Large Values of the Zeta-Function on the Critical Line

This is primarily an overview article (Lecture given during the conference “Number Theory and its Applications Workshop” in Xi’an (China), October 23–28, 2014.) dealing with the large values of | ζ ( 1 2 + i t ) | $$\vert \zeta (\frac{1} {2} + it)\vert$$ . This approach allows one to obtain upper bounds for moments (mean values) of | ζ ( 1 2 + i t ) | $$\vert \zeta (\frac{1} {2} + it)\vert$$ , which is one of the fundamental problems of the theory of the Riemann zeta-function. A sketch of the upper bound for the 12th moment of D.R. Heath-Brown (Q J Math (Oxford) 29:443–462, 1978) is presented, together with some recent results of the author. They include upper bounds obtained by the use of large values of E ∗(T), a function closely related to the classical function E(T), the error term in the mean square formula for | ζ ( 1 2 + i t ) | $$\vert \zeta (\frac{1} {2} + it)\vert$$ . A new large values result involving E k (T), the general error-term function in the formula for the 2k-th moment of | ζ ( 1 2 + i t ) | $$\vert \zeta (\frac{1} {2} + it)\vert$$ , is also given.