Zero density of L-functions related to Maass forms

让 f (z) 是为 SL 2 的一种 Hecke-Maass 尖顶形式() ，并且让 L (s， f ) 相应自形的 L 功能被联系到 f。为足够地大的 T，让 N (， T ) 是零的数字 L 的 =+i (s， f ) 与|T，，零根据复合正在被数。在这份报纸，我们为 3/4 得到那 1，在那里存在经常的 C =C() 以便 N (， T ) T 2 (1 )/(logT ) C，它改进以前的结果。

The Generalized Prime Number Theorem for Automorphic L-Functions

Let π and π' be automorphic irreducible cuspidal representations of GLm(QA) and GLm'(QA),respectively,and L(s,π×■) be the Rankin-Selberg L-function attached to π and π'.Without assuming the Generalized Ramanujan Conjecture(GRC),the author gives the generalized prime number theorem for L(s,π×■) when π≌π'.The result generalizes the corresponding result of Liu and Ye in 2007.

On the first negative Hecke eigenvalue of an automorphic representation of GL_(2)(A_(Q))

Letπbe a self-dual irreducible cuspidal automorphic representation of GL_(2)(A_(Q))with trivial central character.Its Hecke eigenvalue λπ(n)is a real multiplicative function in n.We show that λπ(n)<0 for some n<<Q^(2/5)_(π),where Qπdenotes(a special value of)the analytic conductor.The value 2/5 is the first explicit exponent for Hecke-Maass newforms.

Estimates for Fourier Coefficients of Cusp Forms in Weight Aspect

Let f be a holomorphic Hecke eigenform of weight k for the modular groupΓ = SL2(Z) and let λf(n) be the n-th normalized Fourier coefficient. In this paper, by a new estimate of the second integral moment of the symmetric square L-function related to f, the estimate 1λf(n21) x2 k2(log(x + k))6n≤x is established, which improves the previous result.

Waring-Goldbach problem for fourth powers in short intervals

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of almost equal fourth prime powers.