论文标题
pintz的方法是INGHAM的问题，内容涉及$ζ$ -Zeros的分布连接以及在Beurling上下文中PNT中的错误顺序
The method of Pintz for the Ingham question about the connection of distribution of $ζ$-zeros and order of the error in the PNT in the Beurling context
论文作者
论文摘要
我们证明了两个结果，概括了有关Riemann Zeta功能的经典案例及其某些概括的经典案例的长期知识。这些与Ingham的问题有关。在经典案例中，基本上鲜明的结果是由于Pintz的大约40年的工作。在这里，我们考虑一个给定的beurling Primes $ \ Mathcal {p} $的系统$δ_ {\ Mathcal {g}}（x）：=ψ_{\ Mathcal {g}}（x）（x） - x $在beurling的pnt中，其中$ψ_{\ Mathcal {g}}}（x）（x）$是$ beurling of $ o $ us $ c of $ψ（x）$。首先，我们证明，如果beurling zeta函数$ζ_ {\ Mathcal {g}} $不会在$ d（η）$中消失，则Pintz'结果的扩展保持：$ |δ__{\ Mathcal {G}}}}}（x）（x）（x）| \ le x \ exp（ - （1- \ varepsilon）ω_η（x））在第二部分中，我们证明了一个相反的内容：如果$ζ_ {\ Mathcal {g}} $在给定域中的无限为零，则类似于经典的情况，$ |δ__{\ Mathcal {g}}}（x）（x）| \ ge x \ exp（ - （1+ \ varepsilon）ω_η（x））$保持“无限频繁”。这也表明，两个主要结果都与任意的小$ \ varepsilon> 0 $不同。
We prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. These are concerned with the question of Ingham who asked for optimal and explicit order estimates for the error term $Δ(x):=ψ(x)-x$, given any zero-free region $D(η):=\{s=σ+it\in\mathbb{C}~:~ σ:=\Re s \ge 1-η(t)\}$. In the classical case essentially sharp results are due to some 40 years old work of Pintz. Here we consider a given a system of Beurling primes $\mathcal{P}$, the generated arithmetical semigroup $\mathcal{G}$ and the corresponding integer counting function $N(x)$, and the corresponding error term $Δ_{\mathcal{G}}(x):=ψ_{\mathcal{G}}(x)-x$ in the PNT of Beurling, where $ψ_{\mathcal{G}}(x)$ is the Beurling analog of $ψ(x)$. First we prove that if the Beurling zeta function $ζ_{\mathcal{G}}$ does not vanish in $D(η)$, then the extension of Pintz' result holds: $|Δ_{\mathcal{G}}(x)| \le x\exp(-(1-\varepsilon)ω_η(x))~(x>x_0(\varepsilon))$, where $ω_η(y)$ is the naturally occurring conjugate function to $η(t)$, introduced into the field by Ingham. In the second part we prove a converse: if $ζ_{\mathcal{G}}$ has an infinitude of zeroes in the given domain, then analogously to the classical case, $|Δ_{\mathcal{G}}(x)| \ge x\exp(-(1+\varepsilon)ω_η(x))$ holds "infinitely often". This also shows that both main results are sharp apart from the arbitrarily small $\varepsilon>0$.
