The Generalized Riemann Zeta heat flow

We consider the PDE flow associated to Riemann zeta and general Dirichlet $L$-functions. These are models characterized by nonlinearities appearing in classical number theory problems, and generalizing the classical holomorphic Riemann flow studied by Broughan and Barnett. Each zero of a Dirichlet $L$-function is an exact solution of the model. In this paper, we first show local existence of bounded continuous solutions in the Duhamel sense to any Dirichlet $L$-function flow with initial condition far from the pole (as long as this exists). In a second result, we prove global existence in the case of nonlinearities of the form Dirichlet $L$-functions and data initially on the right of a possible pole. Additional global well-posedness and convergence results are proved in the case of the defocusing Riemann zeta nonlinearity and initial data located on the real line and close to the trivial zeros of the zeta. The asymptotic stability of any stable zero is also proved. Finally, in the Riemann zeta case, we consider the ``focusing'' model, and prove blow-up of solutions near the pole $s=1$.Comment: 33 p

DISTRIBUTION OF FROBENIUS ELEMENTS IN FAMILIES OF GALOIS EXTENSIONS

Given a Galois extension L/K of number fields, we describe fine distribution properties of Frobenius elements via invariants from representations of finite Galois groups and ramification theory. We exhibit explicit families of extensions in which we evaluate these invariants, and deduce a detailed understanding and a precise description of the possible asymmetries. We establish a general bound on the generic fluctuations of the error term in the Chebotarev density theorem which under GRH is sharper than the Murty-Murty-Saradha and Bellaïche refinements of the Lagarias-Odlyzko and Serre bounds, and which we believe is best possible (assuming simplicity, it is of the quality of Montgomery's conjecture on primes in arithmetic progressions). Under GRH and a hypothesis on the multiplicities of zeros up to a certain height, we show that in certain families these fluctuations are dominated by a constant lower order term. As an application of our ideas we refine and generalize results of K. Murty and of J. Bellaïche and we answer a question of N. Ng. In particular, in the case where L/Q is Galois and supersolvable, we prove a strong form of a conjecture of K. Murty on the unramified prime ideal of least norm in a given Frobenius set. The tools we use include the Rubinstein-Sarnak machinery based on limiting distributions and a blend of algebraic, analytic, representation theoretic, probabilistic and combinatorial techniques