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<trans data-src="By way of context: it is known that the prime number theorem $\pi(x) \sim x/\log x$ is (nontrivially) equivalent to the statement that $\zeta(s)$ does not vanish on the line $\Re s=1$.">通过上下文：众所周知，素数定理$\pi（x）\sim x/\log x$（非常重要）等价于$\zeta（s）$在$\Re s=1$行上不消失的语句。</trans><trans data-src="I would like to have it clarified what are the analogous equivalent statements for primes in arithmetic progressions. ">我想让它澄清什么是算术级数中质数的类似等价语句。</trans><trans data-src="In particular, I would love to have specific references to papers or books in which the following equivalences are proved (or at least stated explicitly):">特别是，我希望能具体提及证明（或至少明确说明）以下等效性的论文或书籍：</trans><trans data-src="- Dirichlet's theorem, that there are infinitely many primes in any admissible arithmetic progression (mod $q$). ">-Dirichlet定理，在任何可容许的算术级数（mod$q$）中都有无穷多个素数。</trans><trans data-src="is equivalent to the nonvanishing of $L(1,\chi)$ for all Dirichlet characters $\chi\pmod q$. ">等价于所有Dirichlet字符$\chi\pmod q$的$L（1，\chi）$的非Anishing。</trans><trans data-src="[Edit: as has been pointed out, this nonvanishing is more likely equivalent to the equality of the Dirichlet densities of primes among the reduced residue classes (mod $q$). What analytic statement could be equivalent to the mere infinitude of primes in all such classes?]">编辑：正如已经指出的那样，这种非等价性更可能等同于简化剩余类（mod$q$）中素数的Dirichlet密度相等。什么样的分析语句可以等同于所有此类类中素数无穷大</trans><trans data-src="- The prime number theorem in arithmetic progressions $\pi(x;q,a)\sim x/(\phi(q)\log x)$ is equivalent to the statement that $L(s,\chi)$ does not vanish on the line $\Re s=1$ for all Dirichlet characters $\chi\pmod q$.">-算术级数$\pi（x；q，a）\sim x/（\phi（q）\log x）$中的素数定理等价于所有Dirichlet字符$\chi\pmod q$的$L（s，\chi）$不在$\Re s=1$行上消失的语句。</trans><trans data-src="Of course, if these statements themselves are incorrect, I would like to be corrected as well as being pointed to the literature.">当然，如果这些说法本身是错误的，我希望得到纠正，并指出文献。</trans>

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$\开始组$ 我很困惑。Dirichlet定理、算术级数中的PNT以及$L（s，\chi）$对于$\Re（s）=1$的非零化是三个正确的语句。真实的陈述总是等价的。 $\端组$
– 来自MO的GH
评论 2020年4月1日7:07
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$\开始组$ 例如，从某种意义上说，这些陈述是等价的，因为类似的陈述在更广泛的意义上是等价的对于Selberg类中的L函数。您可以通过这种方式推导PNT和AP中PNT的等价性。至于狄利克雷特定理，我怀疑它是否等同于1。相反，我希望非Anishing等价于关于AP中素数的Dirichlet密度的结果。 $\端组$
– 沃约乌
评论 2020年4月1日11:32

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