# Why do we study moment of Riemann zeta function and moment of Dirichlet L-function?

I study Analytic Number theory and in particular L-functions. I began to study moments of Riemann zeta function and Dirichlet L-function. However I do not see why this is important, what is the significance of studying moments, or what do they tell us? It would be great if some one can explain me the purpose of it.

• Obviously your books explain why those are useful for. The asymptotic as $T \to \infty$ of $\int_1^T |\zeta(\sigma+it)|^{k}dt, \sigma \le 1$ encodes the Lindelöf hypothesis and the asymptotic of $\int_1^T |\zeta(\sigma+it)|^{-k}dt$ encodes the Riemann hypothesis. – reuns Aug 15 '17 at 0:45