The de Bruijn–Newman constant, denoted by Λ and named after
Nicolaas Govert de Bruijn
Nicolaas Govert (Dick) de Bruijn (; 9 July 1918 – 17 February 2012) was a Dutch mathematician, noted for his many contributions in the fields of analysis, number theory, combinatorics and logic. and
Charles M. Newman, is a
mathematical constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
defined via the
zeros of a certain
function ''H''(''λ'', ''z''), where ''λ'' is a
real parameter and ''z'' is a
complex variable. More precisely,
:
,
where
is the
super-exponentially decaying function
:
and Λ is the unique real number with the property that ''H'' has only real zeros
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
''λ'' ≥ Λ.
The constant is closely connected with
Riemann's hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pu ...
concerning the zeros of the
Riemann zeta-function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
: since the Riemann hypothesis is equivalent to the claim that all the zeroes of ''H''(0, ''z'') are real, the Riemann hypothesis is equivalent to the
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
that Λ ≤ 0.
[ (announcement post)] Brad Rodgers and
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
proved that Λ < 0 cannot be true, so
Riemann's hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pu ...
is equivalent to Λ = 0.
A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.
History
De Bruijn showed in 1950 that ''H'' has only real zeros if ''λ'' ≥ 1/2, and moreover, that if ''H'' has only real zeros for some λ, ''H'' also has only real zeros if λ is replaced by any larger value.
Newman proved in 1976 the existence of a constant Λ for which the "if and only if" claim holds; and this then implies that Λ is unique. Newman also conjectured that Λ ≥ 0.
Upper bounds
De Bruijn's upper bound of
was not improved until 2008, when Ki, Kim and Lee proved
, making the
inequality strict.
discussion
.
In December 2018, the 15th
Polymath project improved the bound to
.
A manuscript of the Polymath work was submitted to arXiv in late April 2019,
[
(preprint)] and was published in the journal Research In the Mathematical Sciences in August 2019.
This bound was further slightly improved in April 2020 by Platt and Trudgian to
.
[
(preprint)]
Historical bounds
References
External links
*
{{DEFAULTSORT:De Bruijn-Newman constant
Mathematical constants
Analytic number theory