Estimates of generalized Chebyshev function on GL_m

Abstract

In this paper, we study the generalized Chebyshev function related to automorphic L-functions of $GL_m \left( {\mathbb{A}_\mathbb{Q} } \right)$, and estimate its asymptotic behavior with respect to the parameters of the original automorphic objects.

Keywords

Automorphic L-function / Chebyshev function / explicit formula / conductor

References

[1] Gallagher P. X. A large sieve density estimate near σ = 1. Invent Math, 1970, 11: 329-339

[2] Gallagher P. X. Some consequences of the Riemann hypothesis. Acta Arith, 1980, 37: 339-343

[3] Ingham A. E. The Distribution of Prime Numbers, 1932, Cambridge: Cambridge University Press

[4] Iwaniec H., Kowalski E. Analytic Number Theory, 2004, Providence: Amer Math Soc

[5] Liu J., Ye Y. Superpositions of distinct L-functions. Forum Math, 2002, 14: 419-455

[6] Liu J., Ye Y. Perron’s formula and the prime number theorem for automorphic L-functions. Pure Appl Math Q, 2007, 3: 481-497

[7] Luo W., Rudnick Z., Sarnak P. On Selberg’s eigenvalue conjecture. Geom Funct Anal, 1995, 5: 387-401

[8] Qu Y. The prime number theorem for automorphic L-functions for GL_m. J Number Theory, 2007, 122: 84-99

[9] Tenenbaum G. Introduction to Analytic and Probabilistic Number Theory, 1995, Cambridge: Cambridge University Press

[10] Wu J., Ye Y. Hypothesis H and the prime number theorem for automorphic representations. Funct Approx Comment Math, 2007, 37(2): 461-471