Geodesic
Ingham divisor problem with square-free numbers
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (2011), pp. 52-55 
D. V. Goryashin. Ingham divisor problem with square-free numbers. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (2011), pp. 52-55. http://geodesic.mathdoc.fr/item/VMUMM_2011_3_a11/
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     author = {D. V. Goryashin},
     title = {Ingham divisor problem with square-free numbers},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {52--55},
     year = {2011},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2011_3_a11/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For the number $N(x)$ of solutions to the equation $aq-bc=1$ in positive integers $a,b,c$ and square-free numbers $q$ satisfying the condition $aq\leqslant x$ the asymptotic formula $$ N(x)=\sum_{n\leqslant x}2^{\omega(n)}\tau(n-1)=\xi_0 x\ln^2 x + \xi_1 x\ln x + \xi_2 x + O(x^{5/6+\varepsilon}) $$ is obtained for any $\varepsilon>0$, where $\xi_0,\xi_1,\xi_2$ are constants.