Deriving the Meissel–Mertens Constant

 

The analysis naturally continues from Mertens’ second theorem

 

(see The Sum That Needs More Than the Universe Can Count), which gave the asymptotic behavior of the prime–reciprocal sum . That same asymptotic, , now becomes the key ingredient in extracting the Meissel–Mertens constant from the Stieltjes representation of the prime zeta function.

Brace yourself — the next post is so long even the primes will need a coffee break before they finish reading it.

 

Dirichlet series and convolution of sequences

The Riemann Zeta function may be thought of as a special case of a more general series called the Dirichlet series. The Dirichlet series expressed in the form:

where  is a complex variable and  a sequence of complex numbers. When  for all , the associated Dirichlet series is:

which is the Riemann Zeta function !

Generally, regarding convergence for Dirichlet series, if we find a real , for which the series converges, then it also converges for all  with  and diverges for all  such that . We start by assuming its partial sums are bounded for a fixed . That is, for all  

Then, following the approach used in The Riemann Hypothesis Revealed, we obtain the following result:

 

If for all

for a fixed , then for  with  and

 

For example, if  and the partial sums  are bounded then we have for

Letting , the right-hand side becomes infinitely small hence the sequence of the partial sums of the series is a Cauchy sequence and the Dirichlet series converges absolutely for .

The smallest  such that the Dirichlet series converges absolutely is called the abscissa of convergence of the series. Of course, if the partial sums of the series are bounded for some  then .

Dirichlet Convolution

Let  and  be two Dirichlet series

We calculate the product

Each term of the product is of the form . So, the terms with denominator  will be the ones where . This, in return, implies that the coefficient of  is:

The notation  means  is a divisor of . Therefore,

in the half plane where both series converge absolutely.

We define the Dirichlet convolution of two sequences  by

The summation extends over all the divisors of .

Let for example  be the identity function, i.e.,  and  where  returns the sum of all divisors of . Using the formula for the convolution we’ll calculate . For , the divisors of  are .

Therefore, for , the convolution.

In this manner, the convolution of arithmetic functions computes a new arithmetic function that encapsulates the combined effect of  and  as  varies, providing insights into the relationships between different arithmetic functions.

We can write

where  vary over all positive integers whose product is . The definition of  makes the commutative property self-evident. For the associative property

let  then

Similarly, if  then .

Dirichlet's convolution and regular or Cauchy’s convolution of sequences are two different operations. Cauchy’s convolution of sequences ,  also denoted by , is defined for a non-negative integer  as:

assuming that both sequences start from zero index. In what follows, we’ll use the Dirichlet convolution unless noted otherwise.

The Möbius function

The Möbius function  is defined as follows:

If  write . Then if

If  then  otherwise . Note that  if  has a square factor > 1. In essence, Möbius function, encodes information about factorization of integers into prime factors.

The first ten values of the Möbius function are:

,  ,  ,  ,  ,  ,  ,  , ,  .

Adding the values of Möbius function for all divisors of , i.e.,  we realize that the only non-vanishing terms come from  and from those divisors of  which are products of distinct primes. Thus

Therefore, we have:

The sum of  over all divisors of   is

For  . We can define a function  where  when  and  when  or better  so, we can write

 serves for an identity element when convolving since  for  and

This gives us the opportunity to define an inverse of an arithmetical function.

Evidently  which for  reduces to  so we must have  to define  at .

When  and assuming that all values of  for  have been defined, we have

Which obviously gives:

That is,  is defined by the recurrence relation:

 

If an arithmetical function takes non-zero value at , i.e.,  then we can define its inverse by the recurrence relation:

 

Inserting the unity function:  for all , in the identity

we can write

which says:

Since , we have  and .

If  then , i.e.,

But then if  multiplying by  we get . That is, the above statement goes both ways.

Möbius inversion formula:

Möbius and Zeta

Let’s consider the product  extended over all the distinct prime divisors of . Assume  be the primes that divide  with their assigned indices. Then,

where the first sum is over all the reciprocals of prime divisors, the second is over all products of reciprocals taken two at a time and so on. Note that each term is of the form  where  is a divisor of . The numerator  is exactly  . So,

Now starting with

and letting   we get

Obviously using the unity function  for every  we can write:

and since both series are absolutely convergent for  we can multiply and rearrange the terms in any way we please without altering the sum. We chose to collect together those terms for which  is a constant for all possible values of the constant, i.e.,

This is a second prof of

Using the same technique, multiplying  by  where  is an appropriate arithmetic function, yields:

That is, there is an arithmetic function  such that , i.e., . But from Möbius inversion formula we should have  so,

and  are related with Möbius inversion formula.

Generalized Convolutions

We introduced “Generalized Convolutions” in a previous post. For completeness, we briefly revisit the idea here, but since this post builds directly on that discussion, we recommend reading the earlier post first. The reason for this recap is that we will introduce two new generalizations.

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.

Generalizing Inversion Formulas

Let

Then

The converse is also true.

Proof:

Evidently

Thus,

 

Furthermore, from the above, if  is an arithmetic function possessing a Dirichlet inverse , then

The previous formula is considered a special case of this if we take !                               

Now start form

where

assuming that the series involved are absolutely convergent for all .

Plug into  the value of

Now set . Every pair  with  corresponds to a unique  and a divisor . So

But

Thus, only  survives and

So  is the inversion of . Similarly starting from  and plugging into  we get an identity.

Therefore,

Now assume that  is an arithmetic function possessing a Dirichlet inverse , i.e.,

Start from

Plug into

so

Thus,

Clearly, this is because

A useful form of the inversion formula is:

under the usual assumption that the defining series converge absolutely. It is known as Möbius inversion with kernel .

The Prime Zeta function

Define the prime zeta function

Start with Euler's product, valid for :

Taking logarithms,

Now expand the logarithm:

Substituting ,

Interchanging the sums (which is justified since everything converges absolutely),

Applying the Möbius inversion with kernel  we get:

Set up partial summation

Define the step function

This is an increasing step function with jumps

at each prime.

Therefore, choosing the function  the Riemann–Stieltjes integral against a step function simply adds the jumps

Apply partial summation

Applying the Stieltjes integration by parts

It is interesting to investigate what happens when .

Previously we saw that

This causes the boundary term to vanish at infinity.

Thus,

From this

Separate the integrals:

Since

Now consider the error term

It becomes

 

or

 

where for , given an , there exists  such that  for all .

So,

The first integral since  is fixed satisfies

The tail

This blows up unless  decays more slowly than .

Therefore,

if you control the approach so that

Let’s assume that and continue with the final estimation.

So, if the error term is correct along the real axis (and in controlled approach regions), the problem reduces to finding an estimate for

Calculate the Meissel–Mertens constant

Make the substitution  to obtain

Let  

and name  the right-hand side integral.

The dominant contribution comes from large , so replacing the lower limit by  gives

Let . Then

because as we’ve seen in The Riemann Hypothesis Revealed

Recalling that for   inside the right regions

Thus,

On the other hand, Möbius inversion gives

and for  (see: The Riemann Hypothesis Revealed, Zeta’s values near one)

hence

and since  near , here , we have

Thus,

or

which expresses the Meissel–Mertens constant entirely in terms of Euler's constant, the Möbius function, and the values of the Riemann zeta function at the positive integers.

 

And with that, I’ll let the primes rest and everyone else enjoy the sunshine. Have a beautiful summer — may your days be as bright as a well‑behaved asymptotic.