论文标题
算术进展中的素数的第一时刻：超越西格尔 - 瓦尔菲斯范围
The first moment of primes in arithmetic progressions: Beyond the Siegel-Walfisz range
论文作者
论文摘要
我们研究了进度中的数量的第一刻$ \ sum _ {\ ordack {q \ leq x/n \\（q，a）= 1}}} \ big（ψ（x; q，a） - \ frac x {φ（q）} {φ（q）} \ big）我们无条件地表明，当$ a = 1 $时，对负值存在很大的偏见，对于$ n \ leq {\ rm e}^{c \ sqrt {\ log x}} $而言，均一致。该证明将作者的最新结果结合在分散方法中的错误项。更一般而言，对于$ a \ in \ Mathbb z \ setMinus \ {0 \} $，我们证明了估计，这些估计值考虑了Landau-Siegel Zeros的潜在存在（或不存在）。
We investigate the first moment of primes in progressions $$ \sum_{\substack{q\leq x/N \\ (q,a)=1}} \Big(ψ(x; q, a) - \frac x{φ(q)}\Big) $$ as $x, N \to \infty$. We show unconditionally that, when $a=1$, there is a significant bias towards negative values, uniformly for $N\leq {\rm e}^{c\sqrt{\log x}}$. The proof combines recent results of the authors on the first moment and on the error term in the dispersion method. More generally, for $a \in \mathbb Z\setminus\{0\}$ we prove estimates that take into account the potential existence (or inexistence) of Landau-Siegel zeros.
