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One Diophantine inequality with unlike powers of prime variables. (English) Zbl 1428.11068

##### MSC:
 11D75 Diophantine inequalities 11P55 Applications of the Hardy-Littlewood method
##### Keywords:
Diophantine inequality; prime; Davenport-Heilbronn method
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##### References:
 [1] Baker, A., On some Diophantine inequalities involving primes, J. Reine Angew. Math., 228, 166-181, (1967) · Zbl 0155.09202 [2] Baker, R. C.; Harman, G., Diophantine approximation by prime numbers, J. London Math. Soc., 25, 2, 201-215, (1982) · Zbl 0443.10015 [3] J. Bourgain, On the Vinogradov mean value, preprint (2016); . · Zbl 1371.11138 [4] Bourgain, J.; Demeter, C.; Guth, L., Proof of the main conjecture in vinogradov’s Mean value theorem for degrees higher than three, Ann. of Math., 184, 2, 633-682, (2016) · Zbl 1408.11083 [5] Davenport, H.; Heilbronn, H., On indefinite quadratic forms in five variables, J. London Math. Soc., 21, 185-193, (1946) · Zbl 0060.11914 [6] Ge, W. X.; Li, W. P., One Diophantine inequality with unlike powers of prime variables, J. Inequal. Appl., 2016, 8, (2016) · Zbl 1382.11032 [7] Harman, G., Trigonometric sums over primes I, Mathematika, 28, 249-254, (1981) · Zbl 0465.10029 [8] Harman, G., Diophantine approximation by prime numbers, J. London Math. Soc., 44, 2, 218-226, (1991) · Zbl 0754.11010 [9] Languasco, A.; Settimi, V., On a Diophantine problem with one prime, two squares of primes and $$s$$ powers of two, Acta Arith., 154, 4, 385-412, (2012) · Zbl 1306.11031 [10] Languasco, A.; Zaccagnini, A., A Diophantine problem with a prime and three squares of primes, J. Number Theory, 132, 3016-3028, (2012) · Zbl 1306.11032 [11] Languasco, A.; Zaccagnini, A., On a ternary Diophantine problem with mixed powers of primes, Acta Arith., 159, 4, 345-362, (2013) · Zbl 1330.11063 [12] Li, W. P.; Wang, T. Z., Diophantine approximation with one prime and three squares of primes, Ramanujan J., 25, 343-357, (2011) · Zbl 1234.11036 [13] Li, W. P.; Wang, T. Z., Diophantine approximation with two primes and one square of prime, Chin. Quart. J. Math., 27, 3, 417-423, (2012) · Zbl 1274.11111 [14] Liu, Z. X.; Sun, H. W., Diophantine approximation with one prime and three squares of primes, Ramanujan J., 30, 327-340, (2013) · Zbl 1281.11029 [15] Matomäki, K., Diophantine approximation by primes, Glasgow Math. J., 52, 87-106, (2010) · Zbl 1257.11035 [16] Mu, Q. W.; Qu, Y. Y., A Diophantine inequality with prime variables and mixed power, Acta Math. Sinica (Chin. Ser.), 58, 3, 491-500, (2015) · Zbl 1340.11055 [17] Ramachandra, K., On the sums $$\sum_{j = 1}^K f_j(p_j)(\lambda_j)$$ fixed non-zero real numbers, $$f_j(x)$$ fixed polynomials and $$p_j$$ arbitrary primes), J. Reine Angew. Math., 262/263, 158-165, (1973) · Zbl 0266.10017 [18] Titchmarsh, E. C.; Heath-Brown, D. R., The Theory of the Riemann Zeta-Function, (1986), Oxford University Press, Oxford [19] Vaughan, R. C., Diophantine approximation by prime numbers, I, Proc. London Math. Soc., 28, 3, 373-384, (1974) · Zbl 0274.10045 [20] Vaughan, R. C., Diophantine approximation by prime numbers, II, Proc. London Math. Soc., 28, 3, 385-401, (1974) · Zbl 0276.10031 [21] Vaughan, R. C., The Hardy-Littlewood Method, (1997), Cambridge University Press, Cambridge · Zbl 0868.11046
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