Download Some problems of probabilistic number theory. nogerbek Nurbol 11

SOME PROBLEMS OF PROBABILISTIC NUMBER THEORY.
Nogerbek Nurbol 11 “A”.
Zheskazgan, Regional kazakh-turkish high school for gifted boys.
Amirkhanov K.M.
Once, in the study of probability theory in the mathematics lesson, the teacher gave
us a challenge: "What is the probability that any randomly chosen integer has no
square factors?"
We know that if a number has no square factors, then it must not be a multiple of
the square of any prime.
Take any randomly chosen integer n. Suppose that A - the event that any randomly
chosen integer n is a square of a prime number, then - an event the fact that a
randomly chosen integer n is not a square of a prime number.
Location:
P (A) + P (B) = 1.
where P - probability of the event; p - a prime number.
Then:
P (A) =
P (B) =
(see [1])
Now applying the same solution to all integers, we must find what is the
probability that the number has no multipliers
…. and ....
For example: what is the probability that the number is not divisible by 4, 9 and 25
(these numbers are the squares of the first three primes)?
Applying the previously used formula, we obtain:
(
)(
)(
)=
It turns out that the probability that the number is not divisible by 4, 9 and 25
is equal to
. Now the simple calculation reveals that the product of the first 10
terms is 0.61234 ...
This approach determines the number of composite numbers that do not have a
square factor. This composite will, in accordance with the fundamental theorem of
arithmetic, be evaluable as the product of primes, and thus such a number would be
included in this method.
Clearly,
Each term in this product can be evaluated as a geometric series, summed to
infinity.
Consider the formula for the sum to infinity of the infinite series:
when -1<r<1
Applying the formula for the sum of an infinitely decreasing geometric sequence:
S=
Now let r =
(clearly -1<r<1). Substituting this into the formula gives:
So it seems the original component of the product P appears as the denominator
of the right hand side. Therefore:
Consider the expansion of
as products:
Clearly the computation of this product is going to involve the summing of many
products, each of which will contain a term from each of the series bracketed
above. One such product will for example be:
... and in general each denominator will be the product of squares of primes.
Consider the Fundamental Theorem of Arithmetic, which states that every integer
n can be expressed as the product of primes, in an unique solution, such that:
n=
…
Therefore:
=
…
... as in the denominators.
Due to the fact that each representation of
unique denominator can be expressed as
is unique, this means that each
.
Therefore:
Clearly, these sequences are not geometric, as the ratio between the terms is not
constant. Therefore another method is necessary to evaluate the sum of the
reciprocals of the squares.
The search for this sum is not new. Since the time of Euler, a number of methods
have been suggested, one of which is given here. It turns out, perhaps rather
surprisingly, that the sum:
(1)
(See [3])
THE EVALUATION OF:
(see [2])
This sum is easily calculable to a finite degree of accuracy through basic
arithmetic, but for centuries mathematicians have been finding methods of
evaluating the exact sum. Here is one such method.
Trigonometry, as indicated by the appearance of
, is a major component.
Consider the following trigonometric sum A:
Using the identity:
2
+
Combined with the fact that:
=
The sum A can be re-written as:
After simplification we have:
Therefore:
=
Dividing both sides of the equation by:
Gives:
Consider the result of multiplying both sides by x and integrating in the interval
[0, ] with respect to x:
Evaluating the left hand side of expression (2):
Each of the next integrals can be evaluated through integration by parts:
=
Therefore, the sum of the integrals on the previous page will not change for even
values of n. Hence we may assume n is odd, say n = 2t + 1. Making this
substitution into expression (2) gives:
At t the left side of expression (3) is equal to
minus twice the sum of infinity of
the reciprocals of the odd squares. The right side will tend to zero, given the
following theorem of Riemann.
Riemann’ Lemma:
For any function, f, which is continuous and differentiable on
both go to 0 as t
, the integrals:
.
Then the integral:
May be written as:
Since
is continuous and differentiable on [0, ], this can be transformed
into the sum of two integrals as well as in the Riemann’s lemma the addition
formula for sine to
Therefore:
or:
The “missing terms”, ie the sum of the reciprocal of the even powers are a multiple
of the entire sum of the reciprocals of squares:
Also:
Therefore, adding the two equations above we have:
or:
That is:
Therefore:
Therefore:
P - the probability that any commitment made by an integer has no square factors.
Taking
this gives the value of P as 0.608 ..., which is not far away from
the number 0.61234 ..., which I obtained from the earlier numerical simulation.
USED LITERATURE
[1] W.Duivesteijn, «Probabilistic Number Theory» January 21,2005, page 18
Theorem 6.1
[2] Giesy, D. «Mathematics Magazine» 1972
[3] Scientific magazine «»КВАНТ» №2 за 2007 год, стр.33
[4]
В.С.
Лютикас.
«Факультативный
Москва «Просвещение» 1990.
курс
по
математике»,
[5] Gareth A. Jones. «Mathematics Magazine» (66:5), December, 1993.
[6] Tom Apostol. «Introduction to Analytic Number Theory». New York
«Springer Verlag». 1976.
[7] Sue Gordan «Basic concepts in probability». Sydney «University of Sydney»
2005.
[8] J. Kubilius «Probabilistic Methods in the Theory of Numbers» «AMS
Monographs» 1964
[9]
http://xxx.lang.gov
[10]
http://www.arxiv.org
[11]
http://www.rareit.server247.co.uk/trainrare/primes/primes.
[12]
http://www.osp.mit.edu