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Perron Inversion and the Explicit Formula — Examples

1 · Prerequisites

2 · Summary

These calculations isolate the half-jump, the four residue families, and the ordering convention that a sharp zero sum requires.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Perron's kernel at and away from its jump

Example

At $c = 1$ , the limiting kernels for $y = 1 / 2, 1, 2$ are respectively $0, 1 / 2, 1$ .

Verification

Given: the symmetric and truncated Perron kernel formulas.

1.1

Substitute $y = 1 / 2, 1, 2$ in the three branches of the symmetric kernel.

givenalgebra

2.1

The truncated estimate has the separate $1 / T$ branch only at $y = 1$ , exhibiting why the half value cannot be merged into either away-from-jump estimate.

step 1.1algebra∎

ExampleConstruction: AI-generatedVerification: AI-generatedaudited 2026-09-07Open item page →

Perron inversion for a finite Dirichlet polynomial

Example

For $F (s) = 1 + 2^{- s}$ , Perron's integral is $1$ for $1 < x < 2$ , $3 / 2$ for $x = 2$ , and $2$ for $x > 2$ .

Verification

Given: the Perron formula for finite sums.

1.1

Termwise inversion gives $δ (x) + δ (x / 2)$ ; for $x > 1$ , the first term is $1$ .

givenalgebra

2.1

The second term is $0, 1 / 2, 1$ according as $x < 2, x = 2, x > 2$ , proving all three values without a limit interchange.

step 1.1cases∎

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

The von Mangoldt residue table

Example

For $x > 1$ , the contour integrand has contributions $x$ at $1$ , $- x^{ρ} / ρ$ at a nontrivial zero, $- \frac{1}{2} \log (1 - x^{- 2})$ from all negative even zeros, and $- ζ^{'} (0) / ζ (0)$ at $0$ .

Verification

Given: $x > 1$ and the residue ledger.

1.1

The pole at $1$ and each nontrivial zero have the listed local residues, with zero multiplicity multiplying the second entry.

givenalgebra

2.1

Summing the geometric-logarithmic trivial family and retaining the independent pole of $1 / s$ at $0$ gives the remaining two entries.

step 1.1algebra∎

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Smoothed versus sharp explicit formulas

Example

The linear cutoff equals $1$ through $x$ and fades to $0$ on $(x, y)$ , whereas the sharp formula evaluates $ψ_{0} (x)$ and gives half weight at a prime power.

Verification

Given: the smoothed and sharp formulas.

1.1

The two integrations by parts in the smoothed formula supply Mellin decay, hence a declared convergent zero sum.

givenalgebra

2.1

The sharp formula instead has a finite ordinate sum and an error containing $⟨ x ⟩$ ; at a jump its left hand side is explicitly half-weighted.

step 1.1algebra∎

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Selecting an admissible contour height

Example

For $T \geq 3$ , one can choose $T^{'} \in [T, T + 1]$ with distance at least $c / \log T$ from every zero ordinate in $[T - 1, T + 2]$ , for an absolute $c > 0$ .

Verification

Given: the $O (\log (T + 2))$ unit-interval zero count.

1.1

The three adjacent unit-interval bounds show that the number of relevant ordinates is at most $C \log T$ after enlarging an absolute constant $C$ to cover $3 \leq T \leq 5$ . Around each such ordinate remove an interval of radius $c / \log T$ , where $c < 1 / (2 C)$ .

givenconstruct

2.1

The total removed length inside $[T, T + 1]$ is at most $2 C c < 1$ . Consequently some $T^{'}$ remains in that interval, and by construction its distance from every relevant ordinate is at least $c / \log T$ .

step 1.1discharge-construct∎

CounterexampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

An unordered infinite zero sum is not an explicit formula

Statement refuted

For $x > 1$ , the bare unordered expression $\sum_{ρ} x^{ρ} / ρ$ is a defined quantity equal to the sharp explicit-formula zero contribution.

Counterexample

Given: the sharp and smoothed explicit formulas.

1.1

The sharp formula asserts only finite sums $\sum_{∣ ℑ ρ ∣ < T}$ and specifies an error before any limit is taken.

givenalgebra

2.1

The smoothed formula supplies a different convergence mechanism through its Mellin decay. Neither result assigns a value to the displayed unordered bare sum, so the asserted equality has no defined left side.

step 1.1contradiction∎

CounterexampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07Open item page →

Right-continuous psi has the wrong Perron endpoint

Statement refuted

At every prime power $q$ , the right-continuous $ψ (q)$ equals the Perron endpoint value.

Counterexample

Given: $q = p^{k}$ and the definitions of $ψ$ and $ψ_{0}$ .

1.1

The right-continuous sum includes the full last term: $ψ (q) = \sum_{n < q} Λ (n) + \log p$ .

givenalgebra

2.1

Perron's endpoint is $ψ_{0} (q) = \sum_{n < q} Λ (n) + \frac{1}{2} \log p$ , so the two values differ by $\frac{1}{2} \log p \neq 0$ .

step 1.1algebra∎

Perron Inversion and the Explicit Formula — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Perron's kernel at and away from its jump

Example

Verification

Perron inversion for a finite Dirichlet polynomial

Example

Verification

The von Mangoldt residue table

Example

Verification

Smoothed versus sharp explicit formulas

Example

Verification

Selecting an admissible contour height

Example

Verification

An unordered infinite zero sum is not an explicit formula

Statement refuted

Counterexample

Right-continuous psi has the wrong Perron endpoint

Statement refuted

Counterexample

Sources