Gronwall's Theorem
Let 
be the Divisor Function. Then
where
is the Euler-Mascheroni Constant. Ramanujan independently discovered a less precise versionof this theorem (Berndt 1994). Robin (1984) showed that the validity of the inequality
for
is equivalent to the Riemann Hypothesis.
References
Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, p. 94, 1985. Gronwall, T. H. ``Some Asymptotic Expressions in the Theory of Numbers.'' Trans. Amer. Math. Soc. 37, 113-122, 1913. Nicholas, J.-L. ``On Highly Composite Numbers.'' In Ramanujan Revisited: Proceedings of the Centenary Conference (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin). Boston, MA: Academic Press, pp. 215-244, 1988. Robin, G. ``Grandes Valeurs de la fonction somme des diviseurs et hypothèse de Riemann.'' J. Math. Pures Appl. 63, 187-213, 1984.
- Divisibility Tests
- Divisible
- Division
- Division Algebra
- Division Lemma
- Division Ring
- Divisor
- Divisor Function
- Divisor Theory
- Dixon-Ferrar Formula
- Dixon's Factorization Method
- Dixon's Random Squares Factorization Method
- Dixon's Theorem
- D-Number
- Dobinski's Formula
- Dodecadodecahedron
- Dodecagon
- Dodecagram
- Dodecahedral Conjecture
- Dodecahedral Graph
- Dodecahedral Space
- Dodecahedron
- Dodecahedron 2-Compound
- Dodecahedron-Icosahedron Compound
- Dodecahedron Stellations