Use \(N=4\) throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order.
Reading the drawing
Read the rows from denominator through tangent. Every row uses the same order-four normalization. The identities at the bottom recover the reciprocal coefficient sixty-nine and the tangent coefficient thirty-four from the first two zero moments. The prose below gives both routes to the reciprocal coefficient and states the analytic input behind the zero-sum interpretation.
Drawing labels and data 11 labels
Read the original labels as selectable text. Tables preserve grid relationships; other short labels are grouped compactly.
| Description | Series or values |
|---|---|
| denominator | \(1-w/24+w^2/40320+\cdots\) |
| negative logarithm | \(w/24+(17/20160)w^2+\cdots\) |
| zero moments | \(\sigma_1=1/24,\quad\sigma_2=17/10080\) |
| reciprocal | \(1+w/24+69w^2/40320+\cdots\) |
| tangent | \(z^3/3!+34z^7/7!+\cdots\) |
Route one: invert the denominator
The \(w^2\) coefficient of \(\Delta_4^{-1}\) is
Thus \(P_{4,2}(1)=69\). Keeping \(r\) variable gives \(P_{4,2}(r)=35r^2+34r\).
Route two: use the zero moments
The exact reduced zeros are \(4\pi^4(m+1/2)^4\). Their first two power sums are \(\sigma_1=1/24\) and \(\sigma_2=17/10080\). The Bell/Newton identity gives the same reciprocal coefficient \((\sigma_1^2+\sigma_2)/2\). The tangent independently reads off the second moment as \(4\cdot7!\sigma_2=34\). This links coefficients, geometry, and derivatives without a fitted parameter or an inferred zero list.
Read this as
The content is the agreement of precisely normalized descriptions. A matching small example illustrates the general bridges. It is not a proof of all their analytic hypotheses.
Evidence note from the source edition
Every rational value and polynomial identity on this page is checked exactly. The infinite zero-sum interpretation uses the inherited canonical-product theorem and the exact order-four zero formula.