A closer look at some new identities (Q1392712)

In a series of recent papers in the same journal, the author has discussed a class of infinite product identities which he called vpv (visible (from the origin) point vector) identities. Among new identities is: \[ \prod^\infty_{k=3} (1- z^k)^{\phi_3(k)/k}= {1\over 1-z}\exp\Biggl({z^3\over 2(1- z)^2}- {1\over 2} z- {1\over 2z(1- z)}\Biggr),\quad| z|< 1, \] where \(\phi_3(k)\) denotes the number of positive integer solutions of \(\text{gcd}(a, b, k)= 1\), \(a< b< k\).