## A complete Vinogradov 3-primes theorem under the Riemann hypothesis (1997)

Venue: | ERA Am. Math. Soc |

Citations: | 7 - 1 self |

### BibTeX

@ARTICLE{Deshouillers97acomplete,

author = {J. -m. Deshouillers and G. Effinger and H. Te Riele and D. Zinoviev and Communicated Hugh Montgomery},

title = {A complete Vinogradov 3-primes theorem under the Riemann hypothesis},

journal = {ERA Am. Math. Soc},

year = {1997},

pages = {99--104}

}

### OpenURL

### Abstract

Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1.

### Citations

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Some problems of ’Partitio numerorum’; III: on the expression of a number as a sum of primes
- Hardy, Littlewood
- 1923
(Show Context)
Citation Context ...mes Problem,” we mean: can every odd number greater than 5 be written as a sum of three prime numbers? This problem was first successfully attacked by Hardy and Littlewood in their seminal 1923 paper =-=[6]-=-; using their Circle Method and assuming a “Weak Generalized Riemann Hypothesis,” they proved that every sufficiently large odd number could be so written. The second author has calculated [4] directl... |

147 | The book of prime number records - Ribenboim - 1988 |

139 |
The Riemann Zeta-Function
- Ivić
- 1985
(Show Context)
Citation Context ...n − p ≤ 1.615 × 1012 . ] ,102 J.-M. DESHOUILLERS, G. EFFINGER, H. TE RIELE, AND D. ZINOVIEV We note here that the GRH actually implies an estimate on |Θ(n)−n| which has a single log factor; see Ivic =-=[7]-=- for example. However, the second author, in working through the details of such an estimate, found that the constant obtained was large enough so that, at the level n =10 20 , Schoenfeld’s estimate g... |

58 |
Sharper bounds for the chebyshev functions θ(x) and ψ(x
- Rosser, Schoenfeld
- 1975
(Show Context)
Citation Context ...ds and if 6 ≤ n ≤ 1020 , then there exists a prime number p such that 4 ≤ n − p ≤ 1.615 × 1012 . Proof. The conclusion of the lemma obviously holds for n<1012 , say. For larger n, we apply Schoenfeld =-=[13]-=-, equation (6.1). Let Θ(n) = ∑ p≤nlog p; if the GRH holds, and if n ≥ 23 × 108 ,wehave |Θ(n) − n| < 1 √ n(log n − 2) log n. 8π Just suppose that there is no prime in the interval (n − h, n] except pos... |

47 |
Representation of an odd number as the sum of three primes
- Vinogradov
- 1937
(Show Context)
Citation Context ...s approximately 10 50 . In 1926 Lucke [11], in an unpublished doctoral thesis under Edmund Landau, had already shown that with some refinements the figure could be taken as 10 32 . In 1937 Vinogradov =-=[15]-=- used his ingenious methods for estimating exponential sums to establish the desired asymptotic result while avoiding the GRH entirely. However, the numerical implications of avoiding the GRH are subs... |

29 | Basic analytic number theory - Karatsuba - 1993 |

17 | On strong pseudoprimes to several bases
- Jaeschke
- 1993
(Show Context)
Citation Context ...ound for which e − p is prime, but in general this p is neither the smallest nor the largest such prime. For the actual generation of k primes close to a we have used Jaeschke’s computational results =-=[8]-=-, stating that if a positive integer n<2152302898747 is a strong pseudoprime with respect to the first five primes 2, 3, 5, 7, 11, then n is prime; corresponding bounds for the first six and seven pri... |

9 |
On the odd Goldbach problem
- Chen, Wang
- 1989
(Show Context)
Citation Context ...1956 Borodzkin [1] showed that sufficiently large in Vinogradov’s proof meant numbers greater than 3315 ≈ 107000000 . This figure has since been improved significantly, most recently by Chen and Wang =-=[2]-=-, who have established a bound of 1043000 , but in any case this figure is far beyond hope of “checking the remaining cases by computer.” If, however, we return to the original stance of Hardy and Lit... |

