Related papers: Distribution of prime geodesic traces

We study the distribution of closed geodesics for the modular surface. We improve the error term in the prime geodesic theorem, and obtain results on prime geodesics in very short intervals conditionally on the generalized Riemann…

We reduce the exponent in the error term of the prime geodesic theorem for compact Riemann surfaces from $\frac{3}{4}$ to $\frac{7}{10}$ outside a set of finite logarithmic measure.

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…

Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L^2(G/H) associated to test…

We strengthen the recent result of Cherubini and Guerreiro on the square mean of the error term in the prime geodesic theorem for $\mathrm{PSL}_2(\mathbb{Z})$. We also develop a short interval version of this result.

We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the…

We give a new proof of the best presently known error term in the prime geodesic theorem for compact Riemann surfaces, without the assumption of excluding a set of finite logarithmic measure. Stronger implications of the Gallagher-Koyama…

The prime geodesic theorem for regular geodesics in a higher rank locally symmetric space is proved. An application to class numbers is given. The proof relies on a Lefschetz formula that is based on work of Andreas Juhl.

We generalize Koyama's $7/10$ bound of the error term in the prime geodesic theorems to the principal congruence subgroups for quaternion algebras. Our method avoids the spectral side of the Jacquet--Langlands correspondences, and relates…

Dirichlet's theorem on arithmetic progressions called as Dirichlet prime number theorem is a classical result in number theory. Atle Selberg\cite{Selberg} gave an elementary proof of this theorem. In this article we give an alternative…

Under the generalized Lindel\"{o}f hypothesis, the exponent in the error term of the prime geodesic theorem for the modular surface is reduced to $\frac{5}{8}+\varepsilon $ outside a set of finite logarithmic measure.

By extending the new supersymmetric localization principle introduced in \cite{Choi:2021yuz}, we present a path integral derivation of the Selberg trace formula on arbitrary compact Riemann surfaces, including the case of arbitrary…

In this paper, we prove a discrete analog of the Selberg Trace Formula for the group $\text{GL}_{3}(\mathbb{F}_q).$ By considering a cubic extension of the finite field $\mathbb{F}_q$, we define an analog of the upper half space and an…

Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet $L$-functions. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square…

A prime geodesic theorem is proven for singular geodesics in quotients of SL(4). This is a case where regularity assumptions of previous papers fail. As a consequence, the analysis becomes much more involved. For applications in number…

We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in terms of quadratic twists of certain…

We present some new sharp constructions for the Szemer\'{e}di-Trotter theorem. These constructions generalize previous work of Erd\H{o}s, Elekes, Sheffer and Silier, Guth and Silier, and the author. In the past, arguments showing the…

We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg…

We present three methodological improvements of the "SCK CEN approach" for Bayesian inference of the radionuclide inventory in radioactive waste drums, from radiological measurements. First we resort to the Dirichlet distribution for the…

A prime geodesic theorem for singular geodesics in a locally symmetric space is proved. As an application, an asymptotic formula for units in number fields is given.