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 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 should be more naturally through the series , rather than through the Euler product because of the occurrence of in ; whereas can only be developed through since the numerator in the Dirichlect series of was seemingly an `unknowable’ function of .

**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 (eg. ), as perfect if it equalled the sum of all its proper divisors, so that:

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

is perfect if 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 multiple equals the sum of the power of all their proper divisors ^{[3]}, so that:

**B: Multiplicative -perfect numbers**

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

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

and so .

(iii) Now if , then , and so:

(a) either , and ,

(b) or , and ;

so the number is trivially of the form or , where and are distinct primes.

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

**C: Extended multiplicative -perfect numbers**

(i) An intriguing question, however arises:

Why should ?

The answer lies in extending the concept, as has been done in the additive case, to *-perfect* numbers that equal the product of the power of all their proper divisors, so that:

(ii) It follows that:

and so , which can now take any integral value provided admits of rational values.

(iii) Thus every composite number is *-perfect* in the extended sense, of fractional degree .

(iv) The converse question now becomes interesting:

Which are the *-perfect* numbers of a given fractional degree , when is an integer?

(v) This too can be answered completely:

For every factorisation:

of the given integer into factors, numbers of the form:

are *-perfect* of degree , the ‘s being distinct primes.

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

and, since:

we have:

**D: Unrestricted factorisations of **

(i) However, this last result raises an unusually interesting, non-trivial question of the number of forms of *-perfect* numbers corresponding to a suitably given degree , since each form corresponds uniquely to a particular factorisation of the given integer of the form .

(ii) Stated simply:

In how many distinct ways can any given integer be unrestrictedly factorised, where 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 **

We consider:

In how many distinct ways can any given integer (or ) 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 , and the generating function for such representation is the well-known Riemann Zeta Function ^{[5]}:

where we note that, for :

**Case 2: The Eta Function**

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

If represents the number of distinct ways in which can be represented as a product of its factors, where is not a factor and the order of factors is not important, we have the generating function for :

where we note that:

and that, for :

Also, if denotes the number of distinct ways in which can be represented as a product of factors, where is not a factor and the order of factors is not important, we have:

However, if denotes the number of ways in which can be represented as a product of factors, where may be a repeated factor and the order of factors is important, the generating function is the power of the Zeta function ^{[6]}:

**Case 3: The Partition Function**

If we restrict the form of to a prime power , there are as many distinct factorisations of as there are unrestricted partitions of .

If represents the number of factorisations of into distinct factors (equivalent to the number of partitions of into parts), is generated by ^{[7]}:

while the generating function for , the total number of distinct factorisations of ( being the total number of unrestricted partitions of ) is ^{[8]}:

**Case 4: The Homogeneous Function**

If we restrict the form of to a product of distinct primes , then the number of distinct factorisations of into factors obeys the functional equation:

for , and is seen to be the Homogeneous Function (product) over the set of degree :

where each term in the summation is distictly different.

The generating function for is then:

while the generating function for the total number of factorisations:

is the formal series:

**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 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.

Return to 6: Ti51, p.4.

Return to 7: Di20, p.104.

Return to 8: Di20, p.104; HW59, p.274.

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