Research proof index

H₂(it) > 0 for all t > 0. Blueprint: Lemma 6.43 - H₂ is positive on the imaginary axis. Proof strategy: Each term exp(-π(n+1/2)²t) > 0, so Θ₂(it) > 0, hence H₂ = Θ₂^4 > 0.

Use jacobiTheta₂_functional_equation

These three transformation laws follow directly from tsum definition.

H₄(it) > 0 for all t > 0. Blueprint: Corollary 6.43 - H₄ is positive on the imaginary axis. Proof strategy: Use the modular S-transformation relating H₄ and H₂. From H₄_S_action: (H₄ ∣[2] S) = -H₂ From ResToImagAxis.SlashActionS: relates values at t and 1/t. This gives H₂(i/t) = t² * H₄(it), so H₄(it) > 0 follows from H₂(i/t) > 0.

Associated primes of the kernel of the localization map. This identifies the associated primes of ker (LocalizedModule.mkLinearMap S M) : Submodule R M with the associated primes of M that do meet the multiplicative set S. Equivalently, these are the associated primes of M after removing those disjoint from S. This lemma is used as the “kernel...

One-sided inclusion for associated primes of the kernel of the localization map. If p ∈ associatedPrimes R (ker (mkLinearMap S M)), then p is an associated prime of M and p is not disjoint from the multiplicative set S. This is one direction of associatedPrimes_ker_mkLinearMap_eq.