# Ghost-Witt sequences vs. ghost-Polya-Burnside sequences?

If you're in a hurry scroll down until the questions:

First the known part:

A sequence $\left(b_1,b_2,b_3,...\right)$ of integers will be called a ghost-Witt sequence if there exists a sequence $\left(x_1,x_2,x_3,...\right)$ of integers such that $\left(b_n=\sum\limits_{d\mid n}dx_d^{n/d}\text{ for every positive integer }n\right)$. (The reason for this naming lies in the big/universal Witt polynomials, of course.)

There exist several equivalent criteria for when a sequence $\left(b_1,b_2,b_3,...\right)$ of integers is a ghost-Witt sequence. One of them is that $b_{n/p}\equiv b_n\mod p^{v_p\left(n\right)}$ for any positive integer $n$ and any prime divisor $p$ of $n$, where $v_p\left(n\right)$ denotes the $p$-adic valuation of $n$. Another is that $n\mid \sum\limits_{d\mid n}\phi\left(d\right)b_{n/d}$ for every positive integer $n$, where $\phi$ is the Euler totient function. There are more (I listed some of them in Witt#5 Theorem 15), but there are yet more - I plan to write them up yet.

This can be generalized several times; besides, the above criteria (well, some of them) easily show that the sum and the product of two ghost-Witt sequences are ghost-Witt sequences again (which leads to the integrality of big/universal Witt polynomials, but only if one generalizes this properly to polynomials instead of integers; otherwise you just get that the big Witt polynomials are integral-valued, but this doesn't mean that their coefficients are integers).

Enough ramblings - here is what I am wondering about:

Questions:

A sequence $\left(b_1,b_2,b_3,...\right)$ of integers will be called a ghost-Burnside sequence if $\left(n\mid \sum\limits_{d\mid n}\phi\left(d\right)b_d^{n/d}\text{ for every positive integer }n\right)$.

• Is the sum / the product of two ghost-Burnside sequences another ghost-Burnside sequence?
• Are there any nice equivalent criteria? In particular, is there a criterion similar to the definition of a ghost-Witt sequence?

Why do I care? Because the polynomial $\frac{1}{n}\sum\limits_{d\mid n}\phi\left(d\right)X_d^{n/d}$ is the cycle index of the cyclic group $\mathbb Z / n\mathbb Z$. It is not an integer polynomial, and not even an integer-valued polynomial, but one of the results of Polya enumeration theory states that $\frac{1}{n}\sum\limits_{d\mid n}\phi\left(d\right)\left(Y_1^d+Y_2^d+...+Y_k^d\right)^{n/d}\in\mathbb Z\left[Y_1,Y_2,...,Y_k\right]$ for any $k\geq 0$ (actually, this can be shown using Theorem 13 in Witt#5 as well). This makes every sequence of the form

$\left(u_1^d+u_2^d+...+u_k^d\right)_{d\text{ positive integer}}$

for integers $u_1$, $u_2$, ..., $u_k$ a ghost-Burnside sequence. I hope for a kind of converse; of course, it will require considering finite sequences only and possibly only up to congruence modulo $n$. EDIT: Sorry, this approach is doomed to fail. All the sequences constructed this way are not only ghost-Burnside, but also ghost-Witt, and we can't get the remaining ghost-Burnside sequences. (Yes, being ghost-Witt is stronger than being ghost-Burnside; but I don't have a nice proof.)

Results so far: If we let $m$ be a positive integer and require $n\mid \sum\limits_{d\mid n}\phi\left(d\right)b_d^{n/d}$ to hold for $n\mid m$ only (and not necessarily for all positive integers $n$), then I can prove that the sum and the product of two ghost-Burnside sequences are ghost-Burnside sequences again, if $m$ is a prime power or a squarefree integer. I think I also did it for the case $m=p^2q$, which already required some casebash. The other cases are scaring me off...

I am recording the results for myself:

(1) If $m=p^k$ is a power of a prime $k$, then a sequence $\left(b_1,b_2,b_3,...\right)$ of integers (of course, only the $b_{p^i}$ members matter) is ghost-Burnside if and only if $b_{p^u}\equiv b_{p^{u+1}}\mod p$ for every $0\leq u\leq k-1$. No typo here: it's $\mod p$, not $\mod p^u$ as in the ghost-Witt case!

(2) If $m=pq$ is the product of two distinct primes $p$ and $q$, then a sequence $\left(b_1,b_2,b_3,...\right)$ of integers (of course, it's all in the terms $b_1$, $b_p$, $b_q$, $b_{pq}$) is ghost-Burnside if and only if

$b_1\equiv b_p\mod p$; $b_1\equiv b_q\mod q$; ($b_q\equiv b_{pq}\mod p$ OR $p\mid q-1$); ($b_p\equiv b_{pq}\mod q$ OR $q\mid p-1$).

(3) If $m=p^2q$ for two distinct primes $p$ and $q$, then a sequence $\left(b_1,b_2,b_3,...\right)$ of integers (this times six terms matter) is ghost-Burnside if and only if

$b_1\equiv b_p\equiv b_{p^2}\mod p$; $b_1\equiv b_q\mod q$; ($b_q\equiv b_{pq}\mod p$ OR $p\mid q-1$); ($b_p\equiv b_{pq}\mod q$ OR $q\mid p-1$); ($b_{p^2}\equiv b_{p^2q}\mod q$ OR $q\mid p-1$); [($b_{pq}\equiv b_{p^2q}\mod p$ AND $p\not\mid q-1$) OR ($b_q\equiv b_{pq}\mod p$ AND $p\mid q-1$) OR ($p^2\mid q-1$)].

(4) Generalizing (2): If $m$ is the product of finitely many distinct primes (i. e. if $m$ is squarefree), then a sequence $\left(b_1,b_2,b_3,...\right)$ of integers is ghost-Burnside if and only if

($\phi\left(d\right)b_d\equiv \phi\left(d\right)b_{pd}\mod p$ for every $d\mid m$ and every prime $p$ such that $pd\mid m$).

Dreams:

If this works, we could move on to groups more interesting than cyclic groups.

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Have you tried writing down the Dirichlet generating function of the cycle index polynomials of Z/nZ? – Qiaochu Yuan Mar 3 '10 at 18:00
It's the logarithm of $\prod_{d=1}^{\infty} \left(1-a_dT^d\right)^{-\phi\left(d\right)/d}$. Nice idea, but I don't have a clue how to proceed from here. Reminds me of the Artin-Hasse exponential, though. – darij grinberg Mar 3 '10 at 18:29
Read $a_d$ as $X_d$ in my comment, sorry. – darij grinberg Mar 3 '10 at 18:30
Concerning your dream: there already is a Witt vector construction beyond the case of cyclic groups. In 1988, Dress and Siebeneicher introduced a functor W_G for any profinite group G. For any ring A, the new ring W_G(A) has operations defined in terms of certain universal "G-Witt polynomials", and the key point is that the sum and product of two such polynomials (over independent sets of variables) is again such a polynomial. See A. Dress, C. Siebeneicher, "The Burnside ring of profinite groups and the Witt vector construction", Adv. Math 70 (1988), 87--132. – KConrad Mar 4 '10 at 3:18
Thanks for the reference, but I know the construction (Witt-Burnside functors). I should have mentioned it. However, ghost-Burnside is not the same as ghost-Witt, and their respective generalizations should be even less equivalent. – darij grinberg Mar 4 '10 at 13:15