In mathematics, the Chebyshev function is either of two related functions. The first Chebyshev function ϑ(x) or θ(x) is given by
with the sum extending over all prime numbers p that are less than or equal to x.
The second Chebyshev function ψ(x) is defined similarly, with the sum extending over all prime powers not exceeding x:
where is the von Mangoldt function. The Chebyshev functions, especially the second one ψ(x), are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, π(x) (See the exact formula, below.) Both Chebyshev functions are asymptotic to x, a statement equivalent to the prime number theorem.
Both functions are named in honour of Pafnuty Chebyshev.
The second Chebyshev function can be seen to be related to the first by writing it as
where k is the unique integer such that pk ≤ x and x < pk+1. The values k of are given in A206722. 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 of the integers from 1 to n.
Values of for the integer variable n is given at A003418.
Asymptotics and bounds
- for k ≥ 198,
- for x ≥ 10,544,111,
- for x ≥ exp(22),
Further, under the Riemann hypothesis,
Upper bounds exist for both and such that,
An explanation of the constant 1.03883 is given at A206431.
The exact formula
(The numerical value of ζ'(0)/ζ(0) is log(2π).) Here runs over the nontrivial zeros of the zeta function, and ψ0 is the same as ψ, except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right:
Similarly, the first term, x = x1/1, corresponds to the simple pole of the zeta function at 1. Its being a pole rather than zero accounts for the opposite sign of the term.
A theorem due to Erhard Schmidt states that, for some explicit positive constant K, there are infinitely many natural numbers x such that
and infinitely many natural numbers x such that
In little-o notation, one may write the above as
Relation to primorials
The first Chebyshev function is the logarithm of the primorial of x, denoted x#:
This proves that the primorial x# is asymptotically equal to exp((1+o(1))x), where "o" is the little-o notation (see Big O notation) and together with the prime number theorem establishes the asymptotic behavior of pn#.
Relation to the prime-counting function
The Chebyshev function can be related to the prime-counting function as follows. Define
The transition from to the prime-counting function, , 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 states that all nontrivial zeros of the zeta function have real part 1/2. In this case, , and it can be shown that
By the above, this implies
Good evidence that RH could be true comes from the fact proposed by Alain Connes and others, that if we differentiate the von Mangoldt formula with respect to x make x = exp(u). Manipulating, we have the "Trace formula" for the exponential of the Hamiltonian operator satisfying
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 H = T + V satisfies:
with Z(u) → 0 as u → ∞.
solution to this nonlinear integral equation can be obtained (among others) by in order to obtain the inverse of the potential :
The smoothing function is defined as
It can be shown that
The Chebyshev function evaluated at x = exp(t) minimizes the functional
for c > 0.
- Rosser, J. Barkley; Schoenfeld, Lowell (1962). "Approximate formulas for some functions of prime numbers.". Illinois J. Math. 6: 64–94.
- ^ Pierre Dusart, "Estimates of some functions over primes without R.H.". arXiv:1002.0442
- ^ Pierre Dusart, "Sharper bounds for ψ, θ, π, pk", Rapport de recherche n° 1998-06, Université de Limoges. An abbreviated version appeared as "The kth prime is greater than k(ln k + ln ln k − 1) for k ≥ 2", 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. ISBN 0-387-95097-4. Google Book Search.
- Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001