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Ikuya Kaneko

Publications and source records attributed to Ikuya Kaneko.

At least 19 recordsLinked to original sources
arXiv2025

A Subconvex Metaplectic Prime Geodesic Theorem and the Shimura Correspondence

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.

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arXiv2025

The Prime Geodesic Theorem for the Picard Orbifold

We establish the prime geodesic theorem for the Picard orbifold $\mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathbb{H}^{3}$, wherein the error term shrinks proportionally to improvements in the subconvex exponent for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Our result sheds light on a venerable conjecture by attaining an unconditional exponent of $1.483$ and a conditionally superior exponent of $1.425$ under the generalised Lindelöf hypothesis. The argument synthesises, among other elements, the complete resolution of Koyama's (2001) mean Lindelöf hypothesis over $\mathbb{Q}(i)$, an improved Brun-Titchmarsh-type theorem over short intervals, a bootstrapped multiplicative exponent pair in the limiting regime, and a zero density theorem for the symplectic family of quadratic characters. Notably, despite the theoretical strength of our manifestations towards the mean Lindelöf hypothesis, the fundamental toolbox relies exclusively on the optimal mean square asymptotics for the Fourier coefficients of Maass cusp forms via the pre-Kuznetsov formula.

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arXiv2024

Mean Values and Quantum Variance for Degenerate Eisenstein Series of Higher Rank

We investigate the mean value of the inner product of squared $\mathrm{GL}_{n}$ degenerate maximal parabolic Eisenstein series against a smooth compactly supported function lying in a restricted space of incomplete Eisenstein series induced from a $\mathrm{SL}_{2}(\mathbb{Z})$ Hecke-Maass cusp form $φ$. Our result breaks the fundamental threshold with a polynomial power-saving beyond the pointwise implications of the generalised Lindelöf hypothesis for $L$-functions attached to $φ$. Furthermore, we evaluate the archimedean quantum variance and establish approximate orthogonality, expanding upon Zhang's (2019) work on quantum unique ergodicity for $\mathrm{GL}_{n}$ degenerate maximal parabolic Eisenstein series as well as Huang's (2021) work on quantum variance for $\mathrm{GL}_{2}$ Eisenstein series. Despite the theoretical strength of these manifestations, our argument relies exclusively on the Watson-Ichino-type formula for incomplete Eisenstein series of type $(2, 1, \ldots, 1)$ and Jutila's (1996) asymptotic formula for the second moment of $L$-functions attached to $φ$ in long intervals, supplemented by a standard analytical toolbox.

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arXiv2024

The Prime Geodesic Theorem and Bounds for Character Sums

We establish the prime geodesic theorem for the modular surface with exponent $\frac{2}{3}+\varepsilon$, improving upon the long-standing exponent $\frac{25}{36}+\varepsilon$ of Soundararajan-Young (2013). This was previously known conditionally on the generalised Lindelöf hypothesis for quadratic Dirichlet $L$-functions. Our argument goes through a well-trodden trail via the automorphic machinery, and refines the techniques of Iwaniec (1984) and Cai (2002) to a maximum extent. A key ingredient is an asymptotic for bilinear forms with a counting function in Kloosterman sums via hybrid Weyl-strength subconvex bounds for quadratic Dirichlet $L$-functions due to Young (2017), zero density estimates due to Heath-Brown (1995) near the edge of the critical strip, and an asymptotic for averages of Zagier $L$-series due to Balkanova-Frolenkov-Risager (2022). Furthermore, we strengthen our exponent to $\frac{5}{8}+\varepsilon$ conditionally on the generalised Lindelöf hypothesis for quadratic Dirichlet $L$-functions, which breaks the existing barrier.

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arXiv2023

Towards the Deep Riemann Hypothesis for $\mathrm{GL}_{n}$

We explicate the deep Riemann hypothesis for the general linear group $\mathrm{GL}_{n}$ on the convergence of normalised Euler products of standard $L$-functions on the critical line. It conditionally improves upon the error term in the prime number theorem beyond what the grand Riemann hypothesis predicts. Furthermore, we discuss the Chebyshev bias for Satake parameters on $\mathrm{GL}_{n}$ from the perspective of the deep Riemann hypothesis.

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arXiv2023

On a Finite Field Analogue of van der Corput Differencing

We establish nontrivial bounds for general bilinear forms with a given periodic function, which are thought of as an analogue of van der Corput differencing for exponential sums. The proof employs Poisson summation, Cauchy-Schwarz, and the iteration thereof.

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arXiv2023

Non-Linear Additive Twists of $\mathrm{GL}_{3}$ Hecke Eigenvalues

We bound non-linear additive twists of $\mathrm{GL}_{3}$ Hecke eigenvalues, improving upon the work of Kumar-Mallesham-Singh (2022). The proof employs the DFI circle method with standard manipulations (Voronoi, Cauchy-Schwarz, lengthening, and additive reciprocity). The main novelty includes the conductor lowering mechanism, albeit sacrificing some savings to remove an analytic oscillation, followed by the iteration ad infinitum of Cauchy-Schwarz and Poisson. The resulting character sums are estimated via the work of Adolphson-Sperber (1993). As an application, we prove nontrivial bounds for the first moment of $\mathrm{GL}_{3}$ Hardy's function, which corresponds to the cubic moment of Hardy's function studied by Ivić (2012).

