The Concentration Function of Additive Functions with Special Weight
N. M. Timofeev,
M. B. Khripunova Vladimir State Pedagogical University
Abstract:
Suppose that
$g(n)$ is an additive real-valued function,
$$
W(N)=4+\min_\lambda\lambda^2+\sum_{p<N}\frac1p\min(1,(g(p)-\lambda\log p)^2), \quad
E(N)=4+\sum_{p<N,\ g(p)\ne0}\frac1p.
$$
In this paper, we prove the existence of constants
$C_1$,
$C_2$ such that the following inequalities hold:
$$
\begin{aligned}
&\sup_a|\{n,m,k:m,k\in\mathbb Z,\ n\in\mathbb N,\ n+m^2+k^2=N,\ g(n)\in[a,a+1)\}| \le\frac{C_1N}{\sqrt{W(N)}},
\\
&\sup_a|\{n,m,k:m,k\in\mathbb Z,\ n\in\mathbb N,\ n+m^2+k^2=N,\ g(n)=a\}|
\le\frac{C_2N}{\sqrt{E(N)}}.
\end{aligned}
$$
The obtained estimates are order-sharp.
UDC:
511 Received: 10.11.2001
DOI:
10.4213/mzm105