## 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

### Citations:

Zbl 1353.11066; Zbl 0034.02105
Full Text:

### References:

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