论文标题
Mertens功能的新明确界限和Riemann Zeta功能的倒数
New explicit bounds for Mertens function and the reciprocal of the Riemann zeta-function
论文作者
论文摘要
在本文中，我们为Mertens函数$ M（x）$建立了新的明确界限。特别是，我们将$ m（x）$与Riemann Zeta-function $ζ$的非平凡零的短零比较，我们可以使用最近的计算和显式界限$ζ（S）$绑定其差异。使用这种关系，我们能够证明$ m（x）\ ll x \ exp \ left（-η_1\ sqrt {\ log {x}}} \ right）$和$ m（x）\ ll x \ x \ exp \ left（-η_2（-η_2（-η_2）（-η_2（x} {x} {3/5/5}）） （\ log \ log {x}）^{ - 1/5} \ right）$对于某些$η_i> 0 $。我们以后一种形式的界限是同类的第一个明确结果。在证明这些过程的过程中，我们建立了另一个新颖的结果，即表格$ 1/ζ（σ+ it）\ ll（\ log {t}）^{2/3}（\ log \ log \ log \ log {t}）^{1/4} $。
In this paper, we establish new explicit bounds for the Mertens function $M(x)$. In particular, we compare $M(x)$ against a short-sum over the non-trivial zeros of the Riemann zeta-function $ζ(s)$, whose difference we can bound using recent computations and explicit bounds for the reciprocal of $ζ(s)$. Using this relationship, we are able to prove explicit versions of $M(x) \ll x\exp\left(-η_1 \sqrt{\log{x}}\right)$ and $M(x) \ll x\exp\left(-η_2 (\log{x})^{3/5} (\log\log{x})^{-1/5}\right)$ for some $η_i > 0$. Our bounds with the latter form are the first explicit results of their kind. In the process of proving these, we establish another novel result, namely explicit bounds of the form $1/ζ(σ+ it) \ll (\log{t})^{2/3} (\log\log{t})^{1/4}$.
