The function $ε(x)=\mbox{li}(x)-π(x)$ is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions $ε_θ(x)=\mbox{li}[θ(x)]-π(x)$ and $ε_ψ(x)=\mbox{li}[ψ(x)]-π(x)$ are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $θ(x)=\sum_{p \le x} \log p$ and $ψ(x)=\sum_{n=1}^x Λ(n)$ , respectively, $\mbox{li}(x)$ is the logarithmic integral , $μ(n)$ and $Λ(n)$ are the Möbius and the Von Mangoldt functions). Negative jumps in the above functions $ε$ , $ε_θ$ and $ε_ψ$ may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One denotes $j_p=\mbox{li}(p)-\mbox{li}(p-1)$ and one investigates the jumps $j_p$ , $j_{θ(p)}$ and $j_{ψ(p)}$ . In particular, $j_p<1$ , and $j_{θ(p)}>1$ for $p<10^{11}$ . Besides, $j_{ψ(p)}<1$ for any odd $p \in \mathcal{\mbox{Ch}}$ , an infinite set of so-called { \it Chebyshev primes } with partial list $\{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, \ldots\}$ . We establish a few properties of the set $\mathcal{\mbox{Ch}}$ , give accurate approximations of the jump $j_{ψ(p)}$ and relate the derivation of $\mbox{Ch}$ to the explicit Mangoldt formula for $ψ(x)$ . In the context of RH, we introduce the so-called { \it Riemann primes} as champions of the function $ψ(p_n^l)-p_n^l$ (or of the function $θ(p_n^l)-p_n^l$ ). Finally, we find a { \it good} prime counting function $S_N(x)=\sum_{n=1}^N \frac{μ(n)}{n}\mbox{li}[ψ(x)^{1/n}]$ , that is found to be much better than the standard Riemann prime counting function .
英文摘要：
The function $ε(x)=\mbox{li}(x)-π(x)$ is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions $ε_θ(x)=\mbox{li}[θ(x)]-π(x)$ and $ε_ψ(x)=\mbox{li}[ψ(x)]-π(x)$ are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $θ(x)=\sum_{p \le x} \log p$ and $ψ(x)=\sum_{n=1}^x Λ(n)$, respectively, $\mbox{li}(x)$ is the logarithmic integral, $μ(n)$ and $Λ(n)$ are the Möbius and the Von Mangoldt functions). Negative jumps in the above functions $ε$, $ε_θ$ and $ε_ψ$ may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One denotes $j_p=\mbox{li}(p)-\mbox{li}(p-1)$ and one investigates the jumps $j_p$, $j_{θ(p)}$ and $j_{ψ(p)}$. In particular, $j_p<1$, and $j_{θ(p)}>1$ for $p<10^{11}$. Besides, $j_{ψ(p)}<1$ for any odd $p \in \mathcal{\mbox{Ch}}$, an infinite set of so-called {\it Chebyshev primes } with partial list $\{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, \ldots\}$. We establish a few properties of the set $\mathcal{\mbox{Ch}}$, give accurate approximations of the jump $j_{ψ(p)}$ and relate the derivation of $\mbox{Ch}$ to the explicit Mangoldt formula for $ψ(x)$. In the context of RH, we introduce the so-called {\it Riemann primes} as champions of the function $ψ(p_n^l)-p_n^l$ (or of the function $θ(p_n^l)-p_n^l$ ). Finally, we find a {\it good} prime counting function $S_N(x)=\sum_{n=1}^N \frac{μ(n)}{n}\mbox{li}[ψ(x)^{1/n}]$, that is found to be much better than the standard Riemann prime counting function.
The asymptotic estimation of prime ideals in imaginary quadratic fields and Chebyshev's bias
School of Mathematics, Nanjing University（南京大学数学系）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心；中国科学院大学）；University of the Chinese Academy of Sciences
在两个假设下，我们确定两个函数之间的差的分布，每个函数计算给定的模q的等差数列中的数< x和两个素数的乘积。这两个假设是（i）模q的Dirichlet L-函数的扩展黎曼假设和（ii）这些L-函数的非平凡零点的虚部在有理数上是线性无关的。我们的结果类似于Rubinstein和Sarnak关于等差数列中素数的结果。特别是，我们表明，两个素数的产品的偏见总是从素数的偏见逆转。例如，Rubinstein和Sarnak在（i）和（ii）中表明，在99.6%的情况下，在x之前为3 mod 4的素数比为1 mod 4的素数多，而在89.4%的情况下，两个为1 mod 4的素数的乘积比3 mod 4的素数多。
An expository review of the Chebyshev-Sylvester method in prime number theory
McGill University（麦吉尔大学）；National University of Mongolia（蒙古国立大学）；Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences（蒙古科学院数学与数字技术研究所）
