## Every odd number greater than $1$ is the sum of at most five primes

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## Abstract:

We prove that every odd number $N$ greater than $1$ can be expressed as the sum of at most five primes, improving the result of Ramaré that every even natural number can be expressed as the sum of at most six primes. We follow the circle method of Hardy-Littlewood and Vinogradov, together with Vaughan’s identity; our additional techniques, which may be of interest for other Goldbach-type problems, include the use of smoothed exponential sums and optimisation of the Vaughan identity parameters to save or reduce some logarithmic losses, the use of multiple scales following some ideas of Bourgain, and the use of Montgomery’s uncertainty principle and the large sieve to improve the $L^2$ estimates on major arcs. Our argument relies on some previous numerical work, namely the verification of Richstein of the even Goldbach conjecture up to $4 \times 10^{14}$, and the verification of van de Lune and (independently) of Wedeniwski of the Riemann hypothesis up to height $3.29 \times 10^9$.## References

- E. Bombieri and H. Davenport,
*On the large sieve method*, Number Theory and Analysis (Papers in Honor of Edmund Landau), Plenum, New York, 1969, pp. 9–22. MR**0260703** - Y. Buttkewitz,
*Exponential sums over primes and the prime twin problem*, Acta Math. Hungar.**131**(2011), no. 1-2, 46–58. MR**2776653**, DOI 10.1007/s10474-010-0015-9 - J. Bourgain,
*On triples in arithmetic progression*, Geom. Funct. Anal.**9**(1999), no. 5, 968–984. MR**1726234**, DOI 10.1007/s000390050105 - Jing Run Chen,
*On the estimation of some trigonometrical sums and their application*, Sci. Sinica Ser. A**28**(1985), no. 5, 449–458. MR**813837** - Jing Run Chen and Tian Ze Wang,
*On the Goldbach problem*, Acta Math. Sinica**32**(1989), no. 5, 702–718 (Chinese). MR**1046491** - Tian Ze Wang and Jing Run Chen,
*Estimation of linear trigonometric sums with prime number variables*, Acta Math. Sinica**37**(1994), no. 1, 25–31 (Chinese, with English and Chinese summaries). MR**1272501** - Hédi Daboussi,
*Effective estimates of exponential sums over primes*, Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) Progr. Math., vol. 138, Birkhäuser Boston, Boston, MA, 1996, pp. 231–244. MR**1399341** - Hedi Daboussi and Joël Rivat,
*Explicit upper bounds for exponential sums over primes*, Math. Comp.**70**(2001), no. 233, 431–447. MR**1803131**, DOI 10.1090/S0025-5718-00-01280-1 - J.-M. Deshouillers, G. Effinger, H. te Riele, and D. Zinoviev,
*A complete Vinogradov $3$-primes theorem under the Riemann hypothesis*, Electron. Res. Announc. Amer. Math. Soc.**3**(1997), 99–104. MR**1469323**, DOI 10.1090/S1079-6762-97-00031-0 - Pierre Dusart,
*Inégalités explicites pour $\psi (X)$, $\theta (X)$, $\pi (X)$ et les nombres premiers*, C. R. Math. Acad. Sci. Soc. R. Can.**21**(1999), no. 2, 53–59 (French, with English and French summaries). MR**1697455** - Pierre Dusart,
*Estimates of $\theta (x;k,l)$ for large values of $x$*, Math. Comp.**71**(2002), no. 239, 1137–1168. MR**1898748**, DOI 10.1090/S0025-5718-01-01351-5 - P. Dusart,
*Estimates of some functions over primes without R. H.*, preprint. arxiv:1002.0442. - P. X. Gallagher,
*The large sieve*, Mathematika**14**(1967), 14–20. MR**214562**, DOI 10.1112/S0025579300007968 - X. Gourdon, P. Demichel,
*The first $10^{13}$ zeros of the Riemann Zeta function, and zeros computation at very large height*, http://numbers.computation.free.fr/ Constants/Miscellaneous/zetazeros1e13-1e24.pdf, 2004. - Ben Green and Terence Tao,
*Restriction theory of the Selberg sieve, with applications*, J. Théor. Nombres Bordeaux**18**(2006), no. 1, 147–182 (English, with English and French summaries). MR**2245880** - G. H. Hardy and J. E. Littlewood,
*Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes*, Acta Math.**44**(1923), no. 1, 1–70. MR**1555183**, DOI 10.1007/BF02403921 - D. R. Heath-Brown,
*Zero-free regions for Dirichlet $L$-functions, and the least prime in an arithmetic progression*, Proc. London Math. Soc. (3)**64**(1992), no. 2, 265–338. MR**1143227**, DOI 10.1112/plms/s3-64.2.265 - H. Helfgott,
*Minor arcs for Goldbach’s problem*, preprint. - Henryk Iwaniec and Emmanuel Kowalski,
*Analytic number theory*, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR**2061214**, DOI 10.1090/coll/053 - H. Kadiri,
*Short effective intervals containing primes in arithmetic progressions and the seven cubes problem*, Math. Comp.**77**(2008), no. 263, 1733–1748. MR**2398791**, DOI 10.1090/S0025-5718-08-02084-X - Leszek Kaniecki,
*On Šnirelman’s constant under the Riemann hypothesis*, Acta Arith.**72**(1995), no. 4, 361–374. MR**1348203**, DOI 10.4064/aa-72-4-361-374 - A. F. Lavrik,
*On the twin prime hypothesis of the theory of primes by the method of I. M. Vinogradov*, Soviet Math. Dokl.**1**(1960), 700–702. MR**0157955** - Ming-Chit Liu and Tianze Wang,
