{N}(\lambda)\geq p_{n+1}$, so
we only need to search for partitions of size $\leq n$. But looking at how
partitions of fixed size map under $\widehat{N}$ is not a very useful
classification principle, since partitions of $n>1$ are scattered over the
interval $[p_{n},2^{n}]$ by the supernorm, by Theorem 5. We aim to count
partitions mapping into a relatively small gap $[p_{n},p_{n+1})$.

Speaking naïvely, by (9), all partitions $\lambda$ with norm $N(\lambda)=n$
should map to roughly comparable magnitudes, since
$\widehat{N}(\lambda)\approx n\cdot 2^{m_{1}}\prod_{2\leq i\leq n}(\log
i)^{m_{i}}$ has the common leading factor $n$. So to simplify the problem, let
us count only partitions of norm $n$ and ignore other partitions, as if
norm-$n$ partitions were the only partitions that predictably map just above
$p_{n}$ under $\widehat{N}$.

######  Postulate 1.

We focus on partitions of norm $n$; we assume many of these map above $p_{n}$
on the number line, but not too far above, under $\widehat{N}$.

Wishful thinking suggests, then, that all integers in the interval
$[p_{n},p_{n+1})$ are in bijection with the partitions of norm $n$ under
$\widehat{N}$, and one simply enumerates these to arrive at the exact value of
the prime gap. But the set of partitions of norm $n$ is infinite, since
adjoining arbitrarily many $1$’s to any partition produces new partitions of
the same norm. Moreover, noting lines 10 and 11 of Table 1, even numbers on
$[p_{n},p_{n+1})$ must arise as images of partitions with norms less than $n$,
since factors of $2$ in the supernorm come from parts of size $1$ on the
partition side that do not affect the norm. Deleting partitions with $1$’s
from our count side-steps these issues.

######  Postulate 2.

We focus on partitions with no $1$’s, which map to odd integers under the
supernorm $\widehat{N}$, and disregard partitions containing $1$’s.

This assumption yields an equivalent heuristic: the prime gap is twice the
number of partitions with no $1$’s that map into the interval, i.e., for
$n\geq 2$,

(15) $p_{n+1}-p_{n}\ \ =\ \ 2\ \cdot\ \\#\left\\{\lambda\ \text{with no 1's}\
:\ p_{n}\leq\widehat{N}(\lambda)<p_{n+1}\right\\}.$

But even with the two reasonable simplifications above, we can’t expect a very
realistic model of the odd numbers on the interval $[p_{n},p_{n+1})$ from
these postulates alone. It is just not the case that all partitions with no
$1$’s having norm $n$ will map into this gap; as Theorem 4 shows, these
supernorms can get as large as $n^{\log 3/\log 2}\gg n\log n$. Bertrand’s
postulate guarantees that $p_{n+1}<2p_{n}$; it is easy to see from (9), then,
that norm-$n$ partitions with just a few parts can still have supernorms
exceeding $2p_{n}\sim 2n\log n$, thus potentially mapping above the interval
$[p_{n},p_{n+1})$.

So we still haven’t narrowed our focus to a small enough subset of partitions
to propose even a rough model of prime gaps. We are now going to cut to the
chaff, and disregard all partitions of norm $n$ except those with no $1$’s and
having exactly one or two parts; by (9), most of these have supernorms roughly
near $n\log n$. This rather extreme simplification helps us narrow down to odd
supernorm values that seem likely to lie just above $p_{n}$.

######  Postulate 3.

We focus on partitions of length $\leq 2$, and disregard longer partitions
whose images under $\widehat{N}$ we assume lie mostly above the prime gap.

Together, the above three postulates identify a subset of $\mathcal{P}$ we
intuitively expect maps to “most” odd numbers in the prime gap
$[p_{n},p_{n+1})$. We will assume this scheme resets at $p_{n+1}$ with
partitions of norm $n+1$, so there is not space conceptually for any
length-$2$, norm-$n$ partitions to map above $p_{n+1}$ in our simplistic model
– they are assumed to fit into the gap immediately above $p_{n}$. Moreover, we
assume provisionally that all other partitions are spread out diffusely enough
under $\widehat{N}$ that their contribution to our estimate of the prime gap
is negligible.

These extreme simplifications of the problem allow us to propose a provisional
partition-theoretic model of prime gaps.

######  Conjecture 3.

For $n\geq 1$, the odd numbers on the interval $[p_{n},p_{n+1})$ can be
modeled by the images under the supernorm $\widehat{N}$ of partitions of norm
equal to $n$, with no part equal to $1$, and having exactly one or two parts.

######  Remark.

We will treat this model as a bijection for the sake of making conjectural
predictions. Less exactly, our conjectural model proposes the enumeration of
these partitions dominates the asymptotics of prime gaps.

