A Menon-type identity derived using Cohen-Ramanujan sum

Arya Chandran, K Vishnu Namboothiri
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Abstract

Menon’s identity is a classical identity involving gcd sums and the Euler totient function \(\phi \). We derived the Menon-type identity \(\sum \limits _{\begin{array}{c} m=1\\ (m,n^s)_s=1 \end{array}}^{n^s} (m-1,n^s)_s=\Phi _s(n^s)\tau _s(n^s)\) in [Czechoslovak Math. J., 72(1):165-176 (2022)] where \(\Phi _s\) denotes the Klee’s function and \((a,b)_s\) denotes a a generalization of the gcd function. Here we give an alternate method to derive this identity using the properties of the Cohen-Ramanujan sum defined by E. Cohen.

利用科恩-拉马努扬和推导出的梅农型特性
梅农标识是一个经典标识,涉及 gcd 和及欧拉图腾函数 (\phi \)。我们在[捷克斯洛伐克数学期刊、72(1):165-176 (2022)]中,\(\Phi _s\)表示克利函数,\((a,b)_s\)表示 gcd 函数的一般化。在这里,我们利用科恩定义的科恩-拉玛努扬和的性质给出了另一种方法来推导这一特性。
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