arXiv subjects

Niko Laaksonen

Publications and source records attributed to Niko Laaksonen.

7 recordsLinked to original sources

arXiv2020

On the variance of the nodal volume of arithmetic random waves

Rudnick and Wigman (Ann. Henri Poincaré, 2008; arXiv:math-ph/0702081) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the $d$-dimensional torus is $O(E/\mathcal{N})$, as $E\to\infty$, where $E$ is the energy and $\mathcal{N}$ is the dimension of the eigenspace corresponding to $E$. Previous results have established this with stronger asymptotics when $d=2$ and $d=3$. In this brief note we prove an upper bound of the form $O(E/\mathcal{N}^{1+α(d)-ε})$, for any $ε>0$ and $d\geq 4$, where $α(d)$ is positive and tends to zero with $d$. The power saving is the best possible with the current method (up to $ε$) when $d\geq 5$ due to the proof of the $\ell^{2}$-decoupling conjecture by Bourgain and Demeter.

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arXiv2019

Bykovskii-type theorem for the Picard manifold

We generalise a result of Bykovskii to the Gaussian integers and prove an asymptotic formula for the prime geodesic theorem in short intervals on the Picard manifold. Previous works show that individually the remainder is bounded by $O(X^{13/8+ε})$ and $O(X^{3/2+θ+ε})$, where $θ$ is the subconvexity exponent for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. By combining arithmetic methods with estimates for a spectral exponential sum and a smooth explicit formula, we obtain an improvement for both of these exponents. Moreover, by assuming two standard conjectures on $L$-functions, we show that it is possible to reduce the exponent below the barrier $3/2$ and get $O(X^{34/23+ε})$ conditionally. We also demonstrate a dependence of the remainder in the short interval estimate on the classical Gauss circle problem for shifted centres.

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arXiv2018

Second Moment of the Prime Geodesic Theorem for $\mathrm{PSL}(2, \mathbb{Z}[i])$

The remainder $E_Γ(X)$ in the Prime Geodesic Theorem for the Picard group $Γ= \mathrm{PSL}(2,\mathbb{Z}[i])$ is known to be bounded by $O(X^{3/2+ε})$ under the assumption of the Lindelöf hypothesis for quadratic Dirichlet $L$-functions over Gaussian integers. By studying the second moment of $E_Γ(X)$, we show that on average the same bound holds unconditionally.

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arXiv2018

Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

For $Γ$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=Γ\backslash \mathbb{H}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on $M$. Let $E_Γ(X)$ be the error in the counting of prime geodesics with length at most $\log X$. For the Picard manifold, $Γ=\mathrm{PSL}(2,\mathbb{Z}[i])$, we improve the classical bound of Sarnak, $E_Γ(X)=O(X^{5/3+ε})$, to $E_Γ(X)=O(X^{13/8+ε})$. In the process we obtain a mean subconvexity estimate for the Rankin-Selberg $L$-function attached to Maass-Hecke cusp forms. We also investigate the second moment of $E_Γ(X)$ for a general cofinite group $Γ$, and show that it is bounded by $O(X^{16/5+ε})$.

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arXiv2015

Quantum Limits of Eisenstein Series in H^3

We study the quantum limits of Eisenstein series off the critical line for $\mathrm{PSL}_{2}(\mathcal{O}_{K})\backslash\mathbb{H}^{3}$, where $K$ is an imaginary quadratic field of class number one. This generalises the results of Petridis, Raulf and Risager on $\mathrm{PSL}_{2}(\mathbb{Z})\backslash\mathbb{H}^{2}$. We observe that the measures $\lvert E(p,σ_{t}+it)\rvert^{2}dμ(p)$ become equidistributed only if $σ_{t}\rightarrow 1$ as $t\rightarrow\infty$. We use these computations to study measures defined in terms of the scattering states, which are shown to converge to the absolutely continuous measure $E(p,3)dμ(p)$ under the GRH.

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arXiv2015

Lattice Point Counting in Sectors of Hyperbolic 3-space

Let $Γ$ be a cocompact discrete subgroup of $\mathrm{PSL}_{2}(\mathbb{C})$ and denote by $\mathcal{H}$ the three dimensional upper half-space. For a $p\in\mathcal{H}$, we count the number of points in the orbit $Γp$, according to their distance, $\operatorname{arccosh} X$, from a totally geodesic hyperplane. The main term in $n$ dimensions was obtained by Herrmann for any subset of a totally geodesic submanifold. We prove a pointwise error term of $O(X^{3/2})$ by extending the method of Huber and Chatzakos-Petridis to three dimensions. By applying Chamizo's large sieve inequalities we obtain the conjectured error term $O(X^{1+ε})$ on average in the spatial aspect. We prove a corresponding large sieve inequality for the radial average and explain why it only improves on the pointwise bound by $1/6$.

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arXiv2015

On the Value Distribution of Two Dirichlet L-functions

We look at the values of two Dirichlet $L$-functions at the Riemann zeros (or a horizontal shift of them). Off the critical line we show that for a positive proportion of these points the pairs of values of the two $L$-functions are linearly independent over $\mathbb{R}$, which, in particular, means that their arguments are different. On the critical line we show that, up to height $T$, the values are different for $cT$ of the Riemann zeros for some positive $c$.

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