Prime geodesic theorem

Yingchun Cai

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 1, page 59-72
  • ISSN: 1246-7405

Abstract

top
Let Γ denote the modular group P S L ( 2 , Z ) . In this paper it is proved that π Γ ( x ) = li x + O ( x 71 102 + ϵ ) , ϵ > 0 . The exponent 71 102 improves the exponent 7 10 obtained by W. Z. Luo and P. Sarnak.

How to cite

top

Cai, Yingchun. "Prime geodesic theorem." Journal de théorie des nombres de Bordeaux 14.1 (2002): 59-72. <http://eudml.org/doc/248896>.

@article{Cai2002,
abstract = {Let $\Gamma $ denote the modular group $PSL(2, Z)$. In this paper it is proved that $\pi _\Gamma (x) = \text\{li\} x + O(x^\{\frac\{71\}\{102\} + \epsilon \}), \epsilon &gt; 0$. The exponent $\frac\{71\}\{102\}$ improves the exponent $\frac\{7\}\{10\}$ obtained by W. Z. Luo and P. Sarnak.},
author = {Cai, Yingchun},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {prime geodesic theorem; modular group},
language = {eng},
number = {1},
pages = {59-72},
publisher = {Université Bordeaux I},
title = {Prime geodesic theorem},
url = {http://eudml.org/doc/248896},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Cai, Yingchun
TI - Prime geodesic theorem
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 1
SP - 59
EP - 72
AB - Let $\Gamma $ denote the modular group $PSL(2, Z)$. In this paper it is proved that $\pi _\Gamma (x) = \text{li} x + O(x^{\frac{71}{102} + \epsilon }), \epsilon &gt; 0$. The exponent $\frac{71}{102}$ improves the exponent $\frac{7}{10}$ obtained by W. Z. Luo and P. Sarnak.
LA - eng
KW - prime geodesic theorem; modular group
UR - http://eudml.org/doc/248896
ER -

References

top
  1. [1] D. Burgess, The character sums estimate with r = 3. J. London. Math. Soc.(2) 33 (1986), 219-226. Zbl0593.10033MR838632
  2. [2] D. Hejhal, The Selberg Trace Formula for PSL(2, R), I. Lecture Notes in Math.548, Berlin- New York, 1976. Zbl0347.10018
  3. [3] D. Hejhal, The Selberg Trace Formula and the Riemann Zeta-function. Duke Math. J.43 (1976), 441-482. Zbl0346.10010MR414490
  4. [4] H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen II. Math. Ann.142 (1961), 385-398 and 143 (1961), 463-464. Zbl0101.05702MR154980
  5. [5] H. Iwaniec, Prime geodesic theorem. J. Reine Angew. Math.349 (1984), 136-159. Zbl0527.10021MR743969
  6. [6] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms. Biblioteca de la Revista Matemática Iberoamericana, 1995 Zbl0847.11028MR1325466
  7. [7] N.V. Kuznetzov, An arithmetical form of the Selberg Trace formula and the distribution of the norms of primitive hyperbolic classes of the modular group. Akad. Nauk. SSSR., Khabarovsk, 1978. Zbl0381.10022
  8. [8] N.V. Kuznetzov, The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums (Russian). Mat. Sb. (N.S.)111(153) (1980), no. 3, 334-383, 479. Zbl0427.10016MR568983
  9. [9] Z. Luo, P. Sarnak, Quantum ergodicity of eigenfunctions on PSL2(Z). Inst. Hautes. Etudes Sci. Publ. Math.81 (1995), 207-237. Zbl0852.11024
  10. [10] Z. Luo, Z. Rudnick, P. Sarnak, On Selberg's eigenvalue Conjecture. Geom. Funct. Anal.5 (1995), 387-401. Zbl0844.11038MR1334872
  11. [11] R.A. Rankin, Contribution to the theory of Ramanujan's function τ(n) and similar arithmetical functions. Proc. Cambridge. Phil. Soc.35 (1939), 351—372. Zbl0021.39201
  12. [12] P. Sarnak, Class number of infinitely binary quadratic forms. J. Number Theory15 (1982), 229-247. Zbl0499.10021MR675187
  13. [13] A.B. Venkov, Spectral theory of automorphic functions, (in Russian). Trudy Math. Inst. Steklova153 (1981), 1-171. Zbl0483.10029MR665585

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.