Hilbert's inequality
Encyclopedia
In complex analysis
, a branch of mathematics, Hilbert's inequality was first demonstrated by David Hilbert
and later improved by Issai Schur
, who replaced Hilbert's constant
with the sharp constant 
.
Let
be complex numbers, Hilbert's inequality is
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, a branch of mathematics, Hilbert's inequality was first demonstrated by David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
and later improved by Issai Schur
Issai Schur
Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...
, who replaced Hilbert's constant
with the sharp constant .Let
be complex numbers, Hilbert's inequality is-
In 1973, Montgomery & Vaughan reported several generalizations of Hilbert's inequality, considering the bilinear forms
and
where
are distinct real numbers modulo 1 (i.e. they belong to distinct classes in the quotient groupQuotient groupIn mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
) and 
are distinct real numbers. Montgomery & Vaughan's generalizations of Hilbert's inequality are then given by
-
and
-
where
and 
(where 
is the distance from 
to the nearest integer and 
denotes the smallest positive value). Moreover, if 
and 
then the following inequalities holds:
-
and
-
In their paper they give conditions by which the above sums are allowed to be infinite sum.
See also
- Hugh Montgomery (mathematician)
- David HilbertDavid HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
- Online book chapter Hilbert’s Inequality and Compensating Difficulties extracted from .
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