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Chebyshev function

In mathematics, the Chebyshev function is either of two related functions. The first Chebyshev function ϑ(x) or θ(x) is given by

vartheta(x)=sum_{ple x} log p

with the sum extending over all prime numbers p that are less than x. The second Chebyshev function psi(x) is defined by

psi(x) = sum_{n leq x} Lambda(n),

where Lambda 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, pi(x). Both functions are asymptotic to x, a statement equivalent to the prime number theorem.

Both functions are named in honour of Pafnuty Lvovich Chebyshev.

Relationships

The second Chebyshev function can be seen to be related to the first by writing it as
psi(x)=sum_{ple x} k log p

where k is the unique integer such that p^kle x but p^{k+1}>x. A more direct relationship is given by

psi(x)=sum_{n=1}^infty vartheta left(x^{1/n}right).

Note that this last sum has only a finite number of non-vanishing terms, as

vartheta left(x^{1/n}right) = 0 for n>log_2 x.

The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to n.
operatorname{lcm}(1,2,dots n)=e^{psi(n)}.

Asymptotics and bounds

Pierre Dusart proved the following bounds for the Chebyshev functions:

vartheta(p_k)ge kleft(ln k+lnln k-1+frac{lnln k-2.0553}{ln k}right) for k' ≥ exp(22)

vartheta(p_k)le kleft(ln k+lnln k-1+frac{lnln k-2}{ln k}right) for k ≥ 198

psi(p_k)le kleft(ln k+lnln k-1+frac{lnln k-2}{ln k}right) + 1.43sqrt x for k ≥ 198

|vartheta(x)-x|le0.006788frac{x}{ln x} for x ≥ 10,544,111

|psi(x)-x|le0.006409frac{x}{ln x} for x ≥ exp(22)

psi(x)-vartheta(x)<0.0000132frac{x}{ln x} for x ≥ exp(30)

Along with psi(x)ge vartheta(x), 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 psi(x) as a sum over the nontrivial zeros of the Riemann zeta function:

psi_0(x) = x - sum_{rho} frac{x^{rho}}{rho} - frac{zeta'(0)}{zeta(0)} - frac{1}{2} log (1-x^{-2}).

Here rho runs over the nontrivial zeros of the zeta function, and

psi_0(x) = begin{cases} psi(x) - frac{1}{2} Lambda(x) & x = p^m mbox{, p prime, m an integer} psi(x) & mbox{otherwise.} end{cases}

From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of -x^{omega}/{omega} over the trivial zeros of the zeta function, omega = -2, -4, -6, ldots, i.e.

sum_{k=1}^{infty} frac{x^{2k}}{2k} = frac{1}{2} log (1 - x^{-2} ).

Properties

A theorem due to Erhard Schmidt states that, for any real, positive K, there are values of x such that

psi(x)-x < -Ksqrt{x}

and

psi(x)-x > Ksqrt{x}

infinitely often. On big-O notation, one may write the above as

psi(x)-x ne Oleft(sqrt{x}right).

Hardy and Littlewood prove the stronger result, that

psi(x)-x ne Oleft(sqrt{x}logloglog xright).

Relation to primorials

The first Chebyshev function is the logarithm of the primorial of x, denoted x#:

vartheta(x)=sum_{ple x} log p=log prod_{ple x} p = log 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

Pi(x) = sum_{n leq x} frac{Lambda(n)}{log n}.

Then

Pi(x) = sum_{n leq x} Lambda(n) int_n^x frac{dt}{t log^2 t} + frac{1}{log x} sum_{n leq x} Lambda(n) = int_2^x frac{psi(t), dt}{t log^2 t} + frac{psi(x)}{log x}.

The transition from Pi to the prime-counting function, pi, is made through the equation

Pi(x) = pi(x) + frac{1}{2} pi(x^{1/2}) + frac{1}{3} pi(x^{1/3}) + cdots.

Certainly pi(x) leq x, so for the sake of approximation, this last relation can be recast in the form

pi(x) = Pi(x) + O(sqrt x).

The Riemann hypothesis

The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part 1/2. In this case, |x^{rho}|=sqrt x, and it can be shown that

sum_{rho} frac{x^{rho}}{rho} = O(sqrt x log^2 x).

By the above, this implies

pi(x) = operatorname{li}(x) + O(sqrt x log x).

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

zeta(1/2+i hat H )|n ge zeta(1/2+iE_{n})=0,

sum_{n}e^{iu E_{n}}=Z(u)=e^{u/2}-e^{-u/2} frac{dpsi _{0}}{du}-frac{e^{u/2}}{e^{3u}-e^{u}} = operatorname{Tr}(e^{iuhat H }).

Where the "trigonometric sum" can be considered to be the trace of the operator (statistical mechanics) e^{iu hat H} ,which is only true if rho =1/2+iE(n) .

Using the semiclassical approach the potential of H=T+V satisfies:

frac{Z(u)u^{1/2}}{sqrt pi }sim int_{-infty}^{infty}e^{i (uV(x)+ pi /4 )},dx

with Z(u) → 0 as u → ∞.

Smoothing function

The smoothing function is defined as

psi_1(x)=int_0^x psi(t),dt.

It can be shown that

psi_1(x) sim frac{x^2}{2}.

Variational formulation

The Chebyshev function evaluated at x = exp(t) minimizes the functional

J[f]=int_{0}^{infty}frac{f(s)zeta' (s+c)}{zeta(s+c)(s+c)},ds-int_{0}^{infty}!!!int_{0}^{infty} e^{-st}f(s)f(t),ds,dt,

so

f(t)= psi (e^t)e^{-ct},,

for c > 0.

References

  • Pierre Dusart, "Sharper bounds for ψ, θ, π, p_k", 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.

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