## Functiones et Approximatio Commentarii Mathematici

### A remark on the Goldbach-Vinogradov theorem

Yingchun Cai

#### Abstract

Let $N$ denote a sufficiently large odd integer. In this paper it is proved that $N$ can be represented as the sum of three primes, one of which is $\leq N^{\frac{11}{400}+\varepsilon}$ for any $\varepsilon>0$. This result constitutes an improvement upon that of K.C. Wong, who obtained the exponent $\frac{7}{216}$.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 48, Number 1 (2013), 123-131.

Dates
First available in Project Euclid: 25 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.facm/1364222834

Digital Object Identifier
doi:10.7169/facm/2013.48.1.10

Mathematical Reviews number (MathSciNet)
MR3086965

Zentralblatt MATH identifier
1329.11104

Keywords
prime sieve mean value theorem

#### Citation

Cai, Yingchun. A remark on the Goldbach-Vinogradov theorem. Funct. Approx. Comment. Math. 48 (2013), no. 1, 123--131. doi:10.7169/facm/2013.48.1.10. https://projecteuclid.org/euclid.facm/1364222834

#### References

• R.C. Baker, G. Harman and J. Pintz, The exceptional set for Goldbach's problem in short intervals, in: Sieves, Exponetial Sums and Their Applications in Number Theory, 1-54, Cambridge University Press, 1997.
• H. Davenport, Multiplicative Number Theory, 2nd ed., Springer, Berlin, 1980.
• P.X. Gallagher, A large sieve density estimate near $\sigma=1$, Invent. Math. 11 (1970), 329–339.
• G. Harman, Primes in short intervals, Math. Z. 180 (1982), no. 3, 335–348.
• G. Harman, Prime-Detecting Sieves, Princeton Press, 2007.
• C.H. Jia, Almost all short intervals containing prime numbers, Acta Arith. 76 (1996), 21–84.
• A. Kumchev, The difference between consecutive primes in an arithmetic progression, Quart. J. Math. 53 (2002), 479–501.
• C.D. Pan, Some new results in additive number theory, Acta Math Sinica. 9 (1959), 315–329.
• I.M. Vinogradov, Representation of an odd integer as a sum of three primes, Dokl. Akad. Hauk. SSSR. 15 (1937), 169–172.
• K.C. Wong, Contribution to analytic number theory, Ph. Thesis, Cardiff, 1996.
• T. Zhan, A generalization of the Goldbach–Vinogradov theorem, Acta Arith. LXXI.2 (1995), 95–106.