Abstract

In 1924 Littlewood showed that, assuming the Riemann hypothesis, for large t , there is a constant C such that |ॐ(1/2+ it )|<exp( C log t /log log t ). In this note we show how the problem of bounding |ॐ(1/2+ it )| may be framed in terms of minorizing the function log ((4+ x 2 )/ x 2 ) by functions whose Fourier transforms are supported in a given interval, and drawing upon recent work of Carneiro and Vaaler we find the optimal such minorant. Thus we establish that any C >(log 2)/2 is permissible in Littlewood's result.