Geodesic
The Concentration Function of Additive Functions with Special Weight
Matematičeskie zametki, Tome 76 (2004) no. 2, pp. 265-285.

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Suppose that $g(n)$ is an additive real-valued function, $$ W(N)=4+\min_\lambda\lambda^2+\sum_{p}\frac1p\min(1,(g(p)-\lambda\log p)^2), \quad E(N)=4+\sum_{p,\ g(p)\ne0}\frac1p. $$ 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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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/

[1] Ruzsa I., “On the concentration of additive functions”, Acta Math. Aca. Sci. Hungar., 36 (1980), 215–232 | DOI | MR | Zbl

[2] Elliott P. D. T. A., “The concentration function of additive functions on shifted primes”, Acta Math., 173 (1994), 1–35 | DOI | MR | Zbl

[3] Timofeev N. M., “Gipoteza Erdesha–Kubilyusa o raspredelenii znachenii additivnykh funktsii na posledovatelnosti sdvinutykh prostykh chisel”, Acta Arithmetica, 58:2 (1991), 113–131 | MR | Zbl

[4] Timofeev N. M., Khripunova M. B., “O funktsii kontsentratsii additivnoi funktsii s nemultiplikativnym vesom”, Matem. zametki, 75:6 (2004), 877–894 | MR | Zbl

[5] Timofeev N. M., “Problema Khardi i Littlvuda dlya chisel, imeyuschikh zadannoe chislo prostykh delitelei”, Izv. RAN. Ser. matem., 59:6 (1995), 181–206 | MR | Zbl

[6] Tenenbaum G., Introduction à la théorie analytique et probabiliste des nombres, Institut Elie Cartan, 13. Université de Nancy I, 1990

[7] Karatsuba A. A., Osnovy analiticheskoi teorii chisel, 2-e izd., Nauka, M., 1983

[8] Halberstam H., Richert H. E., Sieve Methods, Academic Press, London–New York, 1974 | Zbl

[9] Hall R., Tenenbaum G., 90 Divisors, Cambridge University Press, Cambridge, 1988

[10] Khooli K., Primenenie metodov resheta v teorii chisel, Nauka, M., 1987