Psi(x) Plotter
The applet below animates the partial sums (in
order of increasing ) of Von
Mangoldt's formula for :
where the appearing in the formula are zeros
of the Riemann Zeta
Function; more on the significance of
and below.

To use the applet, enter starting and ending values and click the ``ok'' button  this sets the viewing
range. Don't go below 1.5 or above 5000 for ,
and don't make the difference between starting and ending values
any more than 100, otherwise you can't see the grid or the graph
detail. The applet will stop once the first 5000 zeta function
zeros are used. Convergence is fairly fast for small values, and considerably slower once you set starting and
ending values up in the thousands. Place the mouse
pointer over the various buttons for a short description of their
function in the messages area.
The function above was first introduced by
Chebyshev as
where the are the primes (i.e. the function which
jumps by at each prime power) and is
closely related to the distribution of prime numbers. The
appearing in the formula are the zeros
of the Riemann Zeta
Function which lie in the region (called the critical strip), and
the sum should be considered as
The prime number theorem states that (the
number of primes less than ) is asymptotic to
, that is,
This last statement is equivalent to
and it was which ultimately led to the proof
of the prime number theorem by Hadamard and De la Vallée
Poussin (independently) in 1896. The formula for is actually a variation of an explicit formula
proposed by Riemann in his groundbreaking 1859 paper On the
Number of Primes Less Than a Given Magnitude. Riemann's
formula expresses the distribution of the primes in terms of the
zeros of the Riemann Zeta
Function. It was in this paper that Riemann conjectured the
now famous Riemann Hypothesis: all zeros of the zeta function lie
on the line in the complex plane, real.
If you spot any booboos or have any comments, please send me
some mail.