REVIEW
Even and Odd Integer Values of the Secondary Zeta Function
General Mathematics · Submitted 13 Oct 2026 · 9 references
Pith's verdict: Nice integral-operator idea, but the product representation (2.1) is mathematically wrong, and the derivation of even values depends on it.
T0 · AI review
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Referee report
The paper
The paper claims
Convergent integral formulas for all integer values of the secondary zeta function, derived from the Riemann ξ-function via a differential operator that also yields the first zero ordinate.
The reading
What carries the argument
The centered Euler operator D_k = x d/dx ∏_{j=1}^k (x d/dx − 2j) annihilates all even monomials $x^{{2m}}$ for 1 ≤ m ≤ k, thereby removing the low-order Taylor terms from L(x) that would otherwise cause singularities in the Mellin integrands. Applied to each logarithmic zero factor log(1 + $x^{2}$/$γ^{2}$), D_k produces the rational function (−1)^k $2^{{k+1}}$ k! $x^{{2k+2}}$/($γ^{2}$ + $x^{2}$)^{k+1}, whose integral from 0 to ∞ yields exactly $γ^{{-2k}}$ or $γ^{{-(2k+1)}}$ after summation over the ordinates γ.
Extended reading notes
Core claim
For every integer k ≥ 1, the secondary zeta function at even and odd arguments can be written as absolutely convergent integrals: Z(2k) = ((-1)^k)/(2^k (k-1)!) ∫_0^∞ (D_k L(x))/$x^{{2k+1}}$ dx and Z(2k+1) = ((-1)^k)/(π (2k-1)!!) ∫_0^∞ (D_k L(x))/$x^{{2k+2}}$ dx, where L(x) = log(ξ(x+1/2)/ξ(1/2)) and D_k = x d/dx ∏_{j=1}^k (x d/dx − 2j). These representations replace earlier series expansions with integrals that involve only the ξ-function and its derivatives, and they unify the treatment of even and odd arguments.
Load-bearing premise
The Riemann hypothesis is assumed throughout, because the central product representation of ξ(x+1/2)/ξ(1/2) over zeros requires all non-trivial zeros to lie on the critical line with ordinates ±γ.
Editorial extensions
If this is right
- The integral formulas provide a direct, ξ-function-based method for computing Z(k) at any integer, avoiding the need for summation over zeros or analytic continuation from the definition.
- The expressions in terms of derivatives of ξ may facilitate the study of asymptotics of Z(k) as k → ∞, with potential connections to moments of the ξ-function.
- The limit γ₁ = lim_{k→∞} Z(2k)/Z(2k+1) gives a new, convergent representation of the first nontrivial zero ordinate that can be approximated from high-order Z values.
- The same differential operator technique could be extended to other Dirichlet series whose terms arise as sums over spectral parameters with known product representations.
Reading between the lines
- Because the final integrals involve only the ξ-function and its derivatives, they might be evaluated numerically for fixed k without relying on the Riemann hypothesis, provided the centered logarithm L(x) is defined via the functional equation rather than the product over zeros; this could offer a consistency check of the RH assumption.
- The operator D_k effectively acts as a Mellin-transform regularizer, suggesting a general principle: for sums over parameters γ that appear quadratically in a logarithm, an Euler operator of matching order can convert the Taylor remainder into an integrable kernel.
- The high-precision convergence of Z(2k)/Z(2k+1) to γ₁ shown in the tables offers a practical route to computing γ₁ to many digits using only the ξ-function, independent of root-finding algorithms.
- These integral representations may inspire new sum rules for the moments of the zero ordinates, since they express Z(k) as an integral of a positive kernel (after sign normalization) involving the ξ-function.
Original abstract
The secondary zeta function is defined by $$ Z(s)=\sum_{\gamma>0}\frac{1}{\gamma^{s}}, $$ where $\tfrac12+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. In this article, we derive convergent integral representations of $Z(s)$ at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann $\xi$-function. Define $$ L(x)=\log\left(\frac{\xi(x+\tfrac12)}{\xi(\tfrac12)}\right), \qquad \mathcal{E}=x\frac{\mathrm{d}}{\mathrm{d}x}, \qquad \mathcal{D}_k=\mathcal{E}\prod_{j=1}^{k}(\mathcal{E}-2j). $$ Then, for $k\geq1$, $$ Z(2k)=\frac{(-1)^k}{2^k(k-1)!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+1}}\,\mathrm{d}x $$ and $$ Z(2k+1)=\frac{(-1)^k}{\pi\,(2k-1)!!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+2}}\,\mathrm{d}x. $$ In the last part, we write these formulas out explicitly for $Z(k)$, $2\leq k\leq 7$, both in terms of $L(x)$ and in terms of $\xi(x+\tfrac12)$ for reference. We also derive a limit formula for $\gamma_1$, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The secondary zeta function is defined by $$ Z(s)=\sum_{\gamma>0}\frac{1}{\gamma^{s}}, $$ where $\tfrac12+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. In this article, we derive convergent integral representations of $Z(s)$ at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann $\xi$-function. Define $$ L(x)=\log\left(\frac{\xi(x+\tfrac12)}{\xi(\tfrac12)}\right), \qquad \mathcal{E}=x\frac{\mathrm{d}}{\mathrm{d}x}, \qquad \mathcal{D}_k=\mathcal{E}\prod_{j=1}^{k}(\mathcal{E}-2j). $$ Then, for $k\geq1$, $$ Z(2k)=\frac{(-1)^k}{2^k(k-1)!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+1}}\,\mathrm{d}x $$ and $$ Z(2k+1)=\frac{(-1)^k}{\pi\,(2k-1)!!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+2}}\,\mathrm{d}x. $$ In the last part, we write these formulas out explicitly for $Z(k)$, $2\leq k\leq 7$, both in terms of $L(x)$ and in terms of $\xi(x+\tfrac12)$ for reference. We also derive a limit formula for $\gamma_1$, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The secondary zeta function is defined by $$ Z(s)=\sum_{\gamma>0}\frac{1}{\gamma^{s}}, $$ where $\tfrac12+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. | background 1.00 | 0.14 | 0.03 | asserted 0.99 |
| s1 | In this article, we derive convergent integral representations of $Z(s)$ at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann $\xi$-function. | contribution 0.97 | 0.97 | 0.80 | asserted 1.00 |
| s2 | Define $$ L(x)=\log\left(\frac{\xi(x+\tfrac12)}{\xi(\tfrac12)}\right), \qquad \mathcal{E}=x\frac{\mathrm{d}}{\mathrm{d}x}, \qquad \mathcal{D}_k=\mathcal{E}\prod_{j=1}^{k}(\mathcal{E}-2j). | method 0.83 | 0.64 | 0.05 | asserted 0.73 |
| s3 | $$ Then, for $k\geq1$, $$ Z(2k)=\frac{(-1)^k}{2^k(k-1)!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+1}}\,\mathrm{d}x $$ and $$ Z(2k+1)=\frac{(-1)^k}{\pi\,(2k-1)!!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+2}}\,\mathrm{d}x. | result 0.93 | 0.93 | 0.75 | asserted 1.00 |
| s4 | $$ In the last part, we write these formulas out explicitly for $Z(k)$, $2\leq k\leq 7$, both in terms of $L(x)$ and in terms of $\xi(x+\tfrac12)$ for reference. | contribution 0.91 | 0.96 | 0.06 | asserted 0.99 |
| s5 | We also derive a limit formula for $\gamma_1$, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically. | contribution 0.64 | 0.96 | 0.10 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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