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## Section: New Results

### Computing Jacobi's theta function in quasi-linear time

Participant : Hugo Labrande [contact] .

We designed a new algorithm that improves the complexity of computing the value of the Jacobi theta function, $\theta \left(z,\tau \right)$ to arbitrary precision [23] . The algorithm uses a quadratically convergent sequence similar to the complex AGM, as well as Newton's method; its complexity is $O\left(ℳ\left(n\right)logn\right)$ for computing the value up to an error bounded by ${2}^{-n}$, which is an improvement over the state-of-the-art complexity of $O\left(ℳ\left(n\right)\sqrt{n}\right)$. Here, $ℳ\left(n\right)$ denotes the time taken by a multiplication of two $n$-bit numbers. We provide bounds on the loss of significant digits incurred during the computation. The algorithm was implemented using GNU MPC, showing practical improvement over (our optimized implementation of) existing algorithms for precision above approximately $300,000$ bits. The paper was submitted to Mathematics of Computation.