相关论文: On Koyama's refinement of the prime geodesic theor…

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 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…

In his recent research, the author improved the error term in the prime geodesic theorem for compact, even-dimensional, rank one locally symmetric spaces. It turned out that the obtained estimate $O(x^{2\rho-\frac{\rho}{n}}(\log x)^{-1})$…

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…

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…

Taking the Iwaniec explicit formula as a starting point, we give a short proof of a more precise $\frac{2}{3}$ bound for the exponent in the error term of the Gallagher-type prime geodesic theorem for the modular surface.

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.

Through the Selberg zeta approach, we reduce the exponent in the error term of the prime geodesic theorem for cocompact Kleinian groups or Bianchi groups from Sarnak's $\frac{5}{3}$ to $\frac{3}{2}$. At the cost of excluding a set of finite…

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 develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.

This note complements a recent paper of Chatzakos, Harcos and Kaneko \cite{CHK}. We use a Dirichlet style Prime Geodesic Theorem to improve the error term estimate in loc. cit. at the cost of lowering the resolution. The proof relies on the…

Assuming the Riemann Hypothesis, we derive explicit bounds for the error terms in short interval analogues of the prime number theorem and Mertens' theorems using a smoothing argument. Our results improve upon previous bounds in both…

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…

Soundararajan and Young (2013) proposed a new approach to improve the error term of the prime geodesic theorem for the modular group and actually obtained the exponent 25/36 of the error term. In the present paper, we sharpen it to 9/13.

We develop a new Riemannian descent algorithm that relies on momentum to improve over existing first-order methods for geodesically convex optimization. In contrast, accelerated convergence rates proved in prior work have only been shown to…

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the…

We show a Prime Geodesic Theorem for the group SL3(Z), counting those geodesics whose lifts lie in the split Cartan subgroup. This is the first arithmetic Prime Geodesic Theorem of higher rank for a non-cocompact group.

We show a smoothed version of Goldston-Pintz-Yildirim's sifting argument to detect small gaps between primes, which has a higly flexible error term. Our argument is applicable to high dimensional Selberg sieve situations as well, although…

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.

The master theorem, introduced by Richter-Gebert and generalized by Fomin and the first author, provides a method for proving incidence theorems of projective geometry using triangular tilings of surfaces. We investigate which incidence…