Geodesic
Arithmetic properties of functions associated with Dirichlet $L$-functions
Matematičeskie zametki, Tome 52 (1992) no. 1, pp. 120-127.

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This article considers functions of the form $E_d(x,j)=\sum^{\infty}_{k=1}\bigl(\frac kd\bigr)x^hk^{-j}$, where $\bigl(\frac kd\bigr)$ is the Jacobian quadratic symbol, $d$ runs through all natural divisors of a given number $r,j=1,\ldots,s$. Linear independence is proved over the field of rationals for the values of these functions on small rational $x$. Effective lower bounds are obtained for linear forms with rational integral coefficients. The results, in particular, strengthen known bounds for polylogarithms. Hermite–Padé approximations of the second kind are used. 
@article{MZM_1992_52_1_a16,
     author = {V. N. Sorokin},
     title = {Arithmetic properties of functions associated with {Dirichlet} $L$-functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {120--127},
     publisher = {mathdoc},
     volume = {52},
     number = {1},
     year = {1992},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1992_52_1_a16/}
}
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V. N. Sorokin. Arithmetic properties of functions associated with Dirichlet $L$-functions. Matematičeskie zametki, Tome 52 (1992) no. 1, pp. 120-127. http://geodesic.mathdoc.fr/item/MZM_1992_52_1_a16/