Anyone seen these conclusions about the Riemann xi function or see any errors here?

With $\xi(s)$ the entire Landau Riemann xi function defined by the Hadamard product representation

$$\xi(s) = (1/2) \prod_{\rho_m} [1 - \frac{s}{\rho_m}]$$

where the product ranges over the nontrivial zeros of the Riemann zeta function, let

$$\vartheta(z) = 2\xi(\frac{1}{1-z}) $$

$$= \Xi(\frac{1}{1-z}) $$

$$= \prod_{\rho_m}[1- \frac{1}{\rho_m} \frac{1}{(1-z)}].$$

Let $u = 1/(1-z)$, then $z = (u-1)/u$ and

$$\vartheta(\frac{u-1}{u}) = \Xi(u)$$

$$= \Xi(1-u) = \vartheta(\frac{u}{u-1}),$$

so

$$\vartheta(z) = \vartheta(\frac{1}{z})$$ $$ = \Xi[\frac{1}{1-(1/z)}]= \Xi[\frac{z}{z-1}]$$

$$= \prod_{\rho_k} [1 - \frac{1}{\rho_k} \frac{z}{z-1} ].$$

(Edit June 10: Corrected transcription error. At last moment I changed notation from $\Omega$ in my notes to $\Xi$ but missed the change in this last set of equalities since I read them in my mind as the same Hadamard product. Sorry for the confusion.)

We then have the two equivalent expressions

$$\vartheta(z) = \prod_{\rho_m}[1- \frac{1}{\rho_m} \frac{1}{(1-z)}] $$

$$ = \prod_{\rho_k} [1 - \frac{1}{\rho_k} \frac{z}{z-1} ].$$

The zeros of the last product are given implicitly by

$$1 = \frac{1}{\rho_k} \frac{z_k}{z_k-1},$$

so

$$z_k= \rho_k/(\rho_k-1)$$

and the other product gives a null factor when

$$1 = \frac{1}{\rho_m}\frac{1}{1-z_k},$$

so

$$z_k = (\rho_m-1)/\rho_m.$$

Each zero is paired with its complex conjugate so that both equations are simultaneously satisfied only if $\rho_k = \bar{\rho}_m$ giving

$$\frac{\rho_m-1}{\rho_m} = \frac{\bar{\rho}_m}{\bar{\rho}_m-1}$$

implying

$$(\rho_m-1)(\bar{\rho}_m-1) = \rho_m \bar{\rho}_m$$

and, therefore, that

$$Real(\rho_m) = 1/2.$$

Edit to address some comments:

If $\vartheta$ were an entire function with real zeros only, e.g., a polynomial with real zeros only, or the reciprocal of the gamma function written as it's Weierstrass factorization, the result would not apply since the necessary pairing that each zero's complex conjugate is also a zero, which is distinct but related to $\rho = 1 - \bar{\rho}$ since $ 3 + i \alpha$ and $3 - i\alpha$ are a conjugate pair that do not satisfy that relation, nor does the symmetry relation $f(s) = f(1-s)$ apply. The symmetry relation and factorization are indeed enough (explicit values for the zeros are not needed) to ensure the critical line--that's the point. Check that $\vartheta(z) = \vartheta(1/z)$ is a severe restriction that when evaluated at $z =0$ giving $\vartheta(0)) = \vartheta(\infty) = 1 = \prod_{\rho_k}[1+(1/\rho_k]$ enforces that $\rho_m = 1/2 + i\alpha$ and $ \rho_k = 1/2 - i\alpha$ are indeed solns.

(See, I believe Bump and others on the local Riemann hypothesis of other functions with critical lines.)

The two conditions of the general form of the factorization and the symmetry are necessary and sufficient to prove the critical line hypothesis without resort to calculating the actual values of the imaginary parts of the zeros (regardless of how interesting they are and the associations to the PNT) is the crux of the matter. (Anticlimactic, if true.)

Edit to address a comment:

From "Remark on Dirichlet Series Satisfying Functional Equations" by Eugenio P. Balanzario:

"By a theorem of H. Hamburger (see [6], page 31) the zeta function of Riemann is determined by its functional equation (FE). Hence, if we want to produce other Dirichlet series satisfying a functional equation, then it is necessary to change (FE) somehow."

[6] Titchmarsh, E. C. The theory of the Riemann zeta function, Oxford, 1986.

I should say, since I use this in the formal analysis, that the FE, the Hadamard factorization, and the fact that if $\rho$ is a zero then so is its complex conjugate are necessary (but not necessarily independent) conditions. I do not assume that $\rho = 1 - \bar{\rho}$, that is to be determined from the conditions.

Edit Jun 11;

Prompted by the initial responses, I thought more carefully about my assumptions and concluded before looking at the extended and revised responses that I indeed made a mistake by omitting potential solutions of the form of multiple factors, say $$(1-\frac{1}{p_k}\frac{z}{z-1}) (1- \frac{1}{p_m}\frac{z}{z-1}) = 0,$$ so, in effect, tantamount to assuming the RH is true, as LI did, to arrive at his criteria. Thanks to the responders for the input, the attention, and effort to show me the errors of my way--I learned some things along the way. I'll look over your last responses to see if I can learn some more.

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