Let S(t) = 1 π Im log ζ ( 1 2 + it ) . Using Soundararajan’s resonance method we prove an unconditional lower bound on the size of the tails of the distribution of S(t). In particular we reproduce the best unconditional Ω result for S(t) which is due to Tsang, S(t) = Ω± (

We introduce a resonance method to produce large values of the Riemann zeta-function on the critical line, and large and small central values of L-functions.

Following Selberg it is known that as T → ∞, [formula] uniformly for Δ ≤ (log log log T)^((1/2) - e). We extend the range of Δ to Δ « (log log T)^((1/10) - e). We also speculate on the size of the… Expand

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects… Expand

We combine our version of the resonance method with certain convolution formulas for $$\zeta (s)$$ζ(s) and $$\log \, \zeta (s)$$logζ(s). This leads to a new $$\Omega $$Ω result for $$|\zeta… Expand

Let $S(t)$ denote the argument of the Riemann zeta-function \begin{equation*} S(t):=\pi^{-1}\Im(\log \zeta(1/2+it)). \end{equation*} Under Riemann Hypothesis, we obtain the following $\Omega_\pm$… Expand