Problems and Theorems in Analysis · 6

Consider all possible factorizations of an even integer of the form $n=\alpha \beta$, such that $\alpha$ is odd (possibly 1) and $\beta$ is even. Then

is equal to the sum of the divisors of $n/2$.

Let $P(n,k)$ denote the number of decompositions of $n$ into a product of $k$ factors; only positive integers greater than 1 are admissible as factors, and two decompositions are regarded as equal if, and only if, they involve the same factors in the same order. For instance, $P(12,2)=4$ and $P(12,3)=3$ since

Show that, for $n\geq 2$,

the difference between the number of even'' and odd'' decompositions, that is, decompositions into an even and an odd number of factors, respectively. For instance

(continued). Show that

equals $1/m$ if $n=p^m$, the $m$th power of a prime number $p$, but equals 0 if $n$ is divisible by more than one prime. In the case $n=12$ we have

(continued). If $n$ is the product of $m$ different prime factors, then

(cf. the introduction to Problem 186 of Part I).

If $p$ is a prime and $m$ a positive integer, then

(For this particular case, verify 58.1 by using the binomial theorem; also cf. 58.2 with Problem 38 of Part I.)

For $n>1$ let $Q(n)$ denote the number of different decompositions of the integer $n$ into a product of integers greater than 1; two decompositions are not regarded as different if they involve the same factors, and so the order of factors does not matter (in contrast to 58.1); define $Q(1)=1$. For instance $Q(12)=4$. If $n$ is the product of $m$ different prime factors, then

(cf. the introduction to Problem 186 of Part I).

(continued). Show that

Let $n$ be a positive integer,

where $2, p_1,\ldots ,p_\mu ,q_1,\ldots ,q_\nu$ are distinct primes, $p_1\equiv \cdots \equiv p_\mu \equiv 1$, $q_1\equiv \cdots \equiv q_\nu \equiv 3 \pmod4$. Set

if $l_1,l_2,\ldots ,l_\nu$ are all even, and $\delta (n)=0$ otherwise. Show that $\delta (n)$ is, in fact, the difference between the numbers of two different kinds of divisors of $n$:

where the summation is extended over the odd divisors $t'$ of $n$.

($4\delta (n)$ is the number of lattice points on the circle $x^2+y^2=n$. This fact was discovered by Gauss.)

Which of the functions $P(n,2), P(n,3), \ldots , Q(n)$ and $\delta (n)$ are multiplicative?