Estimates of short exponential sums with primes in major arcs
For a number of additive problems with almost equal summands, in addition to the estimates for short exponential sums with primes of the form 𝑆𝑘(𝛼; 𝑥, 𝑦) = Σ 𝑥-𝑦<𝑛6𝑥 Λ(𝑛)𝑒(𝛼𝑛𝑘), in minor arcs, we need to have an estimate of these sums in major arcs, except for a small neighborhood of their centers. We also need to have an asymptotic formula on a small neighborhood of the centers of major arcs. In this paper, using the second moment of Dirichlet 𝐿-functions on the critical line, we obtained a nontrivial estimate of the form 𝑆𝑘(𝛼; 𝑥, 𝑦) ≪ 𝑦L-𝐴, for 𝑆𝑘(𝛼; 𝑥, 𝑦) in major arcs 𝑀(L𝑏), = 𝑦5𝑥-2L-𝑏1 , L = ln 𝑥𝑞, except for a small neighborhood of their centers |𝛼 - 𝑞 | > ( 2𝜋𝑘2𝑥𝑘-2𝑦2 ) -1, when > 𝑥1- 1 2𝑘-1+𝜂𝑘 L𝑐𝑘 , where = 2 4𝑘 - 5 + 2 √ (2𝑘 - 2)(2𝑘 - 3) , = 2𝐴 + 22 + ( 2 √ √ 2𝑘-3 2𝑘-2 - 1 ) 𝑏1 2 √ (2𝑘 - 2)(2𝑘 - 3) - (2𝑘 - 3) , and 𝐴, 𝑏1, are arbitrary fixed positive numbers. Furthermore, and we also proved an asymptotic formula on a small neighborhood of the centers of major arcs.