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A formula for the number of solutions of a restricted linear congruence. (English) Zbl 07332741

Summary: Consider the linear congruence equation \(x_1+\dots+x_k\equiv b\pmod{n^s}\) for \(b\in\mathbb{Z}\), \(n,s\in\mathbb{N}\). Let \((a,b)_s\) denote the generalized gcd of \(a\) and \(b\) which is the largest \(l^s\) with \(l\in\mathbb{N}\) dividing \(a\) and \(b\) simultaneously. Let \(d_1,\ldots,d_{\tau(n)}\) be all positive divisors of \(n\). For each \(d_j\mid n\), define \(\mathcal{C}_{j,s}(n)=\{1\leq x\leq n^s\colon(x,n^s)_s=d^s_j\}\). K. Bibak et al. [Int. J. Number Theory 12, No. 8, 2167–2171 (2016; Zbl 1353.11066)] gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on \(x_i\). We generalize their result with generalized gcd restrictions on \(x_i\) and prove that for the above linear congruence, the number of solutions is \[\frac{1}{n^s}\sum\limits_{d\mid n}c_{d,s}(b)\prod\limits_{j=1}^{\tau(n)}\Bigl(c_{{n}/{d_j},s}\Bigl(\frac{n^s}{d^s}\Big)\Big)^{g_j}\] where \(g_j=|\{x_1,\ldots,x_k\}\cap\mathcal{C}_{j,s}(n)|\) for \(j=1,\ldots,\tau(n)\) and \(c_{d,s}\) denotes the generalized Ramanujan sum defined by E. Cohen [Duke Math. J. 16, 85–90 (1949; Zbl 0034.02105)].

MSC:

11D79 Congruences in many variables
11P83 Partitions; congruences and congruential restrictions
11L03 Trigonometric and exponential sums (general theory)
11A25 Arithmetic functions; related numbers; inversion formulas
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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