Estimates of multiple trigonometric sums with prime numbers

The author obtains estimates for multiple trigonometric sums with a general polynomial in the exponent whose variables of summation take the values of the successive prime numbers. The precision of the estimates is similar to that of the estimates in the paper "Multiple trigonometric sums and their applications" by Arkhipov, Karatsuba and Chubarikov (Izv. Akad. Nauk SSSR Ser. Mat., 1980, v. 44, № 4, p. 723–781).
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