The Schonhage-Strassen conjecture that n log n is optimal for integer multiplication on multitape Turing machines

Theorem 1.1: one fixed deterministic multitape Turing machine multiplies two $n$-bit integers exactly in worst-case time $O(n(\lg n)^{1-\kappa})$ with $\kappa=2^{-182}$, for every $n$, so $T(n)/(n\lg n)\to0$ and the $n\log n$ lower bound fails in this model. Corollaries: exact division and integer square root in the same time, and (with Harvey-van der Hoeven's reduction) transposition of $k\times k$ binary matrices in $O(k^2(\lg k)^{1-\kappa})$. Not shown: any practical algorithm (constants are astronomically large), or anything about Boolean circuit size or other machine models; the true complexity of multiplication remains open.

Theorem 1.1: one fixed deterministic multitape Turing machine multiplies two $n$-bit integers exactly in worst-case time $O(n(\lg n)^{1-\kappa})$ with $\kappa=2^{-182}$, for every $n$, so $T(n)/(n\lg n)\to0$ and the $n\log n$ lower bound fails in this model. Corollaries: exact division and integer square root in the same time, and (with Harvey-van der Hoeven's reduction) transposition of $k\times k$ binary matrices in $O(k^2(\lg k)^{1-\kappa})$. Not shown: any practical algorithm (constants are astronomically large), or anything about Boolean circuit size or other machine models; the true complexity of multiplication remains open.