We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences ${\sum }_{i=1}^k{a}_i{x}_i\equiv b(modn)$. In particular, we obtain explicit expressions for the number of solutions, where $x_i$’s are squares modulo $n$. In addition, we obtain expressions for the number of solutions with order restrictions $x_1\ge \cdots \ge x_k$ or with strict order restrictions $x_1>\cdots >x_k$ in some special cases. In these results, the expressions for the number of solutions involve Ramanujan...

Certain generating fuctions for multiple zeta values are expressed as values at some point of solutions of linear meromorphic differential equations. We apply asymptotic expansion methods (like the WKB method and the Stokes operators) to solutions of these equations. In this way we give a new proof of the Euler formula ζ(2) = π²/6. In further papers we plan to apply this method to study some third order hypergeometric equation related to ζ(3).

In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let $\Lambda =b_{2}log{\alpha }_2-b_{1}log{\alpha }_1-b_{3}log{\alpha }_3\ne 0$ with $b_1,b_2,b_3$ positive integers and ${\alpha }_1,{\alpha }_2,{\alpha }_3$ positive multiplicatively independent rational numbers greater than $1$. Let ${\alpha }_{j1}={\alpha }_{j1}/{\alpha }_{j2}$ with ${\alpha }_{j1},{\alpha }_{j2}$ coprime positive integers $(j=1,2,3)$. Let ${\alpha }_j\ge \text{max}\{{\alpha }_{j1},e\}$ and assume that gcd$(b_1,b_2,b_3)=1.$ Let$$b^{\text{'}}=\left(\frac{b_2}{log{\alpha }_1}+\frac{b_1}{log{\alpha }_2}\right)\left(\frac{b_2}{log{\alpha }_3}+\frac{b_3}{log{\alpha }_2}\right)$$and assume that $B\ge \text{max}\{10,log{b}^{\text{'}}\}.$ We prove that either $\{b_1,b_2,b_3\}$ is $\left(c_4,B\right)$-linearly dependent over $\mathbb{Z}$ (with respect to $\left(a_1,a_2,a_3\right)$) or$$\Lambda \>exp\left\{-C{B}^2\left(\prod _{j=1}^{3}log{a}_j\right)\right\},$$where $c_4$ and $C=c_1{c}_{2}log\rho +\delta$ are given in the tables...