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A Diophantine problem with a prime and three squares of primes. (English) Zbl 1306.11032
Let $$\lambda_1, \dots, \lambda_4$$ be nonzero real numbers, not all of the same sign, with $$\lambda_1/\lambda_2 \not \in {\mathbb Q}$$. The authors show that for every real number $$\varpi$$ and every $$\varepsilon > 0$$ the inequality $|\lambda_1 p_1 + \lambda_2 p_2^2 + \lambda_3 p_3^2 + \lambda_4 p_4^2 + \varpi| < (\max p_j)^{-1/18+\varepsilon}$ has infinitely many solutions in primes $$p_1, \dots, p_4$$. This improves on earlier work of W. Li and T. Wang [Ramanujan J. 25, No. 3, 343–357 (2011; Zbl 1234.11036)], in which the exponent 1/18 was replaced by 1/28. The proof uses the Davenport-Heilbronn version of the circle method, and the new savings arise from a widening of the major arc, as in previous work of the first author and V. Settimi [Acta Arith. 154, No. 4, 385–412 (2012; Zbl 1306.11031)]. The minor (or intermediate) arcs are handled using an estimate of A. Ghosh [Proc. Lond. Math. Soc. (3) 42, 252–269 (1981; Zbl 0397.10026)] for the exponential sum over squares of primes, together with a well-known bound of R. C. Vaughan for sums over primes. The authors observe that this refinement can also be applied to improve their earlier result [Acta Arith. 145, No. 2, 193–208 (2010; Zbl 1222.11049)] on the real analogue of the Linnik-Goldbach problem involving two primes and a bounded number of powers of two.

##### MSC:
 11D75 Diophantine inequalities 11J25 Diophantine inequalities 11P32 Goldbach-type theorems; other additive questions involving primes 11P55 Applications of the Hardy-Littlewood method
##### Citations:
Zbl 1234.11036; Zbl 0397.10026; Zbl 1222.11049; Zbl 1306.11031
Full Text:
##### References:
 [1] Ghosh, A., The distribution of $$\alpha p^2$$ modulo one, Proc. lond. math. soc., 42, 252-269, (1981) · Zbl 0447.10035 [2] Languasco, A.; Settimi, V., On a Diophantine problem with one prime, two squares of primes and s powers of two, Acta arith., 154, 385-412, (2012) · Zbl 1306.11031 [3] Languasco, A.; Zaccagnini, A., On a Diophantine problem with two primes and s powers of 2, Acta arith., 145, 193-208, (2010) · Zbl 1222.11049 [4] Li, W.; Wang, T., Diophantine approximation with one prime and three squares of primes, Ramanujan J. math., 25, 343-357, (2011) · Zbl 1234.11036 [5] Liu, T., Representation of odd integers as the sum of one prime, two squares of primes and powers of 2, Acta arith., 115, 97-118, (2004) · Zbl 1080.11072 [6] Parsell, S.T., Diophantine approximation with primes and powers of two, New York J. math., 9, 363-371, (2003) · Zbl 1038.11065 [7] Pintz, J.; Ruzsa, I.Z., On linnikʼs approximation to goldbachʼs problem, I, Acta arith., 109, 169-194, (2003) · Zbl 1031.11060 [8] Rieger, G.J., Über die summe aus einem quadrat und einem primzahlquadrat, J. reine angew. math., 231, 89-100, (1968) · Zbl 0164.05004 [9] Saffari, B.; Vaughan, R.C., On the fractional parts of $$x / n$$ and related sequences. II, Ann. inst. Fourier, 27, 1-30, (1977) · Zbl 0379.10023 [10] Vaughan, R.C., Diophantine approximation by prime numbers. I, Proc. lond. math. soc., 28, 373-384, (1974) · Zbl 0274.10045 [11] Vaughan, R.C., Diophantine approximation by prime numbers. II, Proc. lond. math. soc., 28, 385-401, (1974) · Zbl 0276.10031 [12] Vaughan, R.C., The Hardy-Littlewood method, (1997), Cambridge University Press Cambridge · Zbl 0868.11046
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