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 ( 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.