论文标题
戈德巴赫猜想中振荡的大小
The size of oscillations in the Goldbach conjecture
论文作者
论文摘要
令$ r（n）= \ sum_ {a+b = n}λ（a）λ（b）$，其中$λ（\ cdot）$是von mangoldt函数。经常研究功能$ r（n）$与戈德巴赫的猜想有关。在Riemann假设（rh）上，众所周知，$ \ sum_ {n \ leq x} r（n）= x^2/2-4x^{3/2} g（x） + o（x^{1 + am}）$，其中$ g（x） \ frac {x^{iγ}} {（\ frac {1} {2} +iγ）（\ frac {3} {2} {2} +iγ）} $，总和超过了Riemann Zeta Zeros在上半Plane中的非底线Zeros的序列。我们（在RH上）证明，每个不平等$ g（x）<-0.02093 $和$ g（x）> 0.02092 $无限地保持，并在Zeta函数零的线性独立性的假设下建立改进的界限。我们还表明，我们获得的界限非常接近最佳。
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.
