Geodesic
The sum of the values of the divisor function in arithmetic progressions whose difference is a~power of an odd prime
Izvestiya. Mathematics , Tome 15 (1980) no. 1, pp. 145-160.

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

For $D=p^m$, with $p$ a fixed odd prime, $D\leqslant x^{3/8-\varepsilon}$ and $(l,D)=1$, the asymptotic formula $$ \sum_{\substack{n\leqslant x\\n\equiv l\!\!\!\!\pmod D}}\tau_k(n)=\frac{xQ_k(\log x)}{\varphi(D)}+O\biggl(\frac{x^{1-\varkappa}}{\varphi(D)}\biggr), $$ is proved, where $\tau_k(n)$ is the number of positive integer solutions of $x_1\cdots x_k=n$, $Q_k(z)$ is a polynomial of degree $k-1$ in $z$ with coefficients depending on $k$ and $p$, $\varkappa=\min\{\varepsilon/16,\beta/k^3\}$ with $\beta$ a positive constant depending on $p$, and the constant involved in the order $O$ depends on $k$, $p$ and $\varepsilon$. The proof relies on an idea of A. A. Karatsuba that permits one to solve this problem by means of a scheme for a ternary additive problem. Bibliography: 10 titles. 
@article{IM2_1980_15_1_a5,
     author = {M. M. Petechuk},
     title = {The sum of the values of the divisor function in arithmetic progressions whose difference is a~power of an odd prime},
     journal = {Izvestiya. Mathematics },
     pages = {145--160},
     publisher = {mathdoc},
     volume = {15},
     number = {1},
     year = {1980},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1980_15_1_a5/}
}
TY  - JOUR
AU  - M. M. Petechuk
TI  - The sum of the values of the divisor function in arithmetic progressions whose difference is a~power of an odd prime
JO  - Izvestiya. Mathematics 
PY  - 1980
SP  - 145
EP  - 160
VL  - 15
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1980_15_1_a5/
LA  - en
ID  - IM2_1980_15_1_a5
ER  - 
%0 Journal Article
%A M. M. Petechuk
%T The sum of the values of the divisor function in arithmetic progressions whose difference is a~power of an odd prime
%J Izvestiya. Mathematics 
%D 1980
%P 145-160
%V 15
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1980_15_1_a5/
%G en
%F IM2_1980_15_1_a5
M. M. Petechuk. The sum of the values of the divisor function in arithmetic progressions whose difference is a~power of an odd prime. Izvestiya. Mathematics , Tome 15 (1980) no. 1, pp. 145-160. http://geodesic.mathdoc.fr/item/IM2_1980_15_1_a5/

[1] Vinogradov I. M., Izbrannye trudy, AN SSSR, M., 1952 | MR | Zbl

[2] Vinogradov I. M., Osobye varianty metoda trigonometricheskikh summ, Nauka, M., 1976 | MR

[3] Karatsuba A. A., “Raspredelenie proizvedenii sdvinutykh prostykh chisel v arifmeticheskikh progressiyakh”, Dokl. AN SSSR, 192:4 (1970), 724–727 | Zbl

[4] Karatsuba A. A., “Trigonometricheskie summy spetsialnogo vida i ikh prilozheniya”, Izv. AN SSSR. Ser. matem., 28 (1964), 237–248 | Zbl

[5] Postnikov A. G., “O summe kharakterov po modulyu, ravnomu stepeni prostogo chisla”, Izv. AN SSSR. Ser. matem., 19 (1955), 11–16 | MR | Zbl

[6] Lavrik A. F., “Funktsionalnoe uravnenie dlya $L$-funktsii Dirikhle i zadacha delitelei v arifmeticheskikh progressiyakh”, Izv. AN SSSR. Ser. matem., 30 (1966), 433–448 | MR | Zbl

[7] Chubarikov V. N., “Utochnenie granitsy nulei $L$-ryadov Dirikhle po modulyu, ravnomu stepeni prostogo chisla”, Vestnik MGU, 1973, no. 2, 46–52 | Zbl

[8] Edgorov Zh., “Zadacha delitelei v spetsialnykh arifmeticheskikh progressiyakh”, Izv. AN UzSSR, 1977, no. 2, 9–13 | MR | Zbl

[9] Barban M. B., Linnik Yu. V., Chudakov N. G., “O raspredelenii prostykh chisel v korotkikh progressiyakh $\mod{P^n}$”, Dokl. AN SSSR, 154:4 (1964), 751–753 | MR | Zbl

[10] Titchmarsh E. K., Teoriya dzeta-funktsii Rimana, IL, M., 1953