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The Riemann Hypothesis has the following elementary formulation:
If you generate an infinite sequence of heads and tails by flipping a fair coin, then the difference between the number of heads and number of tails tends to be very small. How small? Well, if you look at the first k terms of the sequence, then no matter what k is, the magnitude of the difference is always less than a constant multiple of
[math]k^{3/4}[/math]
and always less than a constant multiple of
[math]k^{2/3}[/math]
and in fact, always less than a constant multiple of
[math]k^{1/2 + \epsilon}[/math]
for any fixed [math]\epsilon > 0[/math].
This bound is sharp in the sense that the magnitude of the difference can be as big as [math]k^{1/2}[/math].
For natural numbers n that are products of distinct prime numbers, assign n the label ``heads'' if n has an even number of prime factors, and assign it ``tails'' if n has an odd number of prime factors.
The Riemann hypothesis states that this labeling is similar to the labeling coming from a fair coin described above, in the sense that if you look at the first k such natural numbers, no matter what k is, the difference between the number of n labelled with heads and the number of n labelled tails is less than a constant multiple of
[math]k^{1/2 + \epsilon}[/math]
for each fixed [math]\epsilon > 0[/math].
If you generate an infinite sequence of heads and tails by flipping a fair coin, then the difference between the number of heads and number of tails tends to be very small. How small? Well, if you look at the first k terms of the sequence, then no matter what k is, the magnitude of the difference is always less than a constant multiple of
[math]k^{3/4}[/math]
and always less than a constant multiple of
[math]k^{2/3}[/math]
and in fact, always less than a constant multiple of
[math]k^{1/2 + \epsilon}[/math]
for any fixed [math]\epsilon > 0[/math].
This bound is sharp in the sense that the magnitude of the difference can be as big as [math]k^{1/2}[/math].
For natural numbers n that are products of distinct prime numbers, assign n the label ``heads'' if n has an even number of prime factors, and assign it ``tails'' if n has an odd number of prime factors.
The Riemann hypothesis states that this labeling is similar to the labeling coming from a fair coin described above, in the sense that if you look at the first k such natural numbers, no matter what k is, the difference between the number of n labelled with heads and the number of n labelled tails is less than a constant multiple of
[math]k^{1/2 + \epsilon}[/math]
for each fixed [math]\epsilon > 0[/math].
The Riemann hypothesis is difficult to state in layman's terms in any sort of precise way. If you are satisfied with knowing that it is question about where the zeros of a particular function [math] \zeta(s) [/math] are, then Hector Pollastri's answer is pretty much perfect. You can't really say more about that, because to construct this function, you need at the very least the power of calculus, and preferably complex analysis.
That said, there are alternative statements that are equivalent to the Riemann hypothesis (i.e. if you prove one of these statements, you prove RH, and vice versa). Some of these statements appear to be very simple. For example, let me define the following objects:
1.) The n-th harmonic number [math] H_n = 1 + \frac{1}{2} + \ldots + \frac{1}{n} [/math]. So, the first few are: [math] H_1 = 1, \ H_2 = 1 + \frac{1}{2}, \ H_3 = 1 + \frac{1}{2} + \frac{1}{3} [/math].
2.) The sum of divisors function [math] \sigma(n) [/math]. This does precisely what you think it does: it counts the number of divisors a particular number has. So, for example [math] \sigma(1) = 1 [/math], [math] \sigma(2) = 2 [/math], [math] \sigma(10) = 4 [/math], and [math] \sigma(12) = 6 [/math].
With this, we can give a very simple conjecture that turns out to be equivalent to RH: [math] \sigma(n) \leq H_n + e^{H_n} \log(H_n) [/math].
I do want to make something clear though: this statement may look simple, but it is incredibly difficult to work with. The fact of the matter is that almost certainly, RH will be proved first, and then this will be a consequence of that, rather than the other way around.
RH is not a good project to mess around with. If you know something about automorphic forms, then you can try to take stabs at RH, but it does not lend itself as an introductory project. There are far better things to play with.
That said, there are alternative statements that are equivalent to the Riemann hypothesis (i.e. if you prove one of these statements, you prove RH, and vice versa). Some of these statements appear to be very simple. For example, let me define the following objects:
1.) The n-th harmonic number [math] H_n = 1 + \frac{1}{2} + \ldots + \frac{1}{n} [/math]. So, the first few are: [math] H_1 = 1, \ H_2 = 1 + \frac{1}{2}, \ H_3 = 1 + \frac{1}{2} + \frac{1}{3} [/math].
2.) The sum of divisors function [math] \sigma(n) [/math]. This does precisely what you think it does: it counts the number of divisors a particular number has. So, for example [math] \sigma(1) = 1 [/math], [math] \sigma(2) = 2 [/math], [math] \sigma(10) = 4 [/math], and [math] \sigma(12) = 6 [/math].
