## An elementary problem equivalent to the Riemann hypothesis

Venue: | Amer. Math. Monthly |

Citations: | 23 - 2 self |

### BibTeX

@ARTICLE{Lagarias_anelementary,

author = {Jeffrey C. Lagarias},

title = {An elementary problem equivalent to the Riemann hypothesis},

journal = {Amer. Math. Monthly},

year = {},

pages = {534--543}

}

### Years of Citing Articles

### OpenURL

### Abstract

ABSTRACT. The problem is: Let Hn = n∑ n ≥ 1, that with equality only for n = 1. j=1 1 j d ≤ Hn + exp(Hn)log(Hn),

### Citations

290 | Algebraic Number Theory - Lang - 1994 |

236 | Riemann’s Zeta Function - Edwards - 1974 |

177 | An Introduction to the Theory of Numbers. Fifth edition - Hardy, Wright - 1979 |

167 | The Lost Notebook and other unpublished papers, Narosa New Delhi - Ramanujan - 1988 |

132 |
Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsbericht der Berliner Akad
- Riemann
(Show Context)
Citation Context ...‘hard’ would be a better letter to use, since our object is to show the following equivalence. Theorem 1.1 Problem E is equivalent to the Riemann hypothesis. The Riemann hypothesis, stated by Riemann =-=[17]-=- in 1859, concerns the complex zeros of the Riemann zeta function. The Riemann zeta function ζ(s) is defined by the Dirichlet series ζ(s) = ∞∑ n −s , n=1 which converges for ℜ(s) > 1, and it has an an... |

110 | Zeros of zeta functions and symmetries
- Katz, Sarnak
- 1999
(Show Context)
Citation Context ...a function (L-functions) and their connections with problems in number theory, algebraic geometry, topology, representation theory and perhaps even physics, see Berry and Keating [3], Katz and Sarnak =-=[8]-=- and Murty[11]. The connection of the Riemann hypothesis with prime numbers was the original question studied by Riemann [17]. Let π(x) count the number of primes p with 1 < p ≤ x. C. F. Gauss noted e... |

43 | The Riemann zeros and eigenvalue asymptotics
- Berry, Keating
- 1999
(Show Context)
Citation Context ...alizations of the zeta function (L-functions) and their connections with problems in number theory, algebraic geometry, topology, representation theory and perhaps even physics, see Berry and Keating =-=[3]-=-, Katz and Sarnak [8] and Murty[11]. The connection of the Riemann hypothesis with prime numbers was the original question studied by Riemann [17]. Let π(x) count the number of primes p with 1 < p ≤ x... |

25 |
Grandes valeurs de la fonction somme des diviseurs et hypothèse de
- Robin
- 1984
(Show Context)
Citation Context ...to use, since the object of this note is to show the following equivalence. Theorem 1.1 Problem E is equivalent to the Riemann hypothesis. Problem E encodes a modification of a criterion of Guy Robin =-=[6]-=- for the Riemann hypothesis. Our aim was obtain a problem statement as elementary as possible, containing no undefined constants. However the hard work underlying the equivalence really resides in the... |

17 |
An introduction to Ramanujan’s “lost
- Andrews
- 1979
(Show Context)
Citation Context .... 339]. Only the first 52 of 75 sections were printed. The manuscript of the unpublished part was eventually rediscovered among the papers of G. N. Watson, in “Ramanujan’s Lost Notebook” (see Andrews =-=[2]-=- 1 Alaoglu and Erdős [1] use a slightly stronger definition of colossally abundant number; they impose the additional requirement that σ(n) n1+ǫ > σ(k) k1+ǫ must hold for 1≤k < n. With their definitio... |

13 | Petites valeurs de la fonction d’Euler - Nicolas - 1983 |

8 |
Problems and results in number theory
- Erdös
- 1979
(Show Context)
Citation Context ...r the hard work underlying the equivalence really resides in the results of Robin given in §2. The general theme of Robin’s work traces back to Ramanujan’s work on highly composite numbers, see Erdős =-=[2]-=-, via the notion of colossally abundant numbers introduced in Alaoglu and Erdős [1]. 2. Proofs We use two results of Robin [6].Proposition 2.1 (G. Robin) If the Riemann hypothesis is true, then for e... |

6 |
Highly composite numbers. Annotated and with a foreword by J.-L
- Ramanujan
- 1997
(Show Context)
Citation Context ...ǫ > σ(k) k1+ǫ must hold for 1≤k < n. With their definition the colossally abundant numbers are exactly those given by (2.3) below. 4and Ramanujan [15, pp. 280–308]) and finally published in 1997, in =-=[16]-=-. In it superabundant and colossally abundant numbers are considered in Section 59, as special cases of the concepts of generalised highly composite numbers and superior generalised highly composite n... |

3 |
Petites valeurs de la fonction d’Euler et hypothèse de Riemann, Séminaire de Théorie des nombres D.P.P
- Nicolas
- 1983
(Show Context)
Citation Context ...to take any value 1 − b < β < 1 1 2 , where b = ℜ(ρ) for some zero ρ of ζ(s) with ℜ(ρ) > 2 , and C > 0 must be chosen sufficiently small, depending on β. The proof uses ideas from a result of Nicolas =-=[4]-=-, [5, Proposition 3], which itself uses a method of Landau. The extremal numbers are a subset of the colossally abundant numbers studied by Alaoglu and Erdős [1]. We prove two preliminary lemmas. Lemm... |

3 | Répartition des nombres superabondants - Erdős, Nicolas - 1975 |

3 |
A motivated introduction to the Langlands program, Advances in number theory
- Murty
- 1991
(Show Context)
Citation Context ...-functions) and their connections with problems in number theory, algebraic geometry, topology, representation theory and perhaps even physics, see Berry and Keating [3], Katz and Sarnak [8] and Murty=-=[11]-=-. The connection of the Riemann hypothesis with prime numbers was the original question studied by Riemann [17]. Let π(x) count the number of primes p with 1 < p ≤ x. C. F. Gauss noted empirically tha... |

3 |
Highly composite numbers, Proc
- Ramanujan
- 1915
(Show Context)
Citation Context ...teristic smooth shape which can be almost completely described (see (2.3) below). In fact these classes of numbers had been studied earlier, by Ramanujan, in his 1915 work on highly composite numbers =-=[14]-=-. The notes in Ramanjuan’s Collected Papers report: “The London Mathematical Society was in some financial difficulty at the time, and Ramanujan suppressed part of what he had written in order to save... |

2 |
On Highly Composite and Similar
- Alaoglu, Erdős
- 1944
(Show Context)
Citation Context ...iven in §2. The general theme of Robin’s work traces back to Ramanujan’s work on highly composite numbers, see Erdős [2], via the notion of colossally abundant numbers introduced in Alaoglu and Erdős =-=[1]-=-. 2. Proofs We use two results of Robin [6].Proposition 2.1 (G. Robin) If the Riemann hypothesis is true, then for each n ≥ 5041, ∑ d ≤ e γ n log log n , (2.1) where γ is Euler’s constant. Proof. Thi... |

1 | Hilbert’s Tenth Problem: Aspects of a negative solution, in: Mathematical Developments arising from Hilbert - Davis - 1976 |