论文标题
质数定理中误差项的均方根
The mean square of the error term in the prime number theorem
论文作者
论文摘要
我们表明，在Riemann假设上，$ \ limsup_ {x \ to \ infty} i（x）/x^{2} \ leq 0.8603 $，其中$ i（x）= \ int_x^{2x} {2x}（2x}（ψ（ψ（x）-x）-x）-x）^2 \，dx。无条件地说，$ \ frac {1} {5 \，374} \ leq i（x）/x^2 $用于足够大的$ x $，并且$ i（x）/x^{2} $没有限制为$ x \ rightarrow \ rightarrow \ infty $。
We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq 0.8603$, where $I(X) = \int_X^{2X} (ψ(x)-x)^2\,dx.$ This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that $\frac{1}{5\,374}\leq I(X)/X^2 $ for sufficiently large $X$, and that the $I(X)/X^{2}$ has no limit as $X\rightarrow\infty$.
