DEFINING POWER SUMS OF n AND φ(n) INTEGERS

Let n be a positive integer and phi(n) denotes the Euler phi function. It is well known that the power sum of n can be evaluated in closed form in terms of n. Also, the sum of all those phi(n) positive integers that are coprime to n and not exceeding n, is expressible in terms of n and phi(n). Although such results already exist in literature, but here we have presented some new analytical results in these connections. Some functional and integral relations are derived for the general power sums.