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arXiv2023

Mixed Moments of the Riemann Zeta and Dirichlet $L$-Functions

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 $Γ_{0}(q) \backslash \mathbb{H}$. The methods involve a blend of analytic number theory and automorphic forms.

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arXiv2023

Highly Uniform Prime Number Theorems

We prove a highly uniform version of the prime number theorem for a certain class of $L$-functions. The range of $x$ depends polynomially on the analytic conductor, and the error term is expressed in terms of an optimization problem depending explicitly on the available zero-free region. The class contains the Rankin-Selberg $L$-function $L(s,π\times π')$ associated to cuspidal automorphic representations $π$ and $π'$ of $\mathrm{GL}_{m}$ and $\mathrm{GL}_{m'}$, respectively. Our main result implies the first uniform prime number theorems for such $L$-functions (with analytic conductor uniformity) in complete generality.

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arXiv2023

On Multiple Shifted Convolution Sums

We prove strong estimates for averages of shifted convolution sums consisting of quadratic twists of $\mathrm{GL}_{2}$ $L$-functions. The key input involves the circle method together with standard tools such as Vorono\uı, quadratic reciprocity, amplification, and divisor switching.

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arXiv2023

The Prime Geodesic Theorem in Arithmetic Progressions

We address the prime geodesic theorem in arithmetic progressions, and resolve conjectures of Golovchanski\uı-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.

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arXiv2023

On a Mertens-type conjecture for number fields

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 generalized Mertens function of certain dicyclic number fields as consequences of Artin factorization.

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arXiv2023

A New Aspect of Chebyshev's Bias for Elliptic Curves over Function Fields

This work considers the prime number races for non-constant elliptic curves $E$ over function fields. We prove that if $\mathrm{rank}(E) > 0$, then there exist Chebyshev biases towards being negative, and otherwise there exist Chebyshev biases towards being positive. The main innovation entails the convergence of the partial Euler product at the centre that follows from the Deep Riemann Hypothesis over function fields.

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arXiv2022

The Prime Geodesic Theorem for $\mathrm{PSL}_{2}(\mathbb{Z}[i])$ and Spectral Exponential Sums

This work addresses the Prime Geodesic Theorem for the Picard manifold $\mathcal{M} = \mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathfrak{h}^{3}$, which asks for the asymptotic evaluation of a counting function for the closed geodesics on $\mathcal{M}$. Let $E_Γ(X)$ be the error term in the Prime Geodesic Theorem. We establish that $E_Γ(X) = O_{\varepsilon}(X^{3/2+\varepsilon})$ on average as well as many pointwise bounds. The second moment bound parallels an analogous result for $Γ= \mathrm{PSL}_{2}(\mathbb{Z})$ due to Balog et al. and our innovation features the delicate analysis of sums of Kloosterman sums with an explicit manipulation of oscillatory integrals. The proof of the pointwise bounds requires Weyl-strength subconvexity for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Moreover, an asymptotic formula for a spectral exponential sum in the spectral aspect for a cofinite Kleinian group $Γ$ is given. Our numerical experiments visualise in particular that $E_Γ(X)$ obeys a conjectural bound of size $O_ε(X^{1+\varepsilon})$.

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arXiv2022

Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications

A new reciprocity formula for Dirichlet $L$-functions associated to an arbitrary primitive Dirichlet character of prime modulus $q$ is established. We find an identity relating the fourth moment of individual Dirichlet $L$-functions in the $t$-aspect to the cubic moment of central $L$-values of Hecke-Maass newforms of level at most $q^{2}$ and primitive central character $ψ^{2}$ averaged over all primitive nonquadratic characters $ψ$ modulo $q$. Our formula can be thought of as a reverse version of recent work of Petrow-Young. Direct corollaries involve a variant of Iwaniec's short interval fourth moment bound and the twelfth moment bound for Dirichlet $L$-functions, which generalise work of Jutila and Heath-Brown, respectively. This work traverses an intersection of classical analytic number theory and automorphic forms.

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arXiv2021

Correction to: Equidistribution of Eisenstein Series in the Level Aspect

The second author formulated quantum unique ergodicity for Eisenstein series in the prime level aspect in "Equidistribution of Eisenstein series in the level aspect", Commun. Math. Phys. 289(3), 1131-1150 (2009). We point out errors and correct the proofs with partially weakened claims. This article of corrigendum and addendum comprises three sections. Section 1 is devoted to making concise corrections to the claim and the proof of Theorems 1.2 and 1.3, and Sect. 2 serves as a modification of the computation of the contribution from Maass cusp forms. Other relatively marginal mathematical issues and notational/typographical errors are listed afterwards in Sect. 3.

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arXiv2021

Spectral Exponential Sums on Hyperbolic Surfaces

We study an exponential sum over Laplacian eigenvalues $λ_{j} = 1/4+t_{j}^{2}$ with $t_{j} \leqslant T$ for Maass cusp forms on $Γ\backslash \mathbb{H}$, where $Γ$ is a cofinite Fuchsian group acting on the upper half-plane $\mathbb{H}$. The aim is to establish an asymptotic formula which expresses spectral exponential sums in terms of an oscillatory component, von Mangoldt-like functions and Selberg zeta functions. The behaviour is determined by whether $Γ$ is essentially cuspidal or not.

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