An explanation of the importance of analytic formulas representating arithmetic functions related to equivalences to the Riemann Hypothesis

Question

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.

Answer 1

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.

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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.

Answer 2

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