What are we looking for when we are trying to prove Riemann Hypothesis?

Question

My knowledge of mathematics is fairly limited.

But I know that Sieve of Erastothenes can be used to find prime numbers in increasing order. While Riemann Hypothesis enforces that distribution of prime numbers is not random.

But I still don't get it, what are mathematicians looking for exactly? Because when we already have a sieve for prime numbers...

If you google on "riemann hypothesis poular books" you should find what you're looking for. — Barry Cipra, Nov 22 '19 at 14:39
Read this post and Prime Obsession. These two resources are suited towards a layman. — Axion004, Nov 22 '19 at 14:40
@BarryCipra I think you mean that many published works build upon riemann hypothesis and validating riemann hypothesis would validate them but it is not the answer I seek. — Rajat Pundir, Nov 22 '19 at 14:41
@Axion004 I had looked at the medium post you mentioned a year back and I don't think buying the book will definitely answer my question. — Rajat Pundir, Nov 22 '19 at 14:45
@Randall I don't know. My question is fairly fundamental, what do we gain by proving that prime numbers are not random... Why do we go beyond Seive of Erastothenes at all? — Rajat Pundir, Nov 22 '19 at 14:54
The sieve says nothing about the randomness of primes, either way. — Randall, Nov 22 '19 at 14:55
@Randall But we know that sieve isn't a random process. So why would numbers generated by it have random distribution? — Rajat Pundir, Nov 22 '19 at 14:56
The sieve not being a random process says nothing about the distribution of the resulting primes. — Randall, Nov 22 '19 at 14:58
Here is a process that is not random. Consider an integer written out in common English, and delete it if it contains the letter "e." The resulting survivors are pretty random. — Randall, Nov 22 '19 at 15:00
@Randall But we already the seiving process with which we are generating numbers from alphabets here, how is it random in that case.. — Rajat Pundir, Nov 22 '19 at 15:03
@user18646, that is not at all what I meant by my recommendation to google on "popular books." What I meant is that a person with fairly limited knowledge of mathematics, as you describe yourself, would do well to start by reading some of the accounts of the Riemann Hypothesis that have been explicitly written for the lay public. — Barry Cipra, Nov 22 '19 at 15:06

score 0 · Accepted Answer · edited Nov 22 '19 at 16:00

Erastothenes' sieve allows us to find primes, but it says nothing about the distribution of primes. This is where Riemann's Zeta function comes in, as it tell us deep insights about this distribution.

For instance, Riemann gave an explicit formula for the number of primes less than $n$ that involves the non-trivial roots $\rho$ of the Riemann $\zeta$-function:

$${\displaystyle \pi (x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })},$$

with ${\displaystyle \operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} (x^{1/n})}$, $μ(n)$ is the Möbius function and $\operatorname{li}(x)$ the logarithmic integral function. In fact,

$${\displaystyle \pi (x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })-{\frac {1}{\ln {x}}}+{\frac {1}{\pi }}\arctan {\frac {\pi }{\ln {x}}}}.$$

Note that this is an equality, i.e., the exact number of number below $n$. Once we find out whether Riemann's Hypothesis is true, in other words that the $\rho$ are all of the form $\frac12 + bi$, with $b\in\mathbb{R}$, we will discover how the primes behave exactly. As it turns out, this is not trivial at all.

On the other hand, Erastothenes' sieve is more of an algorithm, instead of a deep insight into the nature of the building blocks of the numbers.

If you would like to learn more, a quick search on this website will yield many interesting (and much more concise) answers than mine. I highly recommend you do that.

Thankyou for your time to answer. Here's what I understood: We are looking for insight into the behaviour of the primes then just the algorithm to generate them. But for a lay person like me, I believe Riemann Hypothesis only answers the distribution of primes, I am unsure how Riemann Hypothesis will reveal more on the behaviour part. I will take your recommendation and continue to look for my answer. — Rajat Pundir, Nov 22 '19 at 15:49
In order to answer that question you need an understanding of much more advanced concepts. You could start with complex analysis. — Klangen, Nov 22 '19 at 15:51
@user18646 If you found my answer helpful, don't hesitate to upvote/accept it :) — Klangen, Nov 29 '19 at 09:04

What are we looking for when we are trying to prove Riemann Hypothesis?

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