2024/12/28 by Dong, Anji, Robles, Nicolas, Zaharescu, Alexandru +1
#11L07 #11L20. Secondary: 11P55 #11P82 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11L03
paper · doi:10.48550/arxiv.2412.20101
We examine exponential sums of the form ∑n ≤ X w(n) e2πiαnk, for k=1,2, where α satisfies a generalized Diophantine approximation and where w are different arithmetic functions that might be multiplicative, additive, or neither. A strategy is shown on how to bound these sums for a wide class of functions w belonging within the same ecosystem. Using this new technology we are able to improve current results on minor arcs that have recently appeared in the literature of the Hardy-Littlewood circle method. Lastly, we show how a bound on ∑n ≤ X |μ(n)| e2πiαn can be used to study partitions asymptotics over squarefree parts and explain their connection to the zeros of the Riemann zeta-function.