Although not a foundational investigation, I updated and posted this 1964 curiosity for those with an interest in the elementary theory of numbers. The Eta-Function $\eta(s)$ and the Homogeneous function appeared to be particularly fascinating at the time! The former since it seemed that development of the theory of the Riemann Zeta-Function $\zeta(s)$ should be more naturally through the series $\sum n^{-s}$, rather than through the Euler product because of the occurrence of $\pi(x)$ in $log\ \zeta(s)$; whereas $\eta(s)$ can only be developed through $log\ \eta(s)$ since the numerator in the Dirichlect series of $\eta(s)$ was seemingly an `unknowable’ function of $n$.

Unusual connections between additive and multiplicative number theory, with roots in a trivial multiplicative analogue for the Grecian perfect numbers.

A: Grecian perfect numbers

(i) Ancient Greek mathematicians considered a number $N$ (eg. $6, 28$), as perfect if it equalled the sum of all its proper divisors, so that:

$2N=\sigma(N)=\sum_{d|N}d$

(ii) Euclid, in his Elements, showed that every number of the form:

$2^{(k-1)}(2^{k}-1)$

is perfect if $(2^{k}-1)$ is prime [1], and Euler showed that every even perfect number is necessarily of this form [2].

(iii) Later mathematicians extended the above investigation to numbers whose $m^{th}$ multiple equals the sum of the $k^{th}$ power of all their proper divisors [3], so that:

$m.N+N^{k}=\sigma_{k}(N)=\sum_{d|N}d^{k}$

B: Multiplicative $m$-perfect numbers

(i) A trivial [4] analogous extension has been to consider numbers that are multiplicatively $m$-perfect, and equal the product of all their proper divisors, so that:

$N^{2}=\prod_{d|N}d$

(ii) The triviality lies in the unremarkable completeness with which such numbers can be distinguished for, if $d(N)$ is the number of divisors of such an $N$, then:

$N^{2}.N^{2}=\prod_{d|N}d.\prod_{d|N}(N/d)=N^{d(N)}$

and so $d(N)=4$.

(iii) Now if $N=p_{1}^{a_{1}}.p_{2}^{a_{2}} \ldots p_{r}^{a_{r}}$, then $d(N)=(a_{1}+1).(a_{2}+1) \ldots (a_{r}+1)$, and so:

(a) either $a_{1}=3$, and $a_{2}=a_{3}=\ldots=a_{r}=0$,

(b) or $a_{1}=a_{2}=1$, and $a_{3}=a_{4}=\ldots=a_{r}=0$;

so the number is trivially of the form $p^{3}$ or $p.q$, where $p$ and $q$ are distinct primes.

Conversely, every number of such form is trivially $m$-perfect.

C: Extended multiplicative $m_{_{k}}$-perfect numbers

(i) An intriguing question, however arises:

Why should $d(N) = 4$?

The answer lies in extending the concept, as has been done in the additive case, to $m_{_{k}}$-perfect numbers that equal the product of the $k^{th}$ power of all their proper divisors, so that:

$N.N^{k}=\prod_{d|N}d^{k}$

(ii) It follows that:

$N^{(k+1)}.N^{(k+1)}=\prod_{d|N}d^{k}.\prod_{d|N}(N/d)^{k}=\prod_{d|N}N^{k}=N^{k.d(N)}$

and so $d(N) = 2(k+1)/k$, which can now take any integral value provided $k$ admits of rational values.

(iii) Thus every composite number $N$ is $m_{_{k}}$-perfect in the extended sense, of fractional degree $k = 2/(d(N)-2)$.

(iv) The converse question now becomes interesting:

Which are the $m_{_{k}}$-perfect numbers of a given fractional degree $k$, when $2.(k+1)/k$ is an integer?

(v) This too can be answered completely:

For every factorisation:

$\prod^{r}_{i=1}(l_{i}+1)$

of the given integer $2.(k+1)/k$ into $r$ factors, numbers of the form:

$\prod^{r}_{i=1}(p_{i}^{l_{i}})$

are $m_{_{k}}$-perfect of degree $k$, the $p_{i}$‘s being distinct primes.

(vi) The above follows since, in this case:

$(\prod_{d|N}d^{k})^{2}=\prod_{d|N}d^{k}.\prod_{d|N}N(N/d)^{k}=\prod_{d|N}N^{k}=N^{k.d(N)}$

and, since:

$d(N)=(l_{1}+1).(l_{2}+1) \ldots (l_{r}+1)=2.(k+1)/k$

we have:

$N^{(k+1)}=\prod_{d|N}d^{k}$

D: Unrestricted factorisations of $N$

(i) However, this last result raises an unusually interesting, non-trivial question of the number of forms of $m_{_{k}}$-perfect numbers corresponding to a suitably given degree $k$, since each form corresponds uniquely to a particular factorisation of the given integer of the form $2.(k+1)/k$.

(ii) Stated simply:

In how many distinct ways can any given integer $N$ be unrestrictedly factorised, where $1$ is not a factor, and the order of factors is not important?

(iii) In §E.2 below we show that this too can be answered completely.

E: Factorisations of $N$

We consider:

In how many distinct ways can any given integer $n$ (or $N$) be factorised as a product of its factors?