5 |
On Vinogradov's constant in Goldbach's ternary problem, 1. Number Theory 65
- Zinoviev
- 1997
(Show Context)
Citation Context ...c theorem Theorem (Zinoviev). Assuming the GRH, every odd number greater than 1020 is a sum of three prime numbers. We discuss here briefly the main ideas behind this result; for complete details see =-=[16]-=-. Fix N ≥ 9. We are interested in the number of triples (p1,p2,p3)ofprime numbers which satisfy the equation (1) N = p1 + p2 + p3. Following [10] we introduce the function J(N) = ∑ log(p1)log(p2) log(... |

4 |
Checking the Goldbach conjecture up to 4
- Sinisalo
(Show Context)
Citation Context ...nd te Riele). Every even number 4 ≤ m ≤ 1013 is a sum of two prime numbers. For a complete exposition of this and related results, see [3]. Let pi be the ith odd prime number. The usual approach [5], =-=[14]-=- to verify the Goldbach conjecture on a given interval [a, b] is to find, for every even e ∈ [a, b], the smallest odd prime pi such that e − pi is a prime. An efficient way to do this is to generate t... |

3 |
A new proof of the Goldbach-Vinogradov theorem
- Linnik
- 1946
(Show Context)
Citation Context ...” If, however, we return to the original stance of Hardy and Littlewood by assuming the truth of the GRH while at the same time using some of the refined techniques of primarily Vinogradov and Linnik =-=[10]-=-, and using an extensive computer search, we do indeed arrive at the following: Theorem. Assuming the GRH, every odd number greater than 5 can be expressed as a sum of three prime numbers. Received by... |

2 |
Vinogradov’s constant
- Borodzkin, On
- 1956
(Show Context)
Citation Context ... for estimating exponential sums to establish the desired asymptotic result while avoiding the GRH entirely. However, the numerical implications of avoiding the GRH are substantial: in 1956 Borodzkin =-=[1]-=- showed that sufficiently large in Vinogradov’s proof meant numbers greater than 3315 ≈ 107000000 . This figure has since been improved significantly, most recently by Chen and Wang [2], who have esta... |

1 |
te Riele, and Yannick Saouter, New experimental results concerning the Goldbach conjecture, toappear
- Deshouillers, Herman
(Show Context)
Citation Context ....615 × 1012 .Butformwehave the following: Theorem (Deshouillers and te Riele). Every even number 4 ≤ m ≤ 1013 is a sum of two prime numbers. For a complete exposition of this and related results, see =-=[3]-=-. Let pi be the ith odd prime number. The usual approach [5], [14] to verify the Goldbach conjecture on a given interval [a, b] is to find, for every even e ∈ [a, b], the smallest odd prime pi such th... |

1 |
Some numerical implication of the Hardy and Littlewood analysis of the 3primes problem, submitted for publication
- Effinger
(Show Context)
Citation Context ...3 paper [6]; using their Circle Method and assuming a “Weak Generalized Riemann Hypothesis,” they proved that every sufficiently large odd number could be so written. The second author has calculated =-=[4]-=- directly from that paper that “sufficiently large,” assuming the “full” Generalized Riemann Hypothesis (GRH below, i.e., that all non-trivial zeros of all Dirichlet L-functions have real part equal t... |

1 |
de Lune, and H. te Riele, Checking the Goldbach conjecture on a vector computer
- Granville, van
- 1989
(Show Context)
Citation Context ...ers and te Riele). Every even number 4 ≤ m ≤ 1013 is a sum of two prime numbers. For a complete exposition of this and related results, see [3]. Let pi be the ith odd prime number. The usual approach =-=[5]-=-, [14] to verify the Goldbach conjecture on a given interval [a, b] is to find, for every even e ∈ [a, b], the smallest odd prime pi such that e − pi is a prime. An efficient way to do this is to gene... |

1 |
Zur Hardy-Littlewoodschen Behandlung des Goldbachschen Problems, Doctoral Dissertation
- Lucke
- 1926
(Show Context)
Citation Context ...rge,” assuming the “full” Generalized Riemann Hypothesis (GRH below, i.e., that all non-trivial zeros of all Dirichlet L-functions have real part equal to 1/2), is approximately 10 50 . In 1926 Lucke =-=[11]-=-, in an unpublished doctoral thesis under Edmund Landau, had already shown that with some refinements the figure could be taken as 10 32 . In 1937 Vinogradov [15] used his ingenious methods for estima... |