Xi Product Endpoint Limits

Abstract

The displayed xi product form attains both endpoint values through punctured limits.

Theorem 1.1 (Xi product form tends to one-half at zero).

$$\underset{s\to 0,s\ne 0}{\lim }\frac{1}{2}s(s-1)\Lambda (s)=\frac{1}{2}.$$

Proof. Machine-checked in Lean as D5/S3/Zeros/Endpoints/XiProductEndpointLimits.xi_product_form_tendsto_zero (✓ std3). ∎

Source. Repository-derived.

Commentary.

The frozen Mellin reconstruction gives the pole clause that s times the completed-zeta reading tends to minus one at zero. Multiplication by the continuous factor one-half times s minus one yields the displayed limit.

The approach is punctured at zero. This theorem does not assert the false literal equality obtained by evaluating the raw totalized product there.

Theorem 1.2 (Xi product form tends to one-half at one).

$$\underset{s\to 1,s\ne 1}{\lim }\frac{1}{2}s(s-1)\Lambda (s)=\frac{1}{2}.$$

Proof. Machine-checked in Lean as D5/S3/Zeros/Endpoints/XiProductEndpointLimits.xi_product_form_tendsto_one (✓ std3). ∎

Source. Repository-derived.

Commentary.

The frozen Mellin reconstruction gives the pole clause that s minus one times the completed-zeta reading tends to one at one. Multiplication by the continuous factor one-half times s yields the displayed limit.

The approach is punctured at one. This theorem likewise records limiting attainment rather than literal evaluation of the raw product at the pole.

Theorem 1.3 (Xi product form attains the frozen endpoint values).

$$\left(\underset{s\to 0,s\ne 0}{\lim }\frac{1}{2}s(s-1)\Lambda (s)=\frac{1}{2}\wedge \underset{s\to 1,s\ne 1}{\lim }\frac{1}{2}s(s-1)\Lambda (s)=\frac{1}{2}\right)\wedge \left(\xi (0)=\frac{1}{2}\wedge \xi (1)=\frac{1}{2}\right).$$

Proof. Machine-checked in Lean as D5/S3/Zeros/Endpoints/XiProductEndpointLimits.xi_product_form_attains_endpoint_values (✓ std3). ∎

Source. Repository-derived.

Commentary.

The two punctured limits are paired with the frozen endpoint theorem xiReading zero equals xiReading one equals one-half. Thus continuity of the pole-removed xi reading closes exactly the two points excluded by the frozen off-endpoint product identity.

The source displays the product formula globally, but the repository’s completed-zeta reading is totalized at its poles. At zero and one the raw product evaluates to zero, so the honest endpoint interpretation is the punctured-limit statement recorded here.

References

• Truth anchor: D5/S3/Zeros/Endpoints/XiProductEndpointLimits.xi_product_form_attains_endpoint_values
• Truth anchor: D5/S3/Zeros/Endpoints/XiProductEndpointLimits.xi_product_form_tendsto_one
• Truth anchor: D5/S3/Zeros/Endpoints/XiProductEndpointLimits.xi_product_form_tendsto_zero
• Dependency: D5/S3/Analytic/CompletedZetaMellinReconstruction
• Dependency: D5/S3/Zeros/Endpoints/XiEndpointValues