2017/08/10 by Fokas, Athanassios S.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1708.06607
Lindelöf's hypothesis, one of the most important open problems in the history of mathematics, states that for large t, Riemann's zeta function ζ(1/2+it) is of order O(tε) for any ε>0 . It is well known that for large t, the leading order asymptotics of the Riemann zeta function can be expressed in terms of a transcendental exponential sum. The usual approach to the Lindelöf hypothesis involves the use of ingenious techniques for the estimation of this sum. However, since such estimates can not yield an asymptotic formula for the above sum, it appears that this strategy cannot lead to the proof of Lindelöf's hypothesis. Here, a completely different approach is introduced. In particular, a novel linear integral equation is derived for |ζ(σ+it)|2, 0