The number of solutions of a homogeneous linear congruence (Q2880460)

For given integers \(k,n,b_i \in \mathbb N\) and \(a_i \in \mathbb Z\), let \(N_n(\mathbf a, \mathbf b)\) denote the number of solutions of the congruence NEWLINE\[NEWLINE a_1 x_1 + \dots + a_k x_k \equiv 0 \pmod n\tag{1}NEWLINE\]NEWLINE with \(0 \leq x_i \leq b_i\) for all \(1 \leq i \leq k\). The main result of this paper states thatNEWLINENEWLINENEWLINE\[NEWLINE N_n(\mathbf a, \mathbf b) \geq 2^{1-n} \prod_{i=1}^k (b_i+1).\tag{2}NEWLINE\]NEWLINENEWLINENEWLINEThis result was conjectured by \textit{A. Schinzel} [in: Schlickewei, Hans Peter et al. (eds.), Diophantine approximation. Wien: Springer. Dev. Math. 16, 363--370 (2008; Zbl 1239.11039)] and proved in this and a subsequent paper [in: Chen, W. W. L. et al. (eds.), Analytic number theory. Cambridge Univ. Press. 402--413 (2009; Zbl 1179.11010)] under some restricting hypotheses, which imply the validity of (2) e.g. for \(n<60\) or \(k < 16\).NEWLINENEWLINEIn the present paper the authors give an unconditional proof of (2). They use an idea of Kaczorowski (from the appendix of the last mentioned paper above) to develop an ingenious recursive method to estimate the number of solutions of (1), which applies for \(n \geq 22\). The remaining cases are covered by Schinzel's result.