# zbMATH — the first resource for mathematics

On the exceptional set for the 2$$k$$-twin primes problem. (English) Zbl 0756.11028
The classical Hardy-Littlewood conjecture states, in a slightly modified form, $\psi(N,2k)={\mathfrak S}(2k)(N-2k)+O(N(\ln N)^{-A}),\tag{*}$ where $$k$$ is a fixed positive integer, $\psi(N,2k)=\sum_{\textstyle{{m,n=N}\atop{n- m=2k}}}^{2N}\Lambda(m)\Lambda(n), \qquad {\mathfrak S}(2k)=2\prod_{p>2}\left(1-{1\over(p-1)^ 2}\right)\prod_{\textstyle{{p\mid k}\atop {p>2}}}{p-1 \over p-2},$ and $$A$$ is any positive number. Several “almost all” results are well known [H. L. Montgomery, Topics in multiplicative number theory (Lect. Notes Math. 227) (Springer 1971; Zbl 0216.03501); D. Wolke, Math. Ann. 283, 529-537 (1989; Zbl 0646.10033)].
By a very careful use of the circle method and zero density results for $$L$$-functions the authors prove the following strong statement. Let $$A,B>0$$, $$0\leq V<N/4$$, and $$N^{1/2+\varepsilon}<N<N/4$$ ($$0<\varepsilon<1/2$$). Then (*) holds for all $$V\leq k\leq V+H$$ but $$O(H(\ln N)^{-B})$$ exceptions.
They indicate that a similar result is true for the binary Goldbach problem, i.e. the Hardy-Littlewood formula for the number of binary Goldbach representations is valid for all even numbers $$2n\in I$$ but $$O(| I|(\ln x)^{-B})$$ exceptions, where $$I$$ is the interval $$[x,x+x^{1/2+\varepsilon}]$$.

##### MSC:
 11P32 Goldbach-type theorems; other additive questions involving primes 11N05 Distribution of primes 11P55 Applications of the Hardy-Littlewood method
Full Text:
##### References:
 [1] H. Davenport : Multiplicative Number Theory (2nd edn), Springer-Verlag, 1980. · Zbl 0453.10002 [2] P.X. Gallagher : A large sieve density estimate near \sigma =1 , Invent. Math. 11 (1970), 329-339. · Zbl 0219.10048 · doi:10.1007/BF01403187 · eudml:142061 [3] G.H. Hardy , and J.E. Littlewood : Some problems of ’Partitio Numerorum’ V. A further contribution to the study of Goldbach’s problem , Proc. London Math. Soc. (2)22 (1924), 46-56. · JFM 49.0127.03 [4] D.R. Heath-Brown : The difference between consecutive primes , J. London Math. Soc. (2)18 (1978), 7-13. · Zbl 0387.10025 · doi:10.1112/jlms/s2-18.1.7 [5] M.N. Huxley : On the difference between consecutive primes , Invent. Math. 15 (1972),155-164. · Zbl 0241.10026 · doi:10.1007/BF01418933 · eudml:142126 [6] Yu. V. Linnik : Some conditional theorems concerning the binary Goldbach problem (Russian) , Izv. Akad. Nauk SSSR 16 (1952), 503-520. · Zbl 0049.03104 [7] H.L. Montgomery : Topics in Multiplicative Number Theory , Springer L.N. 227, 1971. · Zbl 0216.03501 [8] H.L. Montgomery and R.C. Vaughan : The exceptional set in Goldbach’s problem , Acta Arith. 27 (1975), 353-370. · Zbl 0301.10043 · eudml:205349 [9] A. Perelli : Local problems with primes I , J. reine angew. Math. 401 (1989), 209-220. · Zbl 0677.10028 · doi:10.1515/crll.1989.401.209 · crelle:GDZPPN002207052 · eudml:153178 [10] K. Prachar : Primzahlverteilung , Springer-Verlag, 1957. · Zbl 0080.25901 [11] B. Saffari and R.C. Vaughan : On the fractional parts of x/n and related sequences II , Ann. Inst. Fourier 27 (1977), 1-30. · Zbl 0379.10023 · doi:10.5802/aif.649 · numdam:AIF_1977__27_2_1_0 · eudml:74318 [12] P. Shiu : A Brun-Titchmarsh theorem for multiplicative functions , J. reine angew. Math. 313 (1980), 161-170. · Zbl 0412.10030 · doi:10.1515/crll.1980.313.161 · crelle:GDZPPN002196921 · eudml:152201 [13] R.C. Vaughan : The Hardy-Littlewood Method , Cambridge Univ. Press, 1981. · Zbl 0455.10034 [14] D. Wolke : Über das Primzahl-Zwillingsproblem , Math. Ann. 283 (1989), 529-537. · Zbl 0646.10033 · doi:10.1007/BF01442852 · eudml:164525
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.