منابع مشابه
Trigonometric sums over primes III
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We investigate the Waring-Goldbach problem of representing a positive integer n as the sum of s kth powers of almost equal prime numbers. Define sk = 2k(k − 1) when k > 3, and put s2 = 6. In addition, put θ2 = 19 24 , θ3 = 4 5 and θk = 5 6 (k > 4). Suppose that n satisfies the necessary congruence conditions, and put X = (n/s). We show that whenever s > sk and ε > 0, and n is sufficiently large...
متن کاملVinogradov’s estimate for exponential sums over primes
1Many of the manipulations of sums in these chapters are hard to follow, and I greatly expand on the calculations in Davenport. The organization of the proof in Davenport seems to be due to Vaughan. I have also used lecture notes by Andreas Strömbergsson, http://www2.math.uu.se/~astrombe/analtalt08/www_notes.pdf, pp. 245–257. Another set of notes, which I have not used, are http://jonismathnote...
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Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that lim inf n→∞ (pn+1 − pn) log pn = 0. We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of exactly two distinct primes. We prove that lim inf n→∞ (qn+1 − qn) ≤ 26. If an appropriate generalization of ...
متن کاملSums of Primes and Squares of Primes in Short Intervals
Let H2 denote the set of even integers n 6≡ 1 (mod 3). We prove that when H ≥ X, almost all integers n ∈ H2 ∩ (X,X + H] can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.
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ژورنال
عنوان ژورنال: Michigan Mathematical Journal
سال: 2006
ISSN: 0026-2285
DOI: 10.1307/mmj/1156345592