Case 1: The Riemann Zeta-Function

If we restrict the factors to being only distinct primes or powers of primes, then we have only one, unique prime decomposition of $n$, and the generating function for such representation is the well-known Riemann Zeta Function [5]:

$\zeta(s) = \prod_{p}(1-\frac{1}{p^{s}})^{-1} = \sum_{n=1}^{\infty}\frac{1}{n^{s}}$

where we note that, for $Re(s) > 1$:

$log_{_{e}} \zeta(s) = s \int_{_{2}}^{^{\infty}}\frac{\pi(x)}{x(x^{s}-1)}dx$

Case 2: The Eta Function

In the general case (which answers the query in §D(ii) above):

$\bullet$ If $F(n)$ represents the number of distinct ways in which $n$ can be represented as a product of its factors, where $1$ is not a factor and the order of factors is not important, we have the generating function for $Re(s) > 1$:

$\eta(s) = \prod_{n=2}^{\infty}(1-\frac{1}{n^{s}})^{-1} = \sum_{n=1}^{\infty}\frac{F(n)}{n^{s}}$

where we note that:

$\eta(1) = \prod_{n=2}^{\infty}(1-\frac{1}{n})^{-1} = \lim_{_{n \rightarrow \infty}} n$

and that, for $Re(s) > 1$:

$log_{_{e}} \eta(s) = s \int_{_{2}}^{^{\infty}}\frac{[x] - 1}{x(x^{s}-1)}dx$

$\bullet$ Also, if $F(n, k)$ denotes the number of distinct ways in which $n$ can be represented as a product of $k$ factors, where $1$ is not a factor and the order of factors is not important, we have:

$\eta_{_{k}}(s) = \prod _{n=2}^{\infty}(1-\frac{x}{n^{s}})^{-1} = \sum_{n=1}^{\infty} \sum_{k=1}^{\infty}\frac{F(n, k)}{n^{s}}.x^{k}$

$\bullet$ However, if $F'(n, k)$ denotes the number of ways in which $n$ can be represented as a product of $k$ factors, where $1$ may be a repeated factor and the order of factors is important, the generating function is the $k^{th}$ power of the Zeta function [6]:

$\zeta^{k}(s) = \prod_{p}(1-\frac{1}{p^{s}})^{-k} = \sum_{n=1}^{\infty}\frac{F'(n, k)}{n^{s}}$

Case 3: The Partition Function

If we restrict the form of $N$ to a prime power $p^{n}$, there are as many distinct factorisations of $N=p^{n}$ as there are unrestricted partitions of $n$.

If $P(n, k)$ represents the number of factorisations of $N=p^{n}$ into $k$ distinct factors (equivalent to the number of partitions of $n$ into $k$ parts), $P(n, k)$ is generated by [7]:

$\prod_{i=1}^{\infty}\frac{1}{(1-ax^{i})} = \sum_{n=1}^{\infty}\sum_{k=1}^{\infty}P(n, k)a^{k}x^{n}$

while the generating function for $P(n)$, the total number of distinct factorisations of $N=p^{n}$ ($\sum_{k=1}^{\infty}P(n, k)$ being the total number of unrestricted partitions of $n$) is [8]:

$\prod_{i=1}^{\infty}\frac{1}{(1-x^{i})} = 1+\sum_{n=1}^{\infty}P(n)x^{n}$

Case 4: The Homogeneous Function

If we restrict the form of $N$ to a product of distinct primes $p_{1}, p_{2}, \ldots, p_{k}$, then the number of distinct factorisations $F''(k, r)$ of $N = p_{1}.p_{2}. \ldots p_{k}$ into $r$ factors obeys the functional equation:

$F''(k, r) = F''(k-1, r-1) + r.F''(k-1, r)$

for $k\geq r\geq 2$, and $F''(k, r)$ is seen to be the Homogeneous Function (product) over the set $R={1, 2, \ldots, r}$ of degree $(k-r)$:

$H(k-r, r) = \sum_{j_{i} \in R, j_{i} \geq j_{i+1} \geq1} j_{1}.j_{2}. \ldots j_{k-r}$

where each term in the summation is distictly different.

The generating function for $H(i, r)$ is then:

$\prod_{i=1}^{r}\frac{1}{(1-ix)} = \sum_{i=0}^{\infty}H(i, r).x^{r}$

while the generating function for the total number of factorisations:

$F''(k) = \sum_{r=1}^{\infty}F''(k, r)$

is the formal series:

$\sum_{r=1}^{\infty}\frac{X^{r}}{(1-X)(1-2X)\ldots(1-rX)} = \sum_{k=0}^{\infty}F(k).X^{k}$

References

Di18 Leonard Eugene Dickson. 1918. History of the Theory of Numbers Volume I. 1952. Chelsea Publishing Company, New York.

Di20 Leonard Eugene Dickson. 1920. History of the Theory of Numbers Volume II. 1952. Chelsea Publishing Company, New York.

HW59 G. H. Hardy and E. M. Wright. 1959.  An Introduction to the Theory of Numbers. Fourth Edition. 1960. Oxford University Press, London.

La27 Edmund Landau. 1927. Elementary Number Theory. Translation into English, by Jacob E. Goodman, of the German language work Elementare Zahlentheorie (Volume $I_{1}$ of Vorlesungen Über Zahlentheorie), by Edmund Landau. 1958. Chelsea Publishing Company, New York.

Ti51 E. C. Titchmarsh. 1951. The Theory of the Riemann Zeta-Function. Oxford University Press, London.

Notes

Return to 1: Di18, p.3; HW59, p.239.

Return to 2: Di18, p.19; HW59, p.240.

Return to 3: Di18, p.38; cf. HW59, p.239, Theorem 274.

Return to 4: Di18, p.58; La27, p.31.

Return to 5: Ti51, pp.1-2; HW59, p.245.