A Study of the Riemann Zeta Function
Khalil Bitar
The well known Riemann zeta function has many interesting properties
and has proven to be useful in many applications in physics. In our
attempt to use one such property, one described by Voronin's
theorems, a new characteristic distribution was discovered numerically
and then calculated analytically. This distribution allows the use of
the Riemann zeta function as a generator of pseudo random numbers.
The Riemann zeta function is complex and according to Voronin's
theorems can approximate *any* complex number when evaluated for an
argument whose real part is in the region 0.5 to 1.0 and for some very
large imaginary part. As one increases the imaginary part one will
encounter other values where this property will repeat itself. In this
`critical strip' if one then chooses a value for the real part
of the argument, 0.75 say, and then varies the imaginary part from say
1.0e6 upwards to larger values, these theorems assure us that one will
generate for some such value *any* pre-chosen number with arbitrary
precision and that this will happen more than once. We have
implemented this process with the aim of studying this characteristic
property.
The aim was to determine whether smooth distributions with which
complex numbers are generated exist and if so whether they may be
calculable. There are several approximations one may use to evaluate
the zeta function for an argument with a large imaginary part and we
have used what is known as the Seigel formula. The formula involves
evaluating a sum of inverse powers with a number of terms that
increases with this imaginary part and a remainder involving the gamma
function with complex argument. On a typical serial machine this could
be programmed to take of the order of a few minutes per evaluation. As
one requires a large number of such evaluations to study distributions
(of the order of several hundred thousands to few millions) the use of
serial machines becomes totally out of the question. On the other hand
a massively parallel machine, where many such evaluations may be done
concurrently, is ideal. With the CM-2 Connection Machine at SCRI we
were able to perform 65536 such evaluations every few minutes, the
time being determined by the largest argument used. We could collect
and study a few million zeta functions very easily.
We have evaluated such sets by taking an initial value for the
imaginary part and then incrementing it by a constant `delta.' A
typical distribution of the zeta functions generated is given in the
figure below. The first interesting property we discovered is that
aside from being smooth, these distributions are independent of both
the initial value we use and the magnitude of the increment
`Delta'. They depend only on the choice of the real part of the
argument. Therefore for a particular such value the distribution is
universal. Furthermore if one computes the autocorrelations of the
numbers generated for any of these distributions this autocorrelation
is consistent with zero for delta of order unity or larger. Thus the
process may be thought of as a pseudo random number generator.
*A three-dimensional plot showing the distribution of a typical set of
Riemann zeta functions where the initial imaginary part is incremented
by a constant delta.*
Using moment integrals over the zeta function we were then able to
derive expressions for the distribution of the absolute value of the
zeta function and its logarithm. These turn out to be expressible as
inverse Mellin transforms over well known functions.
Knowing these distributions allows the use of the zeta function in
evaluating the path integrals for quantum mechanical systems. We have
tested this on simple systems such as the anharmonic oscillator with
good results. Further more since the zeta function is an analytic
function with known properties its use in these applications may lead
to a definition of the path integral in the continuum.
*Dr. Khalil Bitar is a SCRI research scientist in theoretical high
energy physics and is a member of the international High Energy Monte
Carlo Grand Challenge Collaboration.*
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