Extreme values of the Riemann zeta function and its argument

We combine our version of the resonance method with certain convolution formulas for $\zeta(s)$ and $\log\, \zeta(s)$. This leads to a new $\Omega$ result for $|\zeta(1/2+it)|$: The maximum of $|\zeta(1/2+it)|$ on the interval $1 \le t \le T$ is at least $\exp\left((1+o(1)) \sqrt{\log T \log\log\log T/\log\log T}\right)$. We also obtain conditional results for $S(t):=1/\pi$ times the argument of $\zeta(1/2+it)$ and $S_1(t):=\int_0^t S(\tau)d\tau$. On the Riemann hypothesis, the maximum of $|S(t)|$ is at least $c \sqrt{\log T \log\log\log T/\log\log T}$ and the maximum of $S_1(t)$ is at least $c_1 \sqrt{\log T \log\log\log T/(\log\log T)^3}$ on the interval $T^{\beta} \le t \le T$ whenever $0\le \beta < 1$.Comment: This is the final version of the paper which has been accepted for publication in Mathematische Annale

New asymptotic estimates for spherical designs

Let N(n, t) be the minimal number of points in a spherical t-design on the unit sphere S^n in R^{n+1}. For each n >= 3, we prove a new asymptotic upper bound N(n, t) <= C(n)t^{a_n}, where C(n) is a constant depending only on n, a_3 <= 4, a_4 <= 7, a_5 <= 9, a_6 <= 11, a_7 <= 12, a_8 <= 16, a_9 <= 19, a_10 <= 22, and a_n 10.Comment: 12 page