From a
conventional perspective the Riemann Zeta function is identified solely with -
what I refer to as - the Zeta 1 function.
And as is
well known this function can be expressed in two ways, both as a sum over
natural numbers and a product over primes.
So in
general terms,
ζ1(s) = ∑ 1/ns = ∏ 1/(1 –
p–s)
n = 1 p
n = 1 p
So for example, when s = 2,
ζ1(2) = 1/12 + 1/22 + 1/32 + … = 1/(1 – 2– 2) * 1/(1 – 3– 2) * 1/(1 – 5– 2) * …
= 1 + 1/4 + 1/9 + … = 4/3
* 9/8 * 25/24 * … = π2/6 .
However my strong contention throughout is that the Zeta
function can only be properly understood in a dynamic relative manner, entailing
the dynamic interaction of two related aspects, which I refer to as Zeta 1 and
Zeta 2 functions.
Properly understood, this also requires two distinctive types of
mathematical understanding that are analytic (quantitative) and holistic (qualitative)
with respect to each other.
And the analytic aspect (in this newly defined context) relates
to the notion of number as relatively independent (of other numbers) whereas
the holistic aspect relates to the complementary appreciation of number as
relatively interdependent (with other numbers).
So in the dynamics of understanding, the very nature of
mathematical symbols keeps switching as between analytic and holistic
appreciation, i.e. their particle and wave aspects - which are complementary
opposite in nature.
Therefore in conventional mathematical interpretation, a crucial
distortion is at work, whereby the holistic (qualitative) aspect - in every
formal context - is reduced in a merely analytic (quantitative) manner.
When one fully grasps the significance of this observation, then
it becomes apparent that the conventional understanding of number, despite all
the admitted great advances that have been made, fundamentally cannot be fit for purpose.
Whereas I refer to the Zeta 1 as ζ1(s) -
strictly ζ1(s1) - I refer to the Zeta 2 function as ζ2(s),
or again more accurately as ζ2(s2).
In general terms ζ2(s2) in its infinite
expression = 1 + s12 + s22
+ s23 + … =
1/(1 – s2 ).
So in fact it represents an infinite geometric series with common
ratio = s2 .
However the significance here is that each of the individual
terms in both the sum over natural numbers and product over primes expressions
of the Zeta 1 function, can be expressed in terms of the corresponding Zeta 2
aspect.
In this way the Zeta 1 function, seen from one important
perspective can be viewed as representing a collective sum (over all the natural
numbers) or alternatively a collective product (over all the primes)
of individual Zeta 2 functions.
So again to briefly illustrate, let us take the 3rd
term of the Zeta 1 function above (for s1 = 2)!
Now, in the sum over natural numbers expression, this is given as
1/9, which can be stated in terms of the Zeta 2 function as,
{1 + 1/10 + (1/10)2 + (1/10)3 + …} – 1 =
{1/(1 – 1/10)} – 1 = 10/9 – 1 = 1/9.
The link here with the 3rd natural number 3 can be
shown through rearranging the denominator of each term of the expression in the
following manner,
{1 + 1/(32 + 1) + 1/(32 + 1)2 +
1/(32 + 1)3 + …} – 1.
And this approach can be fully generalised.
Therefore the 4th term of the Zeta 1 (sum over
natural numbers expression) = 1/16.
And {1 + 1/17 + (1/17)2 + (1/17)3 + …} – 1
= {1/(1 – 1/17)} – 1 = 17/16 – 1 = 1/16.
Again {1 + 1/17 + (1/17)2 + (1/17)3 + …} –
1
= {1 + 1/(42 +
1) + 1/(42 + 1)2 + 1/(42 + 1)3 + …}
– 1.
In this way the 4th natural number (i.e. 4) is
directly associated with the Zeta 2 expression for this 4th term (of
the Zeta 1 function).
And the Zeta 2 function can equally be used in place of each
individual term of the product over primes expression of the Zeta 1.
So again when s = 2, the 3rd term of the product over
primes expression (for the Zeta 1) = 25/24.
Then in terms of the Zeta 2,
1 + 1/25 + 1/252 + 1/253 + … = 1/(1 – 1/25) = 25/24.
And we can then show directly the link here in the Zeta 2
(with respect to the 3rd prime = 5) by rewriting the expression in
the following manner i.e.
1 + 1/52 + 1/54 + 1/56 + … = 1/(1 – 1/52) = 25/24.
Thus once again, the Zeta 1 function - both with respect to its
sum over natural numbers and product over primes expressions - can be
completely written as the collective sum and product respectively of individual
Zeta 2 functions.
Thus in general terms,
∑ 1/ns = ∏ 1/(1 – p–s) =
ζ1(s) = ∑{ζ2(1/n) – 1}s
= ∏{ζ2(1/p)s}
n = 1 p n = 2 p
n = 1 p n = 2 p
However, though in its own way remarkable, this formulation of
the Zeta 1 function (as the collective sum and product respectively of
individual Zeta 2 functions) is not yet complete.
Though we have been able to express the Zeta 1 function in two
related manners (again as both the sum over natural numbers and product over
primes respectively), so far internally, we have expressed the Zeta 2 function
in just one way as the sum of repeated multiplied terms.
