Geodesic
On the general additive divisor problem
Informatics and Automation, Number theory, algebra, and analysis, Tome 276 (2012), pp. 146-154.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain a new upper bound for the sum $\sum_{h\le H}\Delta_k(N,h)$ when $1\le H\le N$, $k\in\mathbb N$, $k\ge3$, where $\Delta_k(N,h)$ is the (expected) error term in the asymptotic formula for $\sum_{N$, and $d_k(n)$ is the divisor function generated by $\zeta(s)^k$. When $k=3$, the result improves, for $H\ge N^{1/2}$, the bound given in a recent work of Baier, Browning, Marasingha and Zhao, who dealt with the case $k=3$. 
@article{TRSPY_2012_276_a10,
     author = {Aleksandar Ivi\'c and Jie Wu},
     title = {On the general additive divisor problem},
     journal = {Informatics and Automation},
     pages = {146--154},
     publisher = {mathdoc},
     volume = {276},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2012_276_a10/}
}
TY  - JOUR
AU  - Aleksandar Ivić
AU  - Jie Wu
TI  - On the general additive divisor problem
JO  - Informatics and Automation
PY  - 2012
SP  - 146
EP  - 154
VL  - 276
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2012_276_a10/
LA  - en
ID  - TRSPY_2012_276_a10
ER  - 
%0 Journal Article
%A Aleksandar Ivić
%A Jie Wu
%T On the general additive divisor problem
%J Informatics and Automation
%D 2012
%P 146-154
%V 276
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2012_276_a10/
%G en
%F TRSPY_2012_276_a10
Aleksandar Ivić; Jie Wu. On the general additive divisor problem. Informatics and Automation, Number theory, algebra, and analysis, Tome 276 (2012), pp. 146-154. http://geodesic.mathdoc.fr/item/TRSPY_2012_276_a10/

[1] Baier S., Browning T.D., Marasingha G., Zhao L., “Averages of shifted convolutions of $d_3(n)$”, Proc. Edinburgh Math. Soc. (to appear)

[2] Conrey J.B., Gonek S.M., “High moments of the Riemann zeta-function”, Duke Math. J., 107 (2001), 577–604 | DOI | MR | Zbl

[3] Ivić A., The Riemann zeta-function: The theory of the Riemann zeta-function with applications, J. Wiley Sons, New York, 1985 ; 2nd ed., Dover Publ., Mineola, NY, 2003 | MR | Zbl | Zbl

[4] Ivić A., Lectures on mean values of the Riemann zeta function, Tata Inst. Fundam. Res. Lect. Math. Phys., 82, Springer, Berlin, 1991 | MR | Zbl

[5] Ivić A., “On the ternary additive divisor problem and the sixth moment of the zeta-function”, Sieve methods, exponential sums, and their applications in number theory, eds. Ed. by G.R.H. Greaves, G. Harman, M.N. Huxley, Cambridge Univ. Press, Cambridge, 1997, 205–243 | DOI | MR | Zbl

[6] Ivić A., “The general additive divisor problem and moments of the zeta-function”, Analytic and probabilistic methods in number theory, Proc. 2nd Intern. Conf. in Honour of J. Kubilius, Palanga (Lithuania), Sept. 23–27, 1996, New Trends Probab. Stat., 4, eds. Ed. by A. Laurinčikas et al., VSP, Utrecht, 1997, 69–89 | MR | Zbl

[7] Ivić A., Motohashi Y., “On the fourth power moment of the Riemann zeta-function”, J. Number Theory., 51 (1995), 16–45 | DOI | MR | Zbl

[8] Ivić A., Motohashi Y., “On some estimates involving the binary additive divisor problem”, Q. J. Math. Oxford. Ser. 2., 46 (1995), 471–483 | DOI | MR | Zbl

[9] Ivić A., Ouellet M., “Some new estimates in the Dirichlet divisor problem”, Acta arith., 52 (1989), 241–253 | MR | Zbl

[10] Kuznetsov N.V., “Petersson's conjecture for cusp forms of weight zero and Linnik's conjecture. Sums of Kloosterman sums”, Math. USSR. Sb., 39 (1981), 299–342 | DOI | MR | Zbl | Zbl

[11] Kuznetsov N.V., “Convolution of the Fourier coefficients of the Eisenstein–Maass series”, J. Sov. Math., 29:2 (1985), 1131–1159 | DOI | MR | Zbl | Zbl

[12] Meurman T., “On the binary additive divisor problem”, Number theory: Proc. Turku Symp. on Number Theory in Memory of Kustaa Inkeri, Turku, 1999, eds. Ed. by M. Jutila et al., W. de Gruyter, Berlin, 2001, 223–246 | MR | Zbl

[13] Motohashi Y., “The binary additive divisor problem”, Ann. Sci. Éc. Norm. Super. Sér. 4., 27 (1994), 529–572 | MR | Zbl

[14] Motohashi Y., Spectral theory of the Riemann zeta-function, Cambridge Univ. Press, Cambridge, 1997 | MR

[15] Titchmarsh E.C., The theory of the Riemann zeta-function, 2nd ed., Clarendon Press, Oxford, 1986 | MR | Zbl

[16] Vinogradov A.I., “Poincaré series in $\mathrm {SL}(3,\mathbb R)$”, J. Sov. Math., 52:3 (1990), 3022–3025 | DOI | MR

[17] Vinogradov A.I., “Analytic continuation of $\zeta _3(s,k)$ to the critical strip. Arithmetic part”, J. Sov. Math., 46:2 (1989), 1734–1759 | DOI | MR | Zbl

[18] Vinogradov A.I., “$\mathrm {SL}_n$ technique and the density conjecture”, J. Sov. Math., 53:3 (1991), 225–228 | DOI | MR | Zbl | Zbl

[19] Vinogradov A.I., Takhtadzhyan L.A., “Theory of Eisenstein series for the group $SL(3,\mathbb R)$ and its application to a binary problem”, J. Sov. Mat., 18:3 (1982), 293–324 | DOI | MR | Zbl | Zbl

[20] Vinogradov A.I., Takhtadzhyan L.A., “The zeta function of the additive divisor problem and spectral decomposition of the automorphic Laplacian”, J. Sov. Math., 36:1 (1987), 57–78 | DOI | MR | Zbl | Zbl

[21] Zhang W., “On the divisor problem”, Kexue Tongbao, 33 (1988), 1484–1485 | MR