arXiv subjects
Publications and source records attributed to Ikuya Kaneko.
Via the work of Ramanujan, we establish the asymptotic behaviour of partial Euler products for Dirichlet $L$-functions under the Generalised Riemann Hypothesis (GRH). Understanding the behaviour of Euler products on the critical line is called the Deep Riemann Hypothesis (DRH). This work manifests the relation between GRH and DRH.
Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field $L$ has infinitely many nontrivial zeros of multiplicity at least 2 if $L$ has a subfield $K$ for which $L/K$ is a nonabelian Galois extension. We also extend this to zeros of order 3 when $\operatorname{Gal}(L/K)$ has an irreducible representation of degree at least 3, as predicted by the Artin holomorphy conjecture.
For any congruence subgroup of the modular group, we extend the region of convergence of the Euler products of the Selberg zeta functions beyond the boundary Re s = 1, if they are attached with a nontrivial irreducible unitary representation. The region is determined by the size of the lowest eigenvalue of the Laplacian, and it extends to Re s $\geqslant$ 3/4 under Selberg's eigenvalue conjecture. More generally, for any unitary representation we establish the relation between the behavior of partial Euler products in the critical strip and the estimate of the error term in the prime geodesic theorem. For the trivial representation, the proof essentially exploits the idea of the celebrated work of Ramanujan.