Simplex spectral zeta

Specify the positive spectrum

For integer $d\geq2$, consider one eigenvalue per positive integer index, with no additional multiplicity:

$$P_d(n)=S(n,d)=\binom{n+d-1}{d} =\frac{n(n+1)\cdots(n+d-1)}{d!}, \qquad Z_d(s)=\sum_{n=1}^{\infty}P_d(n)^{-s}.$$

Comparison with $n^{-ds}$ proves absolute convergence for $\operatorname{Re}s>1/d$. All eigenvalues in this definition are dimensionless. The integer $d$ labels the polynomial sequence; interpreting an actual spatial operator requires its eigenvalues and multiplicities. The associated positive heat sum is developed in Simplex heat trace.

Continuation by subtracting the large-index terms

Set

$$A_d=\frac{d(d-1)}2, \qquad b_n=\prod_{j=0}^{d-1}\left(1+\frac jn\right).$$

Uniformly for $s$ in a compact set, $b_n^{-s}=1-sA_d/n+O(n^{-2})$. Therefore

$$R_d(s)=\sum_{n=1}^{\infty} n^{-ds} \left[b_n^{-s}-1+\frac{sA_d}{n}\right]$$

converges locally uniformly and defines a holomorphic function for $\operatorname{Re}s>-1/d$. Initially in the convergent half-plane, then by continuation,

$$\boxed{Z_d(s)=(d!)^s \left[\zeta(ds)-sA_d\zeta(ds+1)+R_d(s)\right].}$$

Here $\zeta$ is the Riemann zeta function. The remainder bound follows by expanding $\log b_n=A_d/n+O(n^{-2})$ and its exponential; the tail is $O(n^{-d\operatorname{Re}s-2})$. This also justifies local termwise differentiation.

In this half-plane, the pole at $s=1/d$ has residue $(d!)^{1/d}/d$. At $s=0$, the apparent pole in the second term cancels against its factor $s$.

Value and derivative at zero

Use $\zeta(0)=-1/2$ and $\zeta(1+ds)=1/(ds)+\gamma+O(s)$, where $\gamma$ is Euler's constant. Since $R_d(0)=0$,

$$\boxed{Z_d(0)=-\frac12-\frac{A_d}{d}=-\frac d2.}$$

The derivative of the remainder is an ordinarily convergent series:

$$R_d'(0)=\sum_{n\geq1} \left[-\log b_n+\frac{A_d}{n}\right] =\sum_{j=0}^{d-1}\log\Gamma(1+j)+\gamma A_d.$$

To obtain the last equality, take the logarithm of the gamma-function product:

$$\log\Gamma(1+j)=-\gamma j+ \sum_{n\geq1}\left[\frac jn-\log\left(1+\frac jn\right)\right].$$

Only finitely many $j$ are summed. Differentiating the continuation formula then cancels the $\gamma A_d$ terms and gives

$$\begin{aligned} Z_d'(0) &=-\frac d2\log(d!)+d\zeta'(0) +\sum_{j=0}^{d-1}\log(j!)\\ &=\boxed{-\frac d2\log(2\pi d!) +\sum_{j=0}^{d-1}\log(j!)}. \end{aligned}$$

The Riemann values used here, including $\zeta'(0)=-\tfrac12\log(2\pi)$, are recorded in DLMF §25.6.

Regularized product and scale

Define the zeta determinant by $\det_\zeta P_d=\exp[-Z_d'(0)]$. The result is

$$\boxed{\det\nolimits_\zeta P_d =\frac{(2\pi d!)^{d/2}}{\prod_{j=0}^{d-1}j!}.}$$

This regularized product has the following simple benchmarks:

The first row follows directly from $Z_1=\zeta$. If the physical eigenvalue is $\mu P_d(n)$, choose a fixed comparison unit $E_*$ and set $r=\mu/E_*>0$. Then $Z_{rP_d}(s)=r^{-s}Z_d(s)$ and

$$\log\det\nolimits_\zeta(rP_d) =\log\det\nolimits_\zeta P_d-\frac d2\log r.$$

Thus the normalization convention remains explicit. $Z_d(0)$ is an analytically continued spectral quantity, not a count of spatial directions. A gravitational or vacuum-energy use additionally needs a specified physical operator, action, regularization prescription, and scale matching.

Verification and related

spectral_analysis_checks.py checks convergent sums, remainder bounds, and independent centered-polynomial continuations for $d=2,3$, including numerical derivatives at zero and scale changes. The general formulas follow from the analytic derivation above.

• Simplex reciprocal sum — establishes the ordinary convergent value $Z_d(1)=d/(d-1)$.
• Spectral analysis tools — navigation and scope.