Details

János Pintz ^[a]

An approximate formula for Goldbach's problem with applications

Pages: 181-199
Received: 17 February 2020
Accepted in revised form: 29 June 2020
Mathematics Subject Classification (2010): Primary 11P32; Secondary 11L07.
Keywords: Binary Goldbach problem, Goldbach-Linnik problem, approximations to Goldbach’s problem.
Author address:
[a]: Rényi Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Reáltanoda u. 13-15, H-1053 Hungary

Supported by the National Research Development and Innovation Office, NKFIH, K 119528 and KKP 133819.

Abstract: In the work we present an approximate formula for the contribution of the major arcs in Goldbach’s problem as a function of a relatively small number of zeros of Dirichlet L-functions, lying extreme near to the boundary line Re s=1. This formula plays a crucial role in the proof of various approximate forms of the Goldbach conjecture. The applications include the estimate of the size of the exceptional set in Goldbach’s problem, the so-called Goldbach-Linnik problem and the estimate of the size of the exceptional set in polynomial sequences. The asymptotic formula is quoted without proof while a brief outline is given for the applications. This work is based partly on a joint result of the author and I. Z. Ruzsa, while another part is based on a work in preparation, joint with A. Perelli.

References

[BKW]: J. Brüdern, K. Kawada and T. D. Wooley, Additive representation in thin sequences. II. The binary Goldbach problem, Matematika 47 (2000), 117-125. MR1924492
[Cud]: N. G. Čudakov, On the density of the set of even numbers which are not representable as a sum of two odd primes, Izv. Akad. Nauk SSSR Ser. Mat. 2 (1938), 25-40. Math-Net.Ru | zbMATH
[Est]: T. Estermann, On Goldbach's problem: Proof that almost all even positive integers are sums of two primes, Proc. London Math. Soc. (2) 44 (1938), 307-314. MR1576891
[Gal1]: P. X. Gallagher, A large sieve density estimate near \(\sigma = 1\), Invent. Math. 11 (1970), 329-339. MR0279049
[Gal2]: P. X. Gallagher, Primes and powers of \(2\), Invent. Math. 29 (1975), 125-142. MR0379410
[Gol]: D. A. Goldston, On Hardy and Littlewood's contribution to the Goldbach conjecture, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno, Salerno, 1992, 115-155. MR1220461
[HL]: G. H. Hardy and J. E. Littlewood, Some problems of 'Partitito Numerorum' V: A further contribution to the study of Goldbach's problem, Proc. London Math. Soc. (2) 22 (1924), 46-56. MR1575721
[HP]: D. R. Heath-Brown and J.-C. Puchta, Integers represented as a sum of primes and powers of two, Asian J. Math. 6 (2002), 535-565. MR1946346
[Hel]: H. A. Helfgott, The ternary Goldbach conjecture is true, arXiv:1312.7748v2, preprint, 2014.
[Jut]: M. Jutila, On Linnik's constant, Math. Scand. 41 (1977), 45-62. MR0476671
[Li1]: H. Z. Li, The number of powers of \(2\) in a representation of large even integers by sums of such powers and of two primes, Acta Arith. 92 (2000), 229-237. MR1752027
[Lin1]: Yu. V. Linnik, Prime numbers and powers of two, (Russian), Trudy Nat. Inst. Steklov. 38 (1951), 152-169. MR0050618
[Lin2]: Yu. V. Linnik, Addition of prime numbers with powers of one and the same number, (Russian), Mat. Sbornik N.S. 32(74) (1953), 3-60. MR0059938
[LLW1]: J. Y. Liu, M. C. Liu and T. Z. Wang, The number of powers of \(2\) in a representation of large even integers (I), Sci. China Ser. A 41 (1998), 386-398. MR1663182
[LLW2]: J. Y. Liu, M. C. Liu and T. Z. Wang, The number of powers of \(2\) in a representation of large even integers (II), Sci. China Ser. A 41 (1998), 1255-1271. MR1681935
[LL]: Z. Liu and G. Lü, Density of two squares of primes and powers of two, Int. J. Number Theory 7 (2011), 1317-1329. MR2825974
[Lu]: W. C. Lu, Exceptional set of Goldbach number, J. Number Theory 130 (2010), 2359-2392. MR2660899
[MV]: H. L. Montgomery and R. C. Vaughan, The exceptional set in Goldbach's problem, Collection of articles in memory of Jurii Vladimirovič Linnik, Acta Arith. 27 (1975), 353-370. MR0374063
[Per]: A. Perelli, Goldbach numbers represented by polynomials, Rev. Math. Iberoamericana 12 (1996), 477-490. MR1402675
[Pin1]: J. Pintz, A new explicit formula in the additive theory of primes with applications, I. The explicit formula for the Goldbach and generalized twin prime problems, arXiv:1804.05561, preprint, 2018.
[Pin2]: J. Pintz, A new explicit formula in the additive theory of primes with applications, II. The exceptional set for the Goldbach's problem, arXiv:1804.09084, preprint, 2018.
[PR1]: J. Pintz and I. Z. Ruzsa, On Linnik's approximation to Goldbach's problem, I, Acta. Arith. 109 (2003), 169-194. MR1980645
[PR2]: J. Pintz and I. Z. Ruzsa, On Linnik's approximation to Goldbach's problem, II, Acta Math. Hungar. 161 (2020), 569-582. DOI
[VdC]: J. G. van der Corput, Sur l'hypothèse de Goldbach pour presque tous les nombres pairs, Acta Arith. 2 (1936), 266-290. DOI
[Vau1]: R. C. Vaughan, On Goldbach's problem, Acta Arith. 22 (1972), 21-48. MR0327703
[Vau2]: R. C. Vaughan, The Hardy-Littlewood method, Cambridge University Press, Cambridge-New York, 1981. MR0628618
[Vin]: I. M. Vinogradov, Representation of an odd number as a sum of three prime numbers,(Russian), Doklady Akad. Nauk SSSR 15 (1937), 291-294. zbMATH
[Wan]: T. Z. Wang, On Linnik's almost Goldbach theorem, Sci. China Ser. A 42 (1999), 1155-1172. MR1749863