Reasoning by a mix of analogy and pragmatic first principles led us to this
model. Before we make deductions that are surprisingly accurate given our
simplistic assumptions, we need to point out that the conjecture cannot
strictly be true. First of all, there definitely is not a bijection between
the odd numbers in the prime gap and the partitions counted by Conjecture 3;
that would imply all the odd composite numbers have exactly two prime factors,
since the same heuristic holds for every index $n$ by induction. Indeed, one
observes other miscellaneous partitions mapping to odd numbers in these
intervals, and some of the partitions we counted actually not doing so. Also,
certain prime gaps are not possible, which constrains the problem
further.131313E.g., a predicted gap of 2 may occur after a prime where the
adjacent odd numbers are multiples of 3 and 5, and are thus unavailable for
membership in a twin prime pair; and the interval
$p_{n}\\#-p_{n+1}<x<p_{n}\\#+p_{n+1}$ contains no primes except possibly
$p_{n}\\#\pm 1$.

With those inaccuracies noted, we now go on to draw a few conclusions from
Conjecture 3 that show it to represent a surprisingly strong model.

######  Prediction 1.

Our conjectural model predicts $p_{2}-p_{1}=1$. For $n\geq 2$, our model
predicts $p_{n+1}-p_{n}$ is twice the number of partitions of norm $n$, with
no $1$’s, having one or two parts:

$p_{n+1}-p_{n}\ \ \ \text{is modeled by}\ \ \ \
2\left\lceil\frac{d(n)}{2}\right\rceil,$

where $d(n)$ is the number of divisors of $n$ and $\left\lceil x\right\rceil$
is the ceiling function.

###### Proof.

The partition $\lambda=(1)$ is mapped to $p_{1}=2$ by the supernorm, and
$\lambda=(2)$ is mapped to $p_{2}=3$, with no odd numbers in between. For
$n\geq 2$, the model postulates a bijection between odd integers in
$[p_{n},p_{n+1})$, and partitions with norm $n$, no $1$’s and having one or
two parts. There is an obvious bijection between these partitions and the
pairs $d,d^{\prime}\in\mathbb{N}$ of divisors of $n=dd^{\prime}$: the divisor
pair $d=n,d^{\prime}=1$ maps to the partition $\lambda=(n)$ with one part, and
taking $d\geq d^{\prime}$, the other divisor pairs map to partitions
$\lambda=(d,d^{\prime})$, noting their norms equal $n$. The number of pairs of
divisors of $n$ is $\left\lceil d(n)/2\right\rceil$, where the ceiling
function captures the extra divisor pair $d=d^{\prime}$ when $n$ is a square.
The prime gap length $p_{n+1}-p_{n}$ is twice the number of odd integers in
the gap, which introduces a multiplicative factor of $2$. ∎

Table 2 contrasts the predicted prime gaps from Prediction 1 with actual
values for the primes up to 100. We note that the value of $p_{n}$, actual or
predicted, is obtained by adding $2$ to the sum of rows $1$ through $n-1$ in
the respective column. The reader will see that 16 out of 25 predictions are
accurate – this is surprising, as the model does not claim equality – and only
three inaccurate rows give an error larger than a single over- or under-
counted partition (each partition is weighted by a factor of 2).

$n$ | $p_{n}$ |  $p_{n+1}-p_{n}$ (actual) |  $p_{n+1}-p_{n}$ (prediction)
---|---|---|---
1 | 2 | 1 | 1
2 | 3 | 2 | 2
3 | 5 | 2 | 2
4 | 7 | 4 | 4
5 | 11 | 2 | 2
6 | 13 | 4 | 4
7 | 17 | 2 | 2
8 | 19 | 4 | 4
9 | 23 | 6 | 4
10 | 29 | 2 | 4
11 | 31 | 6 | 2
12 | 37 | 4 | 6
13 | 41 | 2 | 2
14 | 43 | 4 | 4
15 | 47 | 6 | 4
16 | 53 | 6 | 6
17 | 59 | 2 | 2
18 | 61 | 6 | 6
19 | 67 | 4 | 2
20 | 71 | 2 | 6
21 | 73 | 6 | 4
22 | 79 | 4 | 4
23 | 83 | 6 | 2
24 | 89 | 8 | 8
25 | 97 | 4 | 4
Table 2. Comparing actual prime gaps to predictions from Prediction 1; we
highlight entries where the prediction is off.

Furthermore, the model makes an accurate prediction about the average value of
$p_{n+1}-p_{n}$.

######  Prediction 2.

Our conjectural model predicts the average value of $p_{n+1}-p_{n}$ is
asymptotically equal to $\log n$ as $n\to\infty$.

