论文标题
$ω$ bunds for某些修改后的dirichlet字符的部分总和
$Ω$-bounds for the partial sums of some modified Dirichlet characters
论文作者
论文摘要
我们考虑了$ω$的问题，用于修改的字符的部分总和\ textit {i.e。}，一个完全乘法函数$ f $ $ f $，以便除$ f（p）=χ（p）$，除了有限数量的Primes $ p $，其中$χ$是$χ$，其中$χ$是一个原始的Dirichlet dirichlet dirichlet dirichlet cartem。我们证明，在某些特殊情况下，$ \ sum_ {n \ leq x} f（n）=ω（（（\ log x）^{| s |}）$，其中$ s $是一组Primes $ p $，其中$ f（p）\ neqχχ（p）$。这可以证明Klurman等人的猜想的校正版本，Trans。阿米尔。数学。 Soc。，374（11），2021，7967-7990。我们还计算了修改后字符的大$ k $的订单$ k $的riesz平均值，并表明Primes $ P $和$ Q $的表单$ \ log p / \ log p $的非理性数字的Diophantine属性，提供有关这些平均值的信息。
We consider the problem of $Ω$ bounds for the partial sums of a modified character, \textit{i.e.}, a completely multiplicative function $f$ such that $f(p)=χ(p)$ for all but a finite number of primes $p$, where $χ$ is a primitive Dirichlet character. We prove that in some special circumstances, $\sum_{n\leq x}f(n)=Ω((\log x)^{|S|})$, where $S$ is the set of primes $p$ where $f(p)\neq χ(p)$. This gives credence to a corrected version of a conjecture of Klurman et al., Trans. Amer. Math. Soc., 374 (11), 2021, 7967-7990. We also compute the Riesz mean of order $k$ for large $k$ of a modified character, and show that the Diophantine properties of the irrational numbers of the form $\log p / \log q$, for primes $p$ and $q$, give information on these averages.
