The number of powers of 2 in a representation of large even integers II

It is proved that for any integerk≥ 54 000, there isN _k >0 depending onk only such that every even integer ≥N _k is a sum of two odd prime numbers andk powers of 2. Project partially supported by RGC Research Grant (No.HKU 7122/97P) and Post-Doctoral Fellowship of the University of Hong Kong.     

芯片式流通池顺序注射可更新表面反射光谱法用于酶反应检测

将芯片式流通池顺序注射可更新表面反射光谱法用于酶反应检测。HRP催化H2O2氧化BPR底物的反应用于对H2O2的检测。此反应体系与葡萄糖氧化酶联用，用于对血清中葡萄糖的检测。     

Jordan函数r次方Jrk（n）的均值误差项研究

设ｋ，ｒ分别是自然数和非零整数，Ｊｋ（ｎ）是Ｊｏｒｄａｎ函数．文［３］求出了和式∑ｎ≤ｘＪｒｋ（ｎ）的渐近公式．以Ｅ（ｘ；ｋ，ｒ）表示该公式中的误差项，本文研究了Ｅ（ｘ；ｋ，ｒ）的算术均值和积分均值     

Integral points on quadrics with prime coordinates

Let f (x ₁, . . . , x _s ) be a regular indefinite integral quadratic form, and t an integer. Denote by V the affine quadric {x : f (x) = t}, and by \({V(\mathbb {P})}\) the set of \({{\bf x}\in V}\) whose coordinates are simultaneously prime. It is proved that, under suitable conditions, \({V(\mathbb{P})}\) is Zariski dense in V as long as s ≥ 10.     

Integral points on quadrics with prime coordinates

Let f (x ₁, . . . , x _s ) be a regular indefinite integral quadratic form, and t an integer. Denote by V the affine quadric {x : f (x) = t}, and by V(\mathbb P){V(\mathbb {P})} the set of x ? V{{\bf x}\in V} whose coordinates are simultaneously prime. It is proved that, under suitable conditions, V(\mathbbP){V(\mathbb{P})} is Zariski dense in V as long as s ≥ 10.     

A proof of Selberg's orthogonality for automorphic L-functions

Let π and π′ be automorphic irreducible cuspidal representations of GL_m(Q_A) and GL_m′(Q_A), respectively. Assume that π and π′ are unitary and at least one of them is self-contragredient. In this article we will give an unconditional proof of an orthogonality for π and π′, weighted by the von Mangoldt function Λ(n) and 1−n/x. We then remove the weighting factor 1−n/x and prove the Selberg orthogonality conjecture for automorphic L-functions L(s,π) and L(s,π′), unconditionally for m≤4 and m′≤4, and under the Hypothesis H of Rudnick and Sarnak [20] in other cases. This proof of Selberg's orthogonality removes such an assumption in the computation of superposition distribution of normalized nontrivial zeros of distinct automorphic L-functions by Liu and Ye [12].     

On an Error Term Related to the r-th Power of the Euler Totient Function

ＯｎａｎＥｒｒｏｒＴｅｒｍＲｅｌａｔｅｄｔｏｔｈｅｒ－ｔｈＰｏｗｅｒｏｆｔｈｅＥｕｌｅｒＴｏｔｉｅｎｔＦｕｎｃｔｉｏｎＬｉｕＪｉａｎｙａ（刘建亚）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａｎｄｏｎｇＵｎｉｖｅｒｓｉｔｙ，Ｊｉｎａｎ，２５...     

Subconvexity bounds for Rankin–Selberg L-functions for congruence subgroups

Estimation of shifted sums of Fourier coefficients of cusp forms plays crucial roles in analytic number theory. Its known region of holomorphy and bounds, however, depend on bounds toward the general Ramanujan conjecture. In this article, we extended such a shifted sum meromorphically to a larger half plane Res>1/2 and proved a better bound. As an application, we then proved a subconvexity bound for Rankin–Selberg L-functions which does not rely on bounds toward the Ramanujan conjecture: Let f be either a holomorphic cusp form of weight k, or a Maass cusp form with Laplace eigenvalue 1/4+k², for . Let g be a fixed holomorphic or Maass cusp form. What we obtained is the following bound for the L-function L(s,fg) in the k aspect:

L(1/2+it,fg)k^{1−1/(8+4θ)+ε},