I was reading the series of paper on Waring's problem, by Hardy and Littlewood, named "Partitio Numerorum".. I am new to some of their methods.. I was wondering if one knew $\Gamma(k)$ for some positive integer k , can one then calculate G(k)? What about the other way?
G(k) is the smallest 't' such that every large enough positive number is sum of at most t k-th power of positive integers. $\Gamma(k)$ is the least number 's' such that for any natural n, the equation $ x_{1}^{k} + x_{2}^{k} + ... + x_{s} ^{k} = n $ is solvable in each p-adic $\mathbb{Q}_{p}$.
May be I am being too naive here but may be when k is prime or prime power , this is possible? Need some expert help..