Values of binary linear forms at prime arguments

Abstract

value of a given binary linear form at prime arguments. Let λ₁ and λ₂ be positive real numbers such that λ₁/λ₂ is irrational and algebraic. For any (C, c) well-spaced sequence {V} and δ>0, let E({V}, X, δ) denote the number of υ∈{V} with υ≤X for which the inequality

has no solution in primes p₁, p₂. It is shown that for any ε>0,we have E({V}, X, δ) «max(X^{\frac{\mathrm{3}}{\mathrm{5}}+\mathrm{2}\delta +\epsilon },X^{\frac{\mathrm{2}}{\mathrm{3}}+\frac{\mathrm{4}}{\mathrm{3}}\delta +\epsilon }).

Key words： Circle method; Diophantine inequality

References

1 Brüdern J, Cook R J, Perelli A. The values of binary linear forms at prime arguments. In: Greaves G R H, Harman G, Huxley M N, eds. Sieve Methods, Exponential Sums, and Their Applications in Number Theory. London Math Soc Lecture Note Ser, 237. Cambridge: Cambridge Univ Press, 1997, 87−100

2 Cai Y. A remark on the values of binary linear forms at prime arguments. Arch Math (Basel), 2011, 97(5): 431−441

3 Cook R J, Harman G. The values of additive forms at prime arguments. Rocky Mountain J Math, 2006, 36(4): 1153−1164

4 Davenport H, Heilbronn H. On indefinite quadratic forms in five variables. J Lond Math Soc, 1946, 21: 185−193

5 Harman G. Diophantine approximation by prime numbers. J Lond Math Soc (2), 1991, 44(2): 218−226

6 Lu W C. Exceptional set of Goldbach number. J Number Theory, 2010, 130(10): 2359−2392

7 Matomäki K. Diophantine approximation by primes. Glasg Math J, 2010, 52(1): 87−106

8 Montgomery H L, Vaughan R C. The exceptional set in Goldbach’s problem. Acta Arith, 1975, 27: 353−370