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In this survey article, we explain how false theta functions can be embedded into a modular framework and show some of the applications of this modularity. [ABSTRACT FROM AUTHOR]

We prove a Green–Tao type theorem for multiplicative functions. [ABSTRACT FROM AUTHOR]

Abstract: We describe a new large class of Lorentzian Kac–Moody algebras. For all ranks, we classify 2‐reflective hyperbolic lattices S with the group of 2‐reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac–Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac–Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2‐reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings. [ABSTRACT FROM AUTHOR]

We prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the 'smooth' summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As a consequence, assuming the Riemann hypothesis for Dirichlet [ABSTRACT FROM AUTHOR]