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arXiv 2006.14742math.NT
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哥德巴赫猜想中振荡的大小

The size of oscillations in the Goldbach conjecture

Michael J. Mossinghoff, Timothy S. Trudgian

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11 pages
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摘要：

设$R(n) = \sum_{a+b=n} Λ(a)Λ(b)$，其中$Λ(\cdot)$是von Mangoldt函数。函数$R(n)$经常与哥德巴赫猜想联系起来研究。在黎曼假设（RH）上，已知$\sum_{n\leq x} R(n) = x^2/2 - 4x^{3/2} G(x) + O(x^{1+ε})$，其中$G(x)=\Re \sum_{γ>0} \frac{x^{iγ}}{(\frac{1}{2} + iγ)(\frac{3}{2} + iγ)}$和的和是在上半平面中的黎曼ζ函数的非平凡零点的纵坐标上。我们证明（RH上），每个不等式$G(x) < -0.02093$和$G(x)> 0.02092$保持无限经常，并建立改进的边界下的zeta函数的零点线性无关的假设。我们还表明，我们得到的界限是非常接近最优的。

英文摘要：

Let $R(n) = \sum_{a+b=n} Λ(a)Λ(b)$, where $Λ(\cdot)$ is the von Mangoldt function. The function $R(n)$ is often studied in connection with Goldbach's conjecture. On the Riemann hypothesis (RH) it is known that $\sum_{n\leq x} R(n) = x^2/2 - 4x^{3/2} G(x) + O(x^{1+ε})$, where $G(x)=\Re \sum_{γ>0} \frac{x^{iγ}}{(\frac{1}{2} + iγ)(\frac{3}{2} + iγ)}$ and the sum is over the ordinates of the nontrivial zeros of the Riemann zeta function in the upper half-plane. We prove (on RH) that each of the inequalities $G(x) < -0.02093$ and $G(x)> 0.02092$ hold infinitely often, and establish improved bounds under an assumption of linearly independence for zeros of the zeta function. We also show that the bounds we obtain are very close to optimal.

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