In
mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function or is given by
:
where
denotes the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
, with the sum extending over all
prime number
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 way ...
s that are less than or equal to .
The second Chebyshev function is defined similarly, with the sum extending over all prime powers not exceeding
:
where is the
von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
Definition
The von Man ...
. The Chebyshev functions, especially the second one , are often used in proofs related to
prime numbers
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 ...
, because it is typically simpler to work with them than with the
prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ).
History
Of great interest in number theory is ...
, (See
the exact formula, below.) Both Chebyshev functions are asymptotic to , a statement equivalent to the
prime number theorem.
Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalarizing function is used when one has several functions to be minimized and one wants to "scalarize" them to a single function:
:
By minimizing this function for different values of
, one obtains every point on a
Pareto front
In multi-objective optimization, the Pareto front (also called Pareto frontier or Pareto curve) is the set of all Pareto efficient solutions. The concept is widely used in engineering. It allows the designer to restrict attention to the set of eff ...
, even in the nonconvex parts.
[ Often the functions to be minimized are not but for some scalars . Then
All three functions are named in honour of ]Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebysh ...
.
Relationships
The second Chebyshev function can be seen to be related to the first by writing it as
:
where is the unique integer such that and . The values of are given in . A more direct relationship is given by
:
Note that this last sum has only a finite number of non-vanishing terms, as
:
The second Chebyshev function is the logarithm of the least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bo ...
of the integers from 1 to .
:
Values of for the integer variable is given at .
Asymptotics and bounds
The following bounds are known for the Chebyshev functions: (in these formulas is the th prime number , , etc.)
:
Furthermore, under 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 pu ...
,
:
for any .
Upper bounds exist for both and such that,
:
for any .
An explanation of the constant 1.03883 is given at .
The exact formula
In 1895, Hans Carl Friedrich von Mangoldt proved an explicit expression for as a sum over the nontrivial zeros of the Riemann zeta function:
:
(The numerical value of is .) Here runs over the nontrivial zeros of the zeta function, and is the same as , except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right:
:
From the Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
for the logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
, the last term in the explicit formula can be understood as a summation of over the trivial zeros of the zeta function, , i.e.
:
Similarly, the first term, , corresponds to the simple pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
* Pole star, a visible star that is approximately aligned with th ...
of the zeta function at 1. It being a pole rather than a zero accounts for the opposite sign of the term.
Properties
A theorem due to Erhard Schmidt
Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. Schmidt was born in Tartu (german: link=no, Dorpat), in the Gover ...
states that, for some explicit positive constant , there are infinitely many natural numbers such that
:
and infinitely many natural numbers such that
:
In little- notation, one may write the above as
:
Hardy
Hardy may refer to:
People
* Hardy (surname)
* Hardy (given name)
* Hardy (singer), American singer-songwriter Places Antarctica
* Mount Hardy, Enderby Land
* Hardy Cove, Greenwich Island
* Hardy Rocks, Biscoe Islands
Australia
* Hardy, Sout ...
and Littlewood Littlewood is a surname, and may refer to:
* Alison Littlewood, British author
* Angela Littlewood (born 1949), English shot putter
* Barclay Littlewood (born 1978), British entrepreneur
* Chic Littlewood (1930–2015), New Zealand actor
* Clayto ...
prove the stronger result, that
:
Relation to primorials
The first Chebyshev function is the logarithm of the primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...
of , denoted :
:
This proves that the primorial is asymptotically equal to , where "" is the little- notation (see big notation) and together with the prime number theorem establishes the asymptotic behavior of .
Relation to the prime-counting function
The Chebyshev function can be related to the prime-counting function as follows. Define
:
Then
:
The transition from to the prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ).
History
Of great interest in number theory is ...
, , is made through the equation
:
Certainly , so for the sake of approximation, this last relation can be recast in the form
:
The Riemann hypothesis
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 pu ...
states that all nontrivial zeros of the zeta function have real part . In this case, , and it can be shown that
:
By the above, this implies
:
Good evidence that the hypothesis could be true comes from the fact proposed by Alain Connes and others, that if we differentiate the von Mangoldt formula with respect to we get . Manipulating, we have the "Trace formula" for the exponential of the Hamiltonian operator satisfying
:
and
:
where the "trigonometric sum" can be considered to be the trace of the operator ( statistical mechanics) , which is only true if .
Using the semiclassical approach the potential of satisfies:
:
with as .
solution to this nonlinear integral equation can be obtained (among others) by
:
in order to obtain the inverse of the potential:
:
Smoothing function
The smoothing function is defined as
:
Obviously :
Variational formulation
The Chebyshev function evaluated at minimizes the functional
:
so
:
Notes
* Pierre Dusart
Pierre Dusart is a French mathematician at the Université de Limoges who specializes 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 i ...
, "Estimates of some functions over primes without R.H.".
* Pierre Dusart, "Sharper bounds for , , , ", Rapport de recherche no. 1998-06, Université de Limoges. An abbreviated version appeared as "The th prime is greater than for ", ''Mathematics of Computation'', Vol. 68, No. 225 (1999), pp. 411–415.
* Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", ''Mathematische Annalen'', 57 (1903), pp. 195–204.
* G .H. Hardy and J. E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", ''Acta Mathematica'', 41 (1916) pp. 119–196.
* Davenport, Harold (2000). In
Multiplicative Number Theory
'. Springer. p. 104. . Google Book Search.
References
*
External links
*
*
* {{planetmathref, urlname=ChebyshevFunctions, title=Chebyshev functions
with images and movies
Arithmetic functions