*Distribution of zeros of Dirichlet $L$-functions and an explicit formula for $\psi (t,\chi )$*, Acta Arith.**102**(2002), no. 3, 261–293. MR**1884719**, DOI 10.4064/aa102-3-5 - Ming-Chit Liu and Tianze Wang,
*On the Vinogradov bound in the three primes Goldbach conjecture*, Acta Arith.**105**(2002), no. 2, 133–175. MR**1932763**, DOI 10.4064/aa105-2-3 - J. van de Lune, H. J. J. te Riele, and D. T. Winter,
*On the zeros of the Riemann zeta function in the critical strip. IV*, Math. Comp.**46**(1986), no. 174, 667–681. MR**829637**, DOI 10.1090/S0025-5718-1986-0829637-3 - J. H. van Lint and H.-E. Richert,
*On primes in arithmetic progressions*, Acta Arith.**11**(1965), 209–216. MR**188174**, DOI 10.4064/aa-11-2-209-216 - Kevin S. McCurley,
*Explicit estimates for the error term in the prime number theorem for arithmetic progressions*, Math. Comp.**42**(1984), no. 165, 265–285. MR**726004**, DOI 10.1090/S0025-5718-1984-0726004-6 - H. L. Montgomery,
*A note on the large sieve*, J. London Math. Soc.**43**(1968), 93–98. MR**224585**, DOI 10.1112/jlms/s1-43.1.93 - Hugh L. Montgomery,
*Topics in multiplicative number theory*, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin-New York, 1971. MR**0337847** - Hugh L. Montgomery,
*The analytic principle of the large sieve*, Bull. Amer. Math. Soc.**84**(1978), no. 4, 547–567. MR**466048**, DOI 10.1090/S0002-9904-1978-14497-8 - H. L. Montgomery and R. C. Vaughan,
*The large sieve*, Mathematika**20**(1973), 119–134. MR**374060**, DOI 10.1112/S0025579300004708 - T. Oliveira e Silva, http://www.ieeta.pt/˜tos/goldbach.html
- D. Platt,
*Computing degree $1$ $L$-functions rigorously*, Ph.D. Thesis, University of Bristol, 2011. - D. Platt,
*Computing $\pi (x)$ analytically*, preprint. - Olivier Ramaré,
*On Šnirel′man’s constant*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**22**(1995), no. 4, 645–706. MR**1375315** - Olivier Ramaré,
*Eigenvalues in the large sieve inequality*. part 2, Funct. Approx. Comment. Math.**37**(2007), no. part 2, 399–427. MR**2363835**, DOI 10.7169/facm/1229619662 - O. Ramaré,
*On Bombieri’s asymptotic sieve*, J. Number Theory**130**(2010), no. 5, 1155–1189. MR**2607306**, DOI 10.1016/j.jnt.2009.10.014 - Olivier Ramaré and Imre Z. Ruzsa,
*Additive properties of dense subsets of sifted sequences*, J. Théor. Nombres Bordeaux**13**(2001), no. 2, 559–581 (English, with English and French summaries). MR**1879673** - Olivier Ramaré and Robert Rumely,
*Primes in arithmetic progressions*, Math. Comp.**65**(1996), no. 213, 397–425. MR**1320898**, DOI 10.1090/S0025-5718-96-00669-2 - Olivier Ramaré and Yannick Saouter,
*Short effective intervals containing primes*, J. Number Theory**98**(2003), no. 1, 10–33. MR**1950435**, DOI 10.1016/S0022-314X(02)00029-X - Jörg Richstein,
*Verifying the Goldbach conjecture up to $4\cdot 10^{14}$*, Math. Comp.**70**(2001), no. 236, 1745–1749. MR**1836932**, DOI 10.1090/S0025-5718-00-01290-4 - Barkley Rosser,
*Explicit bounds for some functions of prime numbers*, Amer. J. Math.**63**(1941), 211–232. MR**3018**, DOI 10.2307/2371291 - J. Barkley Rosser and Lowell Schoenfeld,
*Approximate formulas for some functions of prime numbers*, Illinois J. Math.**6**(1962), 64–94. MR**137689** - J. Barkley Rosser and Lowell Schoenfeld,
*Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$*, Math. Comp.**29**(1975), 243–269. MR**457373**, DOI 10.1090/S0025-5718-1975-0457373-7 - H. Siebert,
*Montgomery’s weighted sieve for dimension two*, Monatsh. Math.**82**(1976), no. 4, 327–336. MR**424731**, DOI 10.1007/BF01540603 - Robert-C. Vaughan,
*Sommes trigonométriques sur les nombres premiers*, C. R. Acad. Sci. Paris Sér. A-B**285**(1977), no. 16, A981–A983 (French, with English summary). MR**498434** - I. M. Vinogradov,
*Representation of an odd number as a sum of three primes*, Comptes Rendus (Doklady) de l’Academy des Sciences de l’USSR 15 (1937), 191–294. - I. M. Vinogradov, The Method of Trigonometric Sums in Number Theory, Interscience Publ. (London, 1954).
- D. R. Ward,
*Some series involving Euler’s function*, London Math. Soc., 2 (1927), 210–214. - S. Wedeniwski,
*ZetaGrid - Computational verification of the Riemann Hypothesis*, Conference in Number Theory in Honour of Professor H.C. Williams, Banff, Alberta, Canada, May 2003.

## Additional Information

**Terence Tao**- Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1596
- MR Author ID: 361755
- ORCID: 0000-0002-0140-7641
- Email: tao@math.ucla.edu
- Received by editor(s): January 30, 2012
- Received by editor(s) in revised form: July 3, 2012, and July 5, 2012
- Published electronically: June 24, 2013
- © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**83**(2014), 997-1038 - MSC (2010): Primary 11P32
- DOI: https://doi.org/10.1090/S0025-5718-2013-02733-0
- MathSciNet review: 3143702