论文标题
关于riemann zeta函数的平均而言较小的乘法函数
On multiplicative functions which are small on average and zero free regions for the Riemann zeta function
论文作者
论文摘要
在简短的说明中，我们证明了以下结果：如果完全乘法函数$ f：\ mathbb {n} \ to [-1,1] $平均很小，从某种意义上说，$ \ sum_ {n \ leq x} f（n）\ ll x^{1-δ} $ f（1）= 0 $，然后我们为任何$ε> 0 $，$ \ sum_ {p \ leq x}（1+f（p））\ log p \ ll x^{1-δ+ε} $获得。此外，存在这种$ f $的必要条件是Riemann Zeta函数$ζ$在半平面$ re（s）> 1-δ$中没有零。
In this short note we prove the following result: If a completely multiplicative function $f:\mathbb{N}\to[-1,1]$ is small on average in the sense that $\sum_{n\leq x}f(n)\ll x^{1-δ}$, for some $δ>0$, and if the Dirichlet series of $f$, say $F(s)$, is such that $F(1)=0$, then we obtain that for any $ε>0$, $\sum_{p\leq x}(1+f(p))\log p\ll x^{1-δ+ε}$. Moreover, a necessary condition for the existence of such $f$ is that the Riemann zeta function $ζ(s)$ has no zeros in the half plane $Re(s)>1-δ$.
