Square of the Riemann zeta-function in terms of the divisor summatory function.

In summary, the conversation discusses different approaches for expressing the divisor summatory function, D(x), in terms of the Riemann zeta function, \zeta(s). One participant suggests using the Mellin inversion formula, while another mentions seeing D(x) expressed in terms of the roots of \zeta(s). A reference is provided for further reading on the topic.
  • #1
AtomSeven
8
0
Hi,

The divisor summatory function, [tex]D(x)[/tex], can be obtained from [tex]\zeta^{2}(s)[/tex] by [tex]D(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty}\zeta^{2}(w)\frac{x^{w}}{w}dw[/tex] and I was trying to express [tex]\zeta^{2}(s)[/tex] in terms of [tex]D(x)[/tex] but I didnt succeed, could someone help?
 
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  • #2
Use the Mellin inversion formula.
 
  • #3
Hi,

I've done this by a different approach considering that [tex]d(n)=D(n)-D(n-1)[/tex] and [tex] D(0)=0[/tex] it follows that
[tex]
\begin{align}
\zeta^{2}(s)&=\sum_{n=1}^{\infty} \frac{\sigma_{0}(n)}{n^{s}}=\sum_{n=1}^{\infty} \frac{D(n)-D(n-1)}{n^{s}} \nonumber\\
&=\sum_{n=1}^{\infty} \frac{D(n)}{n^{s}}-\sum_{n=1}^{\infty} \frac{D(n-1)}{n^{s}}=\sum_{n=1}^{\infty} \frac{D(n)}{n^{s}}-\sum_{n=1}^{\infty} \frac{D(n)}{(n+1)^{s}} \nonumber\\
&=\sum_{n=1}^{\infty}D(n)\bigg\{ \frac{1}{n^{s}} - \frac{1}{(n+1)^{s}} \bigg\}=\sum_{n=1}^{\infty}D(n)\int_{n}^{n+1}\frac{s}{x^{s+1}} dx \nonumber\\
&=s\sum_{n=1}^{\infty}\int_{n}^{n+1}\frac{D(x)}{x^{s+1}} dx =s\int_{1}^{\infty}\frac{D(x)}{x^{s+1}} dx \nonumber
\end{align}
[/tex]

So it would be interesting to see if anyone could solve this using the Mellin inversion aproach.

--
Seven
 
  • #4
from the properties of Mellin transform i would bet that

[tex] D= \sum_{n\le x}\sigma_{0} = \sum_{n\ge1}[(x/n)] [/tex]

since the Mellin transform of [tex] \sum_{n=1}^{\infty}f(xn) [/tex] is [tex] \zeta (s) F(s) [/tex]here [x] means the floor function
 
  • #5
Hy everyone,

I think that some time ago I've seen [tex]D(x)[/tex] expressed in terms of the roots of the [tex]\zeta(s)[/tex] function. Does anyone knows of references about this?
 
  • #6
Eynstone said:
Use the Mellin inversion formula.

I don't see how to do that. Can you show me (and the OP I assume too) how to do that please?

I did try ok. If I need to show my work, I could but I got to a spot where I tried to represent the integrand in the form that I think I could have inverted it, the inversion didn't come out well.
 
  • #7
For those interested here is a refference:

M. Lukkarinen, The Mellin transform of the square of Riemann’s zeta-function and Atkinson’s formula, Doctoral Dissertation, Annales Acad. Sci. Fennicae, No. 140, Helsinki, 2005
 

1. What is the Riemann zeta-function?

The Riemann zeta-function is a mathematical function that was first introduced by Bernhard Riemann in the 19th century. It is defined as the infinite sum of the reciprocals of the positive integers, raised to a complex power.

2. What is the divisor summatory function?

The divisor summatory function is a mathematical function that counts the number of divisors of a given positive integer. It is closely related to the Riemann zeta-function and is often used in number theory and analytic number theory.

3. What is the square of the Riemann zeta-function in terms of the divisor summatory function?

The square of the Riemann zeta-function in terms of the divisor summatory function is a mathematical relationship that connects the two functions. It states that the square of the Riemann zeta-function can be expressed as a sum of the divisor summatory function multiplied by the natural logarithm of the positive integers.

4. Why is the square of the Riemann zeta-function in terms of the divisor summatory function important?

The square of the Riemann zeta-function in terms of the divisor summatory function is important in number theory and analytic number theory because it helps in understanding the distribution of prime numbers. It is also used in the proof of the Prime Number Theorem, which describes the asymptotic behavior of the prime counting function.

5. How is the square of the Riemann zeta-function in terms of the divisor summatory function related to the Riemann hypothesis?

The square of the Riemann zeta-function in terms of the divisor summatory function is closely related to the Riemann hypothesis, which is one of the most famous unsolved problems in mathematics. The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta-function lie on the critical line with a real part of 1/2. The square of the Riemann zeta-function in terms of the divisor summatory function is used in the proof of the Riemann hypothesis for certain classes of zeta-functions.

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