Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta function; and a special Reulle (aka dynamical systems or Smale) zeta function, the Ihara zeta function for a graph $G$--all can be expressed in the same basic form:

$$\zeta(u)=\exp\left ( \sum_{m\geq 1} \frac{N_mu^m}{m} \right ).$$

For graph zeta functions $\zeta(u,G_n)$ typically $N_m$ is the number of closed walks of $m$ steps (with some qualifications) on the graph $G$ with $n$ vertices and is related to the trace of the power of an edge adjacency matrix. For a vertex adjacency matrix $A_n$, also $N_m = \operatorname{tr}[A_n^m]$ (e.g., A054878 and A092297). (Edited per draks' comment.)

You can use the general heuristic $O=KPK^{-1}\Leftrightarrow P=K^{-1}OK$ to obtain

$$\operatorname{tr}(A)=\ln[\operatorname{det}[\exp(A)]] \Leftrightarrow \operatorname{det}(A)=\exp[\operatorname{tr}[\ln(A)]]$$

and then

$$\operatorname{det}(I-uA_n)=\exp[\operatorname{tr}[\ln(I-uA_n)]]=\exp\left( -\sum_{m\geq 1} \frac{\operatorname{tr}(A_n^m)u^m}{m} \right)$$ $$=\exp\left (-\sum_{m\geq 1} \frac{N_mu^m}{m} \right ),$$

so

$$\zeta(u;G_n)=\frac{1}{\operatorname{det}(I-uA_n)}=\exp\left(\sum_{m\geq 1} \frac{\operatorname{tr}(A_n^m)u^m}{m} \right)=\exp\left(-:\ln(1-ua): \right).$$ where $a^k=a_k=\operatorname{tr}(A_n^k)$ for $k>0$.

This last expression is the umbral form for the exponential generating function for the cycle index polynomials (OEIS-A036039) for the symmetric group (mod signs).

The Appell sequence in MO-Q111165 incorporating the Riemann zeta function reverses the last relation in some sense:

$$\exp\left (-\beta p_{.}(z)\right )=\exp\left [-(z+\gamma)\beta -\sum_{k=2}^{\infty } \frac{\zeta (k)\beta ^k}{k} \right ]=\exp\left [ :\ln(1-b\beta ) :\right ]$$ where $b^1=b_{1}=(z+\gamma)$ and $b^k=b_k=\zeta(k)$ for $k>1$.

For easy reference: $$p_{0}(x)=1$$ $$p_{1}(x)=x+\gamma$$ $$p_2(x)=(x+\gamma)^2-\zeta(2)$$ $$p_3(x)=(x+\gamma)^3-3\zeta(2)(x+\gamma)+2\zeta(3)$$ $$p_4(x)=(x+\gamma)^4-6\zeta(2)(x+\gamma)^2+8\zeta(3)(x+\gamma)+3[\zeta^2(2)-2\zeta(4)]$$

**These polynomials are the first few cycle index polynomials for the symmetric group. I'd like to relate each $p_n(x)$ to the characteristic polynomial of a matrix with a null main diagonal.**

For example, for such a 3x3 matrix the char polynomial is

$$ \sigma^3-(a_{12}a_{21}+a_{13}a_{31}+a_{23}a_{32})\sigma+(a_{12}a_{23}a_{31}+a_{13}a_{32}a_{21}).$$

Picture a triangle with the vertices ($v$) labelled 1 to 3. Make an orbit/cycle/closed loop, or path, traversing the triangle from $v_1$ through $v_2$ and $v_3$ and then to $v_1$. Denote this path of three steps and length three by $a_{12}a_{23}a_{31}$ and assign it the "moment/transition amplitude" of $\zeta(3)$. Likewise, assign the amplitude $\zeta(2)$ to paths of two steps and length one $a_{12}a_{21}$, an amplitude of $\sigma=x+\gamma$ to a self- or null-loop, and so on. This generates $p_3(x)$.

Similarly, consider a square with labeled vertices and edges between all pairs of vertices. With cycles/orbits/closed paths of opposing circulation considered distinct cycles, the associated 4x4 determinant generates six paths each with four steps and length four, e.g., $a_{12}a_{24}a_{43}a_{31}$, that can be assigned an amplitude of $\zeta(4)$ each and three sets of two paths of two steps and length one, e.g., $a_{13}a_{31}a_{24}a_{42}$, that can be assigned an amplitude of $\zeta^{2}(2)$. The algorithm can be continued to the other terms to generate $p_4(x)$.

**How to prove that the algorithm will work for all $p_n(x)$, i.e., that each $p_n(x)$ can be generated in the above manner from an $n$ by $n$ "adjacency" matrix?**

[Nov. 15, 2013 update: Replacing $p_1(x)=x+\gamma$ by $x$ and the $\zeta(n)$ by $1$ gives the characteristic polynomials (mod signs) of the adjacency matrix of the complete n-graph (see A055137).]

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