TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

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The Jacobi theta function satisfies $θ (t) = t^{- 1 / 2} θ (1 / t)$

Statement

For every $t > 0$ ,

$θ (t) = t^{- 1 / 2} θ (1 / t) .$

Facts & Assumptions

Given: A real number $t > 0$ .

[L1]

The Jacobi theta function is $θ (t) = \sum_{n \in Z} e^{- π n^{2} t}$ (The Jacobi theta function $θ (t) = \sum_{n \in Z} e^{- π n^{2} t}$ for $t > 0$ ).

[L2]

The Gaussian integral is $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ (The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ ).

[L3]

The cited zeta sources record the local Fourier/Poisson seam used here: for $g_{t} (x) : = e^{- π t x^{2}}$ , the fixed Fourier normalization gives a Gaussian transform of the form ${\hat{g}}_{t} (ξ) = C_{t} e^{- π ξ^{2} / t},$ and Poisson summation for this Gaussian periodization gives $\sum_{n \in Z} g_{t} (n) = \sum_{m \in Z} {\hat{g}}_{t} (m) .$ This is the same seam recorded in the batch notes.

Proof

technique · direct

1.1

Evaluating the transform in [L3] at $ξ = 0$ gives $C_{t} = {\hat{g}}_{t} (0) = \int_{- \infty}^{\infty} e^{- π t x^{2}} d x .$ With the change of variables $u = \sqrt{π t} x$ and [L2], this integral equals $t^{- 1 / 2}$ . Therefore ${\hat{g}}_{t} (ξ) = t^{- 1 / 2} e^{- π ξ^{2} / t} .$

givenL2L3algebra

2.1

By [L1], $θ (t) = \sum_{n \in Z} g_{t} (n)$ . Poisson summation from [L3] and step 1.1 therefore give $θ (t) = \sum_{m \in Z} {\hat{g}}_{t} (m) = t^{- 1 / 2} \sum_{m \in Z} e^{- π m^{2} / t} = t^{- 1 / 2} θ (1 / t) .$

L1L3step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 6 §2.1 (standard reference, not scraped)
K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 12, The Theta Relation (standard reference, not scraped)

The Jacobi theta function satisfies θ(t)=t−1/2θ(1/t)θ(t)=t−1/2θ(1/t)