With this, we can give a very simple conjecture that turns out to be equivalent to RH: [math] \sigma(n) \leq H_n + e^{H_n} \log(H_n) [/math].
I do want to make something clear though: this statement may look simple, but it is incredibly difficult to work with. The fact of the matter is that almost certainly, RH will be proved first, and then this will be a consequence of that, rather than the other way around.
RH is not a good project to mess around with. If you know something about automorphic forms, then you can try to take stabs at RH, but it does not lend itself as an introductory project. There are far better things to play with.
The Riemann Hypothesis is a statement about the locations of zeros of a certain function. That function is called the Riemann Zeta Function, denoted [math]\zeta(s)[/math], where [math]s\in\mathbb{C}[/math] is a complex variable. For [math] \Re(s)>1[/math], the function is defined as
[math] \zeta(s) := \sum_{n=1}^\infty \frac{1}{n^s}.[/math]
Through a process called meromorphic continuation, one can extend this function to an analytic function on *almost* the entire complex plane [math]\mathbb{C}[/math]. The exception is that [math]\zeta(s)[/math] has a (simple) pole at [math]s=1.[/math] (NB: the series above does *not* give the definition of [math]\zeta(s)[/math] on the rest of the complex plane.)
It turns out that [math]\zeta(s)[/math] is zero at every negative even integer. These are the so-called 'trivial zeros'. The Riemann Hypothesis is simply the statement that all of the other zeros of [math]\zeta(s)[/math] are on the so-called critical line [math]\Re(s)=1/2[/math]. (It is known that all of the other zeros are in the 'critical strip', which is simply [math]\{s\in\mathbb{C}: 0<\Re(s)<1\}.[/math])
[math] \zeta(s) := \sum_{n=1}^\infty \frac{1}{n^s}.[/math]
Through a process called meromorphic continuation, one can extend this function to an analytic function on *almost* the entire complex plane [math]\mathbb{C}[/math]. The exception is that [math]\zeta(s)[/math] has a (simple) pole at [math]s=1.[/math] (NB: the series above does *not* give the definition of [math]\zeta(s)[/math] on the rest of the complex plane.)
It turns out that [math]\zeta(s)[/math] is zero at every negative even integer. These are the so-called 'trivial zeros'. The Riemann Hypothesis is simply the statement that all of the other zeros of [math]\zeta(s)[/math] are on the so-called critical line [math]\Re(s)=1/2[/math]. (It is known that all of the other zeros are in the 'critical strip', which is simply [math]\{s\in\mathbb{C}: 0<\Re(s)<1\}.[/math])
you'll need first to know what Complex numbers are. Riemann Hypothesis is a statement about the values of the complex numbers [math] z [/math] that satisfy
[math] \zeta (z) = 0 [/math]
Where [math] \zeta [/math] is the Riemann zeta function. Specifically, it says that the real part of the nontrivial zeros of that function is [math] \frac{1}{2} [/math]. It's a really dificult problem, and you need to study a lot more to understand fully every word i said. Even when you understand it, solve it is another question. We probably don't have the correct framework yet to solve this problem. If you like math, keep learning, but it's unlikely you'll solve any millenium problem without a good bag of knownledge first, and this'll take you years, not matter how smart you are.
[math] \zeta (z) = 0 [/math]
Where [math] \zeta [/math] is the Riemann zeta function. Specifically, it says that the real part of the nontrivial zeros of that function is [math] \frac{1}{2} [/math]. It's a really dificult problem, and you need to study a lot more to understand fully every word i said. Even when you understand it, solve it is another question. We probably don't have the correct framework yet to solve this problem. If you like math, keep learning, but it's unlikely you'll solve any millenium problem without a good bag of knownledge first, and this'll take you years, not matter how smart you are.
The direct way to state the Riemann hypothesis is in terms of the Riemann zeta function. However, the reason Riemann introduced that function was to study the distribution of prime numbers, and we can state the Riemann hypothesis in those terms as well.
It's well known that, picking an integer at uniform random between 1 and x, it is prime with probability "approximately" [math]\frac{1}{\ln(x)}[/math] if x is large. Specifically, this is in the sense that the ratio between the probability and the approximation gets arbitrarily close to 1 as x gets arbitrarily large. (This is called the Prime Number Theorem). Therefore, in some shaky sense, an arbitrary number n has probability [math]\frac{1}{\ln(n)}[/math] of being prime, and so, in some sense, we should expect the number of primes below x to be [math]\frac{1}{\ln(2)} + \frac{1}{\ln(3)} + ... + \frac{1}{\ln(x)}[/math]; even better, we could replace the discrete sum with a continuous integral [math]\int_{2}^{x} \frac{1}{\ln(n)} \mathrm{d}n[/math], which we give the name Li(x).
The Riemann hypothesis concerns how