In 1904, Vorono\" {\i} formulated a conjecture for arithmetical functions, which is a generalization of the Poisson summation formula. He was able to establish this conjecture for the divisor function $d(n)$, giving a relation between Bessel functions and weighted sums of the divisor function. Such formulas have since been generalized to various arithmetical functions. We report on joint work with Atul Dixit, Bibekananda Maji and A. Sankaranarayanan, giving such a formula for a generalized divisor function.

We study a coupled nonequilibrium system where two species of particles, one lighter (L) and the other heavier (H), move stochastically on an energy landscape, which also fluctuates in time. The particles tend to minimize their energy by moving along the local potential gradient and also by modifying the landscape around their position in such a way that the energy is further lowered. By tuning the coupling parameters that govern the action of the H and L particles on the landscape, we obtain a phase diagram that shows two new kinds of ordered phases in both particle species and also the landscape, along with two earlier found phases, and a disordered phase. The ordered phases show algebraic coarsening to steady state and rich dynamics. The phase diagram remains qualitatively valid for all densities in one and two dimensions. The disordered phase is characterised using a recently developed formalism of non-linear fluctuating hydrodynamics (NLFH) along with mode-coupling theory that predicts diffusive, KPZ, 3/2-Lévy, 5/3-Lévy, and golden mean universality classes for our system. However, we argue that in verifying the predictions through numerical simulations, the conclusions can be masked by finite size effects.

The dynamo effect is an instability that converts kinetic energy of an electrically conducting fluid into magnetic energy. It is the source of the magnetic field observed in most astrophysical objects, Earth and most planets, Sun and other stars, galaxies. This effect was identified by Larmor a hundred years ago and yet many questions are still unanswered, in particular concerning the effect of turbulent fluctuations on the dynamo process. Dynamo instability occurring in nature and in laboratory experiments are driven by a highly turbulent flow. An analytical model that takes into account the turbulent fluctuations was proposed by Kazantsev [1]. Around the same time, Kraichnan [2] proposed a similar model for the passive scalar advection. The Kazantsev model considers a velocity field that is white in time process with a Gaussian distribution. With this hyposthesis the induction equation that governs the evolution of the magnetic field B can be solved. The induction equation is linear in B and its retro-action on the velocity field is not considered (kinematic dynamo). The magnetic field driven by the velocity field grows in a fluctuation manner, i.e., the growth is not monotonic. The growth rate of the n-th moment of the magnetic field is a nonlinear function of n. This implies that each moment of B predicts a different threshold of the instability. Thus the question we ask is, which of these moments predict the actual threshold in the nonlinear system of equations (with the retro-action)? To respond to this question we develop an analytical model using some results of the theory of large deviations [3]. This allows us to find the pdf of the magnetic field for the linear problem. We show that the growth rate of the different moments of B is a nonlinear function of n. We find the theoretical threshold of the different moments of the magnetic field. Then, we use numerical simulations to solve the nonlinear system of equations. We show that the threshold is predicted by the zeroth moment of the magnetic field (the log of B) of the linear problem which is the correct threshold of the system. Notably higher moments like the magnetic energy overestimate the threshold. We verify this for the dynamo instability driven by different types of turbulent flows.

We prove an explicit central value formula for a family of complex L-series of degree 6 for GL2 × GL3 which arise as factors of certain Garret--Rankin triple product L-series associated with modular forms. Our result generalizes a previous formula of Ichino involving Saito--Kurokawa lifts, and as an application, we prove Deligne's conjecture about the algebraicity of the central values of the considered L-series up to the relevant periods. I would also include some other arithmetic applications towards the subconvexity problem, construction of associated p-adic L function, etc. This is joint work with Carlos de Vera Piquero.