Prime geodesic theorem

Cai, Yingchun

Cai, Yingchun

Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 59-72.

Résumé
Abstract

Soit $Γ = P S L (2, Z)$ . On démontre que $π_{Γ} (x) = li x + O (x^{\frac{71}{102} + ϵ}), ϵ > 0$ où l’exposant $\frac{71}{102}$ améliore l’exposant $\frac{7}{10}$ précédemment obtenu par W. Z. Luo et P. Sarnak.

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

MR Zbl

@article{JTNB_2002__14_1_59_0,
     author = {Cai, Yingchun},
     title = {Prime geodesic theorem},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {59--72},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {1},
     year = {2002},
     mrnumber = {1925990},
     zbl = {1028.11030},
     language = {en},
     url = {http://archive.numdam.org/item/JTNB_2002__14_1_59_0/}
}

TY  - JOUR
AU  - Cai, Yingchun
TI  - Prime geodesic theorem
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2002
SP  - 59
EP  - 72
VL  - 14
IS  - 1
PB  - Université Bordeaux I
UR  - http://archive.numdam.org/item/JTNB_2002__14_1_59_0/
LA  - en
ID  - JTNB_2002__14_1_59_0
ER  -

%0 Journal Article
%A Cai, Yingchun
%T Prime geodesic theorem
%J Journal de théorie des nombres de Bordeaux
%D 2002
%P 59-72
%V 14
%N 1
%I Université Bordeaux I
%U http://archive.numdam.org/item/JTNB_2002__14_1_59_0/
%G en
%F JTNB_2002__14_1_59_0

Cai, Yingchun. Prime geodesic theorem. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 59-72. http://archive.numdam.org/item/JTNB_2002__14_1_59_0/

Bibliographie
Cité par

[1] D. Burgess, The character sums estimate with r = 3. J. London. Math. Soc.(2) 33 (1986), 219-226. | MR | Zbl

[2] D. Hejhal, The Selberg Trace Formula for PSL(2, R), I. Lecture Notes in Math. 548, Berlin- New York, 1976. | Zbl

[3] D. Hejhal, The Selberg Trace Formula and the Riemann Zeta-function. Duke Math. J. 43 (1976), 441-482. | MR | Zbl

[4] H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen II. Math. Ann. 142 (1961), 385-398 and 143 (1961), 463-464. | MR | Zbl

[5] H. Iwaniec, Prime geodesic theorem. J. Reine Angew. Math. 349 (1984), 136-159. | MR | Zbl

[6] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms. Biblioteca de la Revista Matemática Iberoamericana, 1995 | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[9] Z. Luo, P. Sarnak, Quantum ergodicity of eigenfunctions on PSL2(Z)\H2. Inst. Hautes. Etudes Sci. Publ. Math. 81 (1995), 207-237. | Numdam | Zbl

[10] Z. Luo, Z. Rudnick, P. Sarnak, On Selberg's eigenvalue Conjecture. Geom. Funct. Anal. 5 (1995), 387-401. | MR | Zbl

[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. | Zbl

[12] P. Sarnak, Class number of infinitely binary quadratic forms. J. Number Theory 15 (1982), 229-247. | MR | Zbl

[13] A.B. Venkov, Spectral theory of automorphic functions, (in Russian). Trudy Math. Inst. Steklova 153 (1981), 1-171. | MR | Zbl