In this note we give a short and elegant proof of the result for not a rational multiple of , uniformly in . This was first proved by Hardy and Littlewood, in 1938. The main ingredient of our proof is Van der Corput's inequality. We then generalize this to obtain , where is a nonnegative constant.
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for 
not a rational multiple of 
, uniformly in 
. This was first proved by Hardy and Littlewood, in 1938. The main ingredient of our proof is Van der Corput's inequality. We then generalize this to obtain 
, where 
is a nonnegative constant.
