Geodesic
The Concentration Function of Additive Functions with Special Weight
Matematičeskie zametki, Tome 76 (2004) no. 2, pp. 265-285 
N. M. Timofeev; M. B. Khripunova. The Concentration Function of Additive Functions with Special Weight. Matematičeskie zametki, Tome 76 (2004) no. 2, pp. 265-285. http://geodesic.mathdoc.fr/item/MZM_2004_76_2_a9/
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     title = {The {Concentration} {Function} of {Additive} {Functions} with {Special} {Weight}},
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     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_2_a9/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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