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Published online by Cambridge University Press: 25 April 2022
We obtain some improved results for the exponential sum
$\sum _{x<n\leq 2x}\Lambda (n)e(\alpha k n^{\theta })$
with
$\theta \in (0,5/12),$
where
$\Lambda (n)$
is the von Mangoldt function. Such exponential sums have relations with the so-called quasi-Riemann hypothesis and were considered by Murty and Srinivas [‘On the uniform distribution of certain sequences’, Ramanujan J. 7 (2003), 185–192].
This work was supported by National Natural Science Foundation of China (Grant No. 11871307).