Extreme Values of |ζ(1 + It)|

Improving on a result of J.E. Littlewood, N. Levinson [3] showed that there are arbitrarily large t for which |ζ(1 + it)| ≥ e log2 t + O(1). (Throughout ζ(s) is the Riemann-zeta function, and logj denotes the j-th iterated logarithm, so that log1 n = logn and logj n = log(logj−1 n) for each j ≥ 2.) The best upper bound known is Vinogradov’s |ζ(1 + it)| (log t). Littlewood had shown that |ζ(1+it)| 2e log2 t assuming the Riemann Hypothesis, in fact by showing that the value of |ζ(1 + it)| could be closely approximated by its Euler product for primes up to log(2 + |t|) under this assumption. Under the further hypothesis that the Euler product up to log(2+ |t|) still serves as a good approximation, Littlewood conjectured that max|t|≤T |ζ(1 + it)| ∼ e log2 T , though later he wrote in [5] (in connection with a q-analogue): “there is perhaps no good reason for believing ... this hypothesis”. Our Theorem 1 evaluates the frequency with which such extreme values are attained; and if this density function were to persist to the end of the viable range then this implies the conjecture that