Ramanujan’s last letter to Hardy concerns the asymptotic properties of modular forms and his ‘mock theta functions’. For the mock theta function $f(q)$, Ramanujan claims that as $q$ approaches an even-order $2k$ root of unity, we have

We prove Ramanujan’s claim as a special case of a more general result. The implied constants in Ramanujan’s claim are not mysterious. They arise in Zagier’s theory of ‘quantum modular forms’. We provide explicit closed expressions for these ‘radial limits’ as values of a ‘quantum’ $q$-hypergeometric function which underlies a new relationship between Dyson’s rank mock theta function and the Andrews–Garvan crank modular form. Along these lines, we show that the Rogers–Fine false $\vartheta $-functions, functions which have not been well understood within the theory of modular forms, specialize to quantum modular forms.

The (real) Grothendieck constant ${K}_{G} $ is the infimum over those $K\in (0, \infty )$ such that for every $m, n\in \mathbb{N} $ and every $m\times n$ real matrix $({a}_{ij} )$ we have

The classical Grothendieck inequality asserts the nonobvious fact that the above inequality does hold true for some $K\in (0, \infty )$ that is independent of $m, n$ and $({a}_{ij} )$. Since Grothendieck’s 1953 discovery of this powerful theorem, it has found numerous applications in a variety of areas, but, despite attracting a lot of attention, the exact value of the Grothendieck constant ${K}_{G} $ remains a mystery. The last progress on this problem was in 1977, when Krivine proved that ${K}_{G} \leqslant \pi / 2\log (1+ \sqrt{2} )$ and conjectured that his bound is optimal. Krivine’s conjecture has been restated repeatedly since 1977, resulting in focusing the subsequent research on the search for examples of matrices $({a}_{ij} )$ which exhibit (asymptotically, as $m, n\rightarrow \infty $) a lower bound on ${K}_{G} $ that matches Krivine’s bound. Here, we obtain an improved Grothendieck inequality that holds for all matrices $({a}_{ij} )$ and yields a bound ${K}_{G} \lt \pi / 2\log (1+ \sqrt{2} )- {\varepsilon }_{0} $ for some effective constant ${\varepsilon }_{0} \gt 0$. Other than disproving Krivine’s conjecture, and along the way also disproving an intermediate conjecture of König that was made in 2000 as a step towards Krivine’s conjecture, our main contribution is conceptual: despite dealing with a binary rounding problem, random two-dimensional projections, when combined with a careful partition of ${ \mathbb{R} }^{2} $ in order to round the projected vectors to values in $\{ - 1, 1\} $, perform better than the ubiquitous random hyperplane technique. By establishing the usefulness of higher-dimensional rounding schemes, this fact has consequences in approximation algorithms. Specifically, it yields the best known polynomial-time approximation algorithm for the Frieze–Kannan Cut Norm problem, a generic and well-studied optimization problem with many applications.