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The Chebyshev function ?(x), with x < 50
The function ?(x) - x, for x < 104
The function ?(x) - x, for x < 107
In mathematics, the Chebyshev function is either of two related functions. The first Chebyshev function?(x) or ?(x) is given by
with the sum extending over all prime numbersp that are less than or equal to x.
The second Chebyshev function?(x) is defined similarly, with the sum extending over all prime powers not exceeding x
(The numerical value of is log(2?).) Here ? runs over the nontrivial zeros of the zeta function, and ?0 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 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, ? = -2, -4, -6, ..., i.e.
Similarly, the first term, x = , corresponds to the simple pole of the zeta function at 1. It being a pole rather than a 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
The first Chebyshev function is the logarithm of the primorial of x, denoted x#:
This proves that the primorial x# is asymptotically equal to e(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
Certainly ?(x) x, 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 . 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 x we get x = eu. 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) eiu?, which is only true if ? = + iE(n).
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 difference of the smoothed Chebyshev function and for x < 106
The smoothing function is defined as
It can be shown that
The Chebyshev function evaluated at x = et minimizes the functional
^ Pierre Dusart, "Sharper bounds for ?, ?, ?, pk", Rapport de recherche no. 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.