Abstract:
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.
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