Ikuya Kaneko’s research while affiliated with California Institute of Technology and other places

We investigate the prime geodesic theorem with an error term dependent on the varying weight and its higher metaplectic coverings in the arithmetic setting, each admitting subconvex refinements despite the softness of our input. The former breaks the $\frac{3}{4}$-barrier due to Hejhal (1983) when the multiplier system is nontrivial, while the latter represents the first theoretical evidence supporting the prevailing consensus on the optimal exponent $1+\varepsilon$ when the multiplier system specialises to the Kubota character. Our argument relies on the elegant phenomenon that the main term in the prime geodesic theorem is governed by the size of the largest residual Laplace eigenvalue, thereby yielding a simultaneous polynomial power-saving in the error term relative to its Shimura correspondent where the multiplier system is trivial.

We prove Motohashi’s formula for a mixed second moment of the Riemann zeta function and a Dirichlet L-function attached to a primitive Dirichlet character modulo $q \in \mathbb {N}$. If q is an odd prime, our reciprocity formula is consistent with Motohashi’s result in the early 1990s. The cubic moment side features two versions of central L-values of automorphic forms on $\Gamma _{0}(q) \backslash \mathbb {H}$. The methods involve a blend of analytic number theory and automorphic forms.

We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous Mertens function, expanding upon work of Ng. Finally, we explore properties of the generalised Mertens function of certain dicyclic number fields as consequences of Artin factorisation.

We address the prime geodesic theorem in arithmetic progressions and resolve conjectures of Golovchanskiĭ–Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the reduced residue classes of a given modulus.

We address the prime geodesic theorem in arithmetic progressions, and resolve conjectures of Golovchanski\u{\i}-Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the reduced residue classes of a given modulus.

This work addresses the prime number races for non-constant elliptic curves E E over function fields. We prove that if r a n k ( E ) > 0 \mathrm {rank}(E) > 0 , then there exist Chebyshev biases towards being negative, and otherwise there exist Chebyshev biases towards being positive. The key input is the convergence of the partial Euler product at the centre, which follows from the Deep Riemann Hypothesis over function fields.

We prove the quantum unique ergodicity conjecture for Eisenstein series over function fields in the level aspect. Adapting the machinery of Luo-Sarnak (1995), we employ the spectral decomposition and handle the cuspidal and Eisenstein contributions separately.