# Logarithm of the Riemann zeta function and entire $L$-functions

Expositor: Andrés Chirre
26/10/2017 , 15:30 | Sala 232

Littlewood showed in $1924$ that the Riemann hypothesis (RH) implies a strong form of the Lindelöf Hypothesis, namely, on RH, for large real number $t$ there is a constant $C$ such that

$$|zeta(tfrac12 +it)|ll expbigg(Cdfrac{log t}{loglog t}bigg). hskip20pt (1)$$
Over the years no improvement has been made on the order of magnitude of the upper bound (1). The advances have rather focused on reducing the value of the admissible constant $C$. In this talk we will show how to obtain the best (up to date) form of this estimate. We extended this bound for the critical strip for the Riemann zeta function and $L$-functions. This involves the use of certain special entire functions of exponential type. This is an aplication of approximation theory in analytic number theory.