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Prime geodesic theorem

Yingchun Cai — 2002

Journal de théorie des nombres de Bordeaux

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.

On the least almost-prime in arithmetic progression

Jinjiang Li;  Min Zhang;  Yingchun Cai — 2023

Czechoslovak Mathematical Journal

Let 𝒫 r denote an almost-prime with at most r prime factors, counted according to multiplicity. Suppose that a and q are positive integers satisfying ( a , q ) = 1 . Denote by 𝒫 2 ( a , q ) the least almost-prime 𝒫 2 which satisfies 𝒫 2 ≡ a ( mod q ) . It is proved that for sufficiently large q , there holds 𝒫 2 ( a , q ) ≪ q 1 . 8345 . This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range 1 . 845 in place of 1 . 8345 .

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