2
$\begingroup$

I'm curios about the following question, from an informational viewpoint. What is the purpose in finding/getting analytic formulas for specific arithmetic functions in the context of the Riemann Hypothesis?

In the literature concerning the Riemann Hypothesis are analytic formulas for different arithmetic functions.

Example 1. The so-called explicit formula is an example of the formula for the second Chebyshev function, involving a summation over all non-trivial zeros of the Riemann Zeta function. See it as (9) in this MathWorld.

Example 2. Also, there was in the literature the reasoning for the Mertens function, see this MathWorld due to Titchmarsh, and also Odlyzko and te Riele.

Imagine that I've got a friend that tells me that he/she wants to find an analytic formula (involving a summation over the nontrivial zeros of $\zeta(s)$) for a specific arithmetic function related to some equivalence to the Riemann Hypothesis. What should be my analysis of such a situation? See my question.

Question Is there some general purpose in the attempt to find analytic formulas for specific arithmetic functions that appear in equivalences to the Riemann Hypothesis? Is the reasoning that such analytic representations should solve the Riemann Hypothesis, or will provide us valuable information about the Riemann Hypothesis? For what arithmetic functions should be interesting to find such analytic representation? I am asking/looking for an answer from an informative viewpoint, thus calculations aren't required or that your answer explicitly shows such formulas. What I am waiting for is an explanation of the why it is interesting to find such analytic representations in the context of the Riemann Hypothesis, and your explanation from an informative viewpoint. Many thanks.

$\endgroup$
  • $\begingroup$ If there are some post in these sites Stack Exchange related to my question, please add those. Many thanks all users. $\endgroup$ – user243301 Aug 12 '17 at 9:43
  • $\begingroup$ Many thanks @user3658307 $\endgroup$ – user243301 Aug 22 '17 at 6:31
  • $\begingroup$ Salut, comment ça va? I've decided accept your answer @reuns . Any case since I was asking many related questions, if you or other users want to expand or clarify some aspect about those questions I will appreciate it. $\endgroup$ – user243301 Aug 25 '17 at 16:49
3
+50
$\begingroup$

See books on the Riemann zeta function where everything is explained. The core of explicit formulas for $\sum_{n \le x} a_n$ where $a_n$ is multiplicative or additive is the inverse Mellin transform and the residue theorem applied to $F(s) = \sum_{n=1}^\infty a_n n^{-s}= s \int_1^\infty (\sum_{n \le x} a_n) x^{-s-1}dx$ assumed to have a meromorphic continuation to the whole complex plane, otherwise we only obtain weaker statements (asymptotic estimates). See those kind of questions and those discussing the proof of the prime number theorem.


The Riemann hypothesis is encoded in the very complicated functions $\frac{1}{\zeta(s)}, \log \zeta(s), \frac{\zeta'(s)}{\zeta(s)}$ and in the corresponding arithmetic functions $\mu(n), \frac{\Lambda(n)}{\log n},\Lambda(n)$. Unfortunately we only have access to $\zeta(s)$ a very simple function defined in term of the integers.

The Riemann hypothesis is hard because it fails for virtually any tiny modification you can do to $\zeta(s)$ (Hurwitz zeta function, linear combinations of Dirichlet L-functions..) that's why the spectral interpretation of the Riemann hypothesis is a good idea : thinking to (the imaginary part of) non-trivial zeros as eigenvalues of some unbounded self-adjoint linear operators, whose tiny modifications give a non-self-adjoint operator to which the spectral theorem doesn't apply.

$\endgroup$
  • $\begingroup$ Many thanks for your answer, and linked post. That I would like to know with this question in the post, is that if it is plausible that finding an analytic representation for such arithmetic functions that I've evoked (and even including in such representation a summation over the non-trivial zeros) one should can to solve or analyze some important fact of the Riemann Hypothesis. $\endgroup$ – user243301 Aug 12 '17 at 9:50
  • $\begingroup$ The Riemann hypothesis [...] fails for virtually any tiny modification of $\zeta(s)$ - This reminds me of chaos theory. $\endgroup$ – Lucian Aug 19 '17 at 18:57
  • $\begingroup$ @Lucian The RH is in some sense equivalent to the statement that the (summatory of the) Möbius function behaves like a Brownian motion. In other words, randomize the integers in some way, and show the randomized Möbius function is ... random. $\endgroup$ – reuns Aug 19 '17 at 19:09
  • $\begingroup$ Perhaps $\zeta(s)$ is not such a simple function after all. $\endgroup$ – bonif Jan 30 at 21:28
0
$\begingroup$

I wish to answer the specific question of user243301 :

Is there some general purpose in the attempt to find analytic formulas for specific arithmetic functions that appear in equivalences to the Riemann Hypothesis? Is the reasoning that such analytic representations should solve the Riemann Hypothesis, or will provide us valuable information about the Riemann Hypothesis? For what arithmetic functions should be interesting to find such analytic representation? ......etc.

My suggested answer: (a) I have found that the function which captures the properties of the zeta function, $\zeta(s)$, required to prove the Riemann Hypothesis, is the function $F(s)= \zeta(2s) / \zeta(s)$. This function F(s) has the property that all the non-trivial zeros of $\zeta(s)$ appear as poles in F(s). Therefore what is then needed to prove (the Riemann Hypothesis) is that all the poles of F(s) lie on the critical line.

(b) The most important arithmetical function is the Liouville function $\lambda(n)$, which is defined as $\lambda(n) = +1$, if n has even number of prime factors and $\lambda(n) =-1$ if n has odd number of prime factors. It so turns out that $F(s)=\Sigma \frac{\lambda(n)}{n^s}$. In order to prove that F(s) has only poles on the critical line one needs to carefully examine the factorization properties of integers. It so turns out that the the $\lambda(n)$ behave like "coin tosses" and it is this behavior of the Liouville function which makes all the poles of $F(s)$ lie on the critical line. For details see the citations below.

All the tasks indicated in (a) and (b) above have been done in: Arxiv: https://arxiv.org/pdf/1609.06971v9.pdf

and I have written up a "Road Map" of the paper in: https://www.researchgate.net/publication/318283850_A_Road_Map_of_the_Paper_on_Coin_Tosses_and_the_Proof_of_the_of_the_Riemann_Hypothesis

K. Eswaran

$\endgroup$
  • $\begingroup$ Many thanks, your are generous since was accepted an answer previously. I am going to study your answer, and try to understand your reasonings of your posts and references. Any case I'm sure that it is useful for many users of this site. $\endgroup$ – user243301 Sep 28 '17 at 17:43
  • $\begingroup$ Why $\frac{\zeta(2s)}{\zeta(s)}$ and not $\frac{1}{\zeta(s)}$ or $\log \zeta(s), \frac{\zeta'(s)}{\zeta(s)}$ or $\frac{\zeta'(s)}{\zeta(s)}+\zeta(s)$ and $\frac{\zeta'(s)}{\zeta(s)}+\frac{1}{s-1}$? Also you don't mention Dirichlet series for which the RH fails in your pdf (unique factorization of the integers is not sufficient, we also need the functional equation of $\zeta(s)$ to expect a RH) $\endgroup$ – reuns Sep 28 '17 at 19:16
  • $\begingroup$ Simply because $\zeta(2s) / \zeta(s)$ worked for me. You will have to read the paper and the "Road Map" cited for you to appreciate this. $\endgroup$ – K. Eswaran Sep 29 '17 at 1:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy