The harmonic mean of two numbers a and b is defined as the reciprocal of the arithmetic mean of their reciprocals, given by
.
The famous harmonic series
earns its name “harmonic”, because each term after the first is
the harmonic mean of
its neighboring terms.
The series 
diverges for 
.
To see why, we can compare it to the harmonic series, which corresponds to 
and is known to diverge.
Since 
, for , the divergence is also true for the sum when .
The proof that the harmonic series diverges, by Oresme [c. 1350] is standard now.
He repeatedly doubled the terms collected in successive groups, obtaining
groups of sums greater than one half and thus enabling the result to grow
beyond all bounds. Indeed,
Following that, people started to
think about what would happen if we replaced the natural numbers with primes.
Intuitively, the set of prime numbers is a subset of natural numbers. Will the
series converge in that case? The answer is that the series of the reciprocals
of primes, called Prime Harmonic Series, diverges.
An attempt to prove the statement was
made by Euler. Euler did not provide correct proof of the divergence of the
series in the modern sense. However, he did offer a heuristic argument that
strongly suggested divergence based on an asymptotic formula.
The formula (see wikipedia) implied that the sum of the reciprocals of the primes, less than 
, is
asymptotic to 
as approaches infinity.
Specifically,
we have the following asymptotic formula:
as 
. 
is the Meissel–Mertens
constant for primes.
Recall that when we say
is little-o of 
as 
then 
in the neighborhood of 
(except possibly at itself) and:
To take the sum equal about 
, we solve
which gives
so
7.7883E+134
or
Using the approximation for
the 
th prime 
this corresponds to
The observable universe is
estimated to contain roughly 
atoms, and
is 
times larger than that!
The reciprocal-prime series
does diverge, but reaching even a modest value like 6 requires summing
reciprocals of roughly the first 
primes!
Let’s see what it takes to
prove the formula
known as Mertens’ second
theorem
Dirichlet convolutions
Given two Dirichlet series,
their product can be written as:
Because absolute convergence (see the
section Dirichlet Series in The Riemann Hypothesis
Revealed, where uniform convergence on compact subsets of the half-plane of
convergence is discussed) allows us to freely multiply and rearrange the terms
of the series without affecting its sum, we group together all terms where the
product 
equals a fixed value 
, considering every possible value of
. This approach leads us to the
following result:
Thus, we define the convolution of
two sequences 
and 
by
This operation, known as Dirichlet
convolution, combines the values of 
and 
over all divisors 
of . It plays a central role in analytic
number theory, especially when expressing products of Dirichlet series as
single series involving convolutions of their coefficients.
As a result, we write:
Convolutions of Arithmetical Functions
So, for
two arithmetical functions 
the Dirichlet product
(convolution) is defined as
We can write
where 
vary over all positive
integers whose product is .
This makes the commutative property self-evident. For the
associative property
Let 
then
Similarly, if 
then 
.
Generalized Convolutions
For any complex or real valued
function defined on the positive real axis with 
for 
and 
an arithmetical function
we define
We call this a generalized convolution of and . As 
may not be an integer, we
replace its divisors with all integers up to .
Clearly, we cannot
prove a commutative property but
Setting 
so, the last sum can be
written as
Therefore
We call this the
associative property for the generalized convolution.
Usually, we denote by 
the partial sum of . For we have 
Let
Then
and obviously 
. What about 
and 
?
Thus for 
Proposition: Given the partial sums 
and 
of and respectively we have
Example: Take 
then 
. Then
Notice that
Thus,
Let
so 
, and the condition is 
. Then
Now group by the product 
. For each integer 
, all pairs 
with 
correspond to divisors 
(with 
).
So:
Proposition:
Applying Generalized Convolutions
There is a well-known arithmetical function the Mangoldt
function 
which returns the value of 
when 
, a power of a prime
number and zero otherwise
, 
, 
, 
, 
, 
, …
In general, assuming that for 
Since the only nonzero terms in 
occur when 
, 
and 
Proposition: 
Using the formula 
and the fact that we get
Proposition:
In The Riemann Hypothesis Revealed (see:
A path from numbers to asymptotic summation) we prove:
If and 
real numbers then
A very useful formula which is known as Euler’s
summation formula.
If, for example, we take 
then considering 
in Euler’s summation
formula:
Simple observation shows that 
yielding
Therefore, we get a formula for the
factorial:
Because 
.
Combining this with 
we get:
Proposition:
Since 
unless is a power of a prime and
for 
, we may rewrite the left-hand side as
But
so
Hence
yielding
But 
hence
Dividing by
Proposition:
This implies
Proposition:
Set up partial summation
Let
be a right-continuous step function. 
is constant on each
interval between consecutive primes, and it jumps by
at each prime 
.It certainly isn’t differentiable.
Let 
and
Then
Using Riemann-Stieltjes integration
because 
is the discrete measure
consisting of point masses 
at the primes.
Then integration by parts for Stieltjes integrals says
(since is 
).
Apply partial summation
So
Now
Thus,
There are several ways to compute the Meissel–Mertens constant
for primes; we leave this for a future post.
The divergence of the reciprocal prime series is one of the most
striking examples of how slowly infinity can reveal itself.
As the Greek summer settles in, the tempo changes.
There will be fewer posts in the coming weeks, though the
mathematics does not stop. Research will continue, calculations will
accumulate, and ideas will quietly mature—just at a pace more suited to the
season.
Summer is also a reminder that not every worthwhile discovery
happens at a desk. Some arrive while reading under a tree, walking along the
shore, or simply taking the time to think.
May you find time for both curiosity and rest. New mathematical
journeys await on the horizon.
Until then, καλό καλοκαίρι!
Comments
Post a Comment