In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, Skewes's number is any of several
large numbers
Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical m ...
used by the
South Africa
South Africa, officially the Republic of South Africa (RSA), is the southernmost country in Africa. It is bounded to the south by of coastline that stretch along the South Atlantic and Indian Oceans; to the north by the neighbouring countri ...
n mathematician
Stanley Skewes
Stanley Skewes (; 1899–1988) was a South African mathematician, best known for his discovery of the Skewes's number in 1933. He was one of John Edensor Littlewood's students at Cambridge University. Skewes's numbers contributed to the refine ...
as
upper bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an eleme ...
s for the smallest
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
for which
:
where is the
prime-counting function and is the
logarithmic integral function
In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
. Skewes's number is much larger, but it is now known that there is a crossing between
and
near
It is not known whether it is the smallest crossing.
Skewes's numbers
J.E. Littlewood
John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations, and had lengthy collaborations with G. H. Hardy, Srinivasa Raman ...
, who was
Skewes's research supervisor, had
proved in that there is such a number (and so, a first such number); and indeed found that the sign of the difference
changes infinitely many times. All numerical evidence then available seemed to suggest that
was always less than
Littlewood's proof did not, however, exhibit a concrete such number
.
proved that, assuming that the
Riemann 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 ...
is true, there exists a number
violating
below
:
In , without assuming the Riemann hypothesis, Skewes proved that there must exist a value of
below
:
Skewes's task was to make Littlewood's existence proof
effective
Effectiveness is the capability of producing a desired result or the ability to produce desired output. When something is deemed effective, it means it has an intended or expected outcome, or produces a deep, vivid impression.
Etymology
The ori ...
: exhibiting some concrete upper bound for the first sign change. According to
Georg Kreisel
Georg Kreisel FRS (September 15, 1923 – March 1, 2015) was an Austrian-born mathematical logician who studied and worked in the United Kingdom and America.
Biography
Kreisel was born in Graz and came from a Jewish background; his family ...
, this was at the time not considered obvious even in principle.
More recent estimates
These upper bounds have since been reduced considerably by using large-scale computer calculations of
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) > ...
. The first estimate for the actual value of a crossover point was given by , who showed that somewhere between
and
there are more than
consecutive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s
with
.
Without assuming the Riemann hypothesis, proved an upper bound of
. A better estimate was
discovered by , who showed there are at least
consecutive integers somewhere near this value where
. Bays and Hudson found a few much smaller values of
where
gets close to
; the possibility that there are crossover points near these values does not seem to have been definitely ruled out yet, though computer calculations suggest they are unlikely to exist. gave a small improvement and correction to the result of Bays and Hudson. found a smaller interval for a crossing, which was slightly improved by . The same source shows that there exists a number
violating
below
. This can be reduced to
assuming the Riemann hypothesis. gave
.
Rigorously, proved that there are no crossover points below
, improved by to
, by to
, by to
, and by to
.
There is no explicit value
known for certain to have the property
though computer calculations suggest some explicit numbers that are quite likely to satisfy this.
Even though the
natural density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the de ...
of the positive integers for which
does not exist, showed that the
logarithmic density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the de ...
of these positive integers does exist and is positive. showed that this proportion is about 0.00000026, which is surprisingly large given how far one has to go to find the first example.
Riemann's formula
Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
gave an
explicit formula for
, whose leading terms are (ignoring some subtle convergence questions)
:
where the sum is over all
in the set of
non-trivial zeros of the Riemann zeta function.
The largest error term in the approximation
(if the
Riemann 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 ...
is true) is negative
, showing that
is usually larger than
. The other terms above are somewhat smaller, and moreover tend to have different, seemingly random complex
arguments
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
, so mostly cancel out. Occasionally however, several of the larger ones might happen to have roughly the same complex argument, in which case they will reinforce each other instead of cancelling and will overwhelm the term
.
The reason why the Skewes number is so large is that these smaller terms are quite a ''lot'' smaller than the leading error term, mainly because the first
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
zero of the zeta function has quite a large
imaginary part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
, so a large number (several hundred) of them need to have roughly the same argument in order to overwhelm the dominant term. The chance of
random complex numbers having roughly the same argument is about 1 in
.
This explains why
is sometimes larger than
and also why it is rare for this to happen.
It also shows why finding places where this happens depends on large scale calculations of millions of high precision zeros of the Riemann zeta function.
The argument above is not a proof, as it assumes the zeros of the Riemann zeta function are random, which is not true. Roughly speaking, Littlewood's proof consists of
Dirichlet's approximation theorem
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers \alpha and N , with 1 \leq N , there exist integers p and q such that 1 \leq q \leq N and
...
to show that sometimes many terms have about the same argument.
In the event that the Riemann hypothesis is false, the argument is much simpler, essentially because the terms
for zeros violating the Riemann hypothesis (with
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
greater than ) are eventually larger than
.
The reason for the term
is that, roughly speaking,
actually counts powers of
primes
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, rather than the primes themselves, with
weighted by
. The term
is roughly analogous to a second-order correction accounting for
squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of primes.
Equivalent for prime ''k''-tuples
An equivalent definition of Skewes' number exists for
prime ''k''-tuples (). Let
denote a prime (''k'' + 1)-tuple,
the number of primes
below
such that
are all prime, let
and let
denote its Hardy-Littlewood constant (see
First Hardy-Littlewood conjecture). Then the first prime
that violates the Hardy-Littlewood inequality for the (''k'' + 1)-tuple
, i.e., the first prime
such that
:
(if such a prime exists) is the ''Skewes number for
''
The table below shows the currently known Skewes numbers for prime ''k''-tuples:
The Skewes number (if it exists) for
sexy prime
In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and .
The term "sexy prime" is a pun stemming from the Latin word for six: .
If o ...
s
is still unknown.
It is also unknown whether all admissible ''k''-tuples have a corresponding Skewes number.
References
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External links
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{{Large numbers
Large numbers
Number theory
Large integers