However to complete the picture, we need to show an alternative
formulation for the Zeta 2 function where the value of the function, in complementary
fashion, results from the sum of repeated added terms.
And this is where the recent entries on the Spectrum of
Mathematics web-site have borne fruit, as they have finally made this latter
piece of the jig-saw readily apparent.
In those entries, I consider - rather in the manner of the
Fibonacci sequence - the unique infinite number sequences associated with the
general polynomial equation,
(x – 1)n = 0.
Now in the case of the Fibonacci, the unique number sequence,
0, 1, 1, 2, 3, 5, 8, 12, 21, … is associated with the equation x2
– x – 1 = 0.
Corresponding unique infinite number sequences are likewise associated
with (x – 1)n = 0.
For example when n = 2, we obtain (x – 1)2 = 0, i.e.
x2 – 2 x + 1 = 0.
And the unique number sequence associated with this equation is
the set of natural numbers, i.e. 0, 1, 2, 3, 4, 5, ….
Then when n = 3, we obtain (x – 1)3 = 0, i.e. x3
– 3x2 + 3x – 1 = 0.
And the unique number sequence associated with this equation is
the set of triangular numbers,
0, 0, 1, 3, 6, 10, 15, ….
There are strong links here with the binomial theorem, and
indeed the diagonal rows (and columns) of Pascal’s triangle can be used as an alternative way of determining the
unique number sequence associated with (x – 1)n for each natural number value of n.
So for n = 4, we obtain the so-called tetrahedral numbers,
0, 0, 0, 1, 4, 10, 20, 35, ….
Now the significance of all these number sequences in the present context, is with
respect to the their (infinite) sum of their reciprocals.
So for example, in the first case when we sum the reciprocals of
the natural numbers, we obtain
1 + 1/2 + 1/3 + 1/4 + …
And of course, this represents the well-known harmonic series,
which is the value of the Zeta 1 function i.e. ζ1(s1), where s1
= 1.
Though the sum of this series diverges to infinity, in all other
cases for (x – 1)n where n > 2,
the sum of reciprocals of the unique number sequences involved, converge to a
finite rational number.
Furthermore a simple general pattern relates to these sums, with
the value depending solely on n and given by the simple expression (n – 1)/(n –
2).
Therefore the sum of reciprocals of the triangular numbers associated with (x – 1)3, i.e.
1 + 1/3 + 1/6 + 1/10 + 1/15 + … = (3 – 1)/(3 – 2) = 2/1.
Now to show that this sum of reciprocals involves all the
natural numbers, we can rewrite it as follows
1 + 1/(1 + 2) + 1/(1 + 2 + 3) + 1/(1 + 2 + 3 + 4) + 1/( 1 + 2 + 3
+ 4 + 5) + …
Therefore the nth term in this manner contains the
sum of the 1st n natural numbers!
Then the denominators in reciprocals of number sequences for (x
– 1)n, where n > 3, contain compound combinations of all the
natural numbers (to n) for the nth term.
The importance (in this context) of these sums of reciprocals is
that they can then be used as the alternative Zeta 2 expressions, where
each individual term of the Zeta 1 - both in its sum over natural numbers and
product over primes expressions - now represents the sum of additive terms with
respect to the Zeta 2 infinite series.
So again with respect to the Zeta 1 function, where s = 2, the 3rd
term of the sum over natural numbers expression = 1/9.
This can now be expressed through the alternative formulation of
the Zeta 2 (representing the sum of compound natural number terms).
So we use here the unique digit sequence associated with (x – 1)11
= 0,
i.e. 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 66, 286, 1001, 3003,
8008, …
There is a relatively quick way of working out the terms in all
these sequences based on the universal ratio of the (t + 1)th to the tth
term = (t + n – 1)/t.
Therefore as 8008 (the last number given) is the 7th term, the next term = 8003 * (7 + 11 – 1)/7 = 19448.
These sequences can all be found, listed to a large number
of terms at
So the sum of reciprocals of the sequence associated with (x –
1)11 = 0, is
1 + 1/11 + 1/66 + 1/286 + 1/1001 + 1/3003 + 1/8008 + 1/19448 + …
= (11 – 1)/11– 2) = 10/9.
So the 3rd term in the Zeta 1 sum over natural
numbers expression (where s = 2) =
(1 + 1/11 + 1/66 + 1/286 + …) – 1
And the denominators 11, 66, 286 represent in turn, ordered
compound combinations of the first 2, first 3 and first 4 natural numbers
respectively.
Then the corresponding 3rd term in the Zeta 1 product over primes expression (where s = 2) = 25/24
This turn is associated with the alternative Zeta 2
functions based relating to the sum of reciprocals of the unique number sequence
associated with (x – 1)26 = 0,
i.e. 1 + 1/26 + 1/351 + 1/3276 + …
= (26 – 1)/26– 2) = 25/24.
And alternative Zeta 2 functions are available for the
individual terms in the corresponding Zeta 1 functions (both in the sum over
natural numbers and product over primes expressions) for all integer values of
s ≥ 2.
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