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 x. The second Chebyshev function is defined by
where is the von Mangoldt function. The Chebyshev function is often used in proofs related to prime numbers, because it is typically simpler to work with than the prime-counting function, . Both functions are asymptotic to , a statement equivalent to the prime number theorem.
Both functions are named in honour of Pafnuty Lvovich 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 but . 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.
Asymptotics and bounds
proved the following bounds for the Chebyshev functions:
- for k' ≥ exp(22)
- for k ≥ 198
- for k ≥ 198
- for x ≥ 10,544,111
- for x ≥ exp(22)
- for x ≥ exp(30)
Along with , this gives a good characterization of the behavior of these two functions.
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
Here runs over the nontrivial zeros of the zeta function, and
From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of over the trivial zeros of the zeta function, , i.e.
A theorem due to Erhard Schmidt
states that, for any
real, positive K
, there are values of x
infinitely often. On big-O notation, one may write the above as
Hardy and Littlewood prove the stronger result, that
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
A good evidence that RH could be true comes from the fact proposed by Alain Connes and others, that if we differentiate V. Mangoldt formula respect to x make x = exp(u) manipulating we have the formula 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 → ∞.
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.
- Pierre Dusart, "Sharper bounds for ψ, θ, π, ", 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.