###### Proof.

The asymptotic comes from Dirichlet’s well-known asymptotic estimate that the
average value of $d(n)$ is asymptotically equal to $\log n$, which is proved
by elementary methods, applied to Prediction 1. ∎

We are encouraged here by noting the $\log n$ asymptotic agrees with the prime
number theorem estimate for $p_{n+1}-p_{n}$. Indeed, as noted above, the $\log
n$ arises in this context from Dirichlet’s well-known elementary proof of the
average value of $d(n)$, independently of the prime number theorem.

######  Prediction 3.

Our conjectural model predicts the prime number theorem.

###### Proof.

The above $\log n$ asymptotic for $p_{n+1}-p_{n}$ is well known to be
equivalent to the main asymptotic term $\pi(n)\sim n/\log n$ of the prime
number theorem. One can also see the model predicts $p_{n}=p_{1}+\sum_{1\leq
i\leq n-1}(p_{i+1}-p_{i})\sim\sum_{1\leq i\leq n-1}d(i)\ \sim\ n\log n$, by
well-known asymptotics; this estimate for $p_{n}$ is an equivalent statement
of the prime number theorem. ∎

In addition to reproducing known asymptotics about prime distribution, the
model of Conjecture 3 predicts a well-known empirical observation.

######  Prediction 4.

Our conjectural model predicts the twin prime conjecture.

###### Proof.

It is clear from the infinitude of prime numbers, that there are infinitely
many prime-indexed primes, i.e., primes of the form $p_{n},\ n\in\mathbb{P}$.
Since $d(n)=2$ when $n\in\mathbb{P}$, treating the model as a bijection,
Prediction 1 predicts infinitely many prime gaps of length $2$, after the
prime-indexed primes. ∎

It is a nice feature of our conjectural model, that the infinitude of twin
primes follows from the infinitude of $\mathbb{P}$ proved by Euclid. Actually,
though, the twin prime conjecture is only that Prediction 1 holds infinitely
often for $n\in\mathbb{P}$, not necessarily always. We note from Table 2 that
the model accurately predicts five out of eight twin prime pairs below 100,
and of the erroneous three, two twin prime predictions (rows 11 and 19) are
adjacent to actual twin prime pairs; only one predicted pair appears to be
entirely erroneous. In any event, our model is not explicitly correct, as we
noted above; perhaps a more detailed model along these lines could provide
insight into twin primes.

Even taken as an approximate or probabilistic statement, Conjecture 3 suggests
a more nuanced description of local density fluctuations of the primes than
appears to be conjectured in the literature.

######  Prediction 5.

Our conjectural model predicts the following qualitatively:

  1. (1)

larger gaps occur near primes $p_{n}$ whose indices $n$ have many divisors;

  2. (2)

smaller gaps occur near primes $p_{n}$ whose indices $n$ have few divisors;

  3. (3)

twin prime pairs occur near primes $p_{n}$ whose indices $n$ are prime.

This partition model of prime distribution is speculative and simplistic. Even
so, it reproduces known asymptotics and observations from elementary first
principles, and implies new conjectures about local fluctuations in prime gaps
that would be difficult to justify without ideas from partition theory.
Moreover, it preserves stochastic heuristics on prime distribution in that
partitions are also treated stochastically.141414For example, in statistical
physics; see e.g. Ch. 16 of [30], as well as [29]. From the standpoint of
scientific modeling, it is a pretty good “first-draft” model of the prime
numbers. The reasonable predictive success of the model provides support for
this appendix’s central heuristic: the problem of prime gaps can be viewed as
a problem about the enumeration of partitions.

######  Remark.

The model itself suggests a possible correction to the overall under-counting
of partitions due to Postulate 3, although not a very accurate or
computationally efficient one. Noting that $p_{n}\sim n\log n$ by Prediction 2
(and the prime number theorem), as an approximate use of (14) one might try
also enumerating partitions $\lambda$ with no $1^{\prime}s$ and lengths $>2$,
which satisfy

(16) $n\log n\ \ \leq\ \ {N}(\lambda)\prod_{i\geq 2}(\log i)^{m_{i}(\lambda)}\
\ <\ \ (n+1)\log(n+1).$

Note by (16), taking $N(\lambda)<n$ may allow the number of parts of $\lambda$
(logarithmic factors) to increase to maintain the inequalities.

## Acknowledgments

The authors are grateful to Krishnaswami Alladi, George Andrews, Maurice
Hendon, Abhimanyu Kumar, Jeffrey Lagarias, Paul Pollack, Ken Ono, Meenakshi
Rana, Maxwell Schneider, A. V. Sills, Ian Wagner, and the first author’s AMS
undergraduate and graduate research students, for conversations that informed
this study. We also thank A. V. Sills and Alexander Walker for computing
numerical examples that informed our intuition.

The first author was partially supported by an AMS-Simons Travel Grant and an
internal grant from the University of Texas at Tyler. The second author was
partially supported by the Research and Training Group grant DMS-1344994
funded by the National Science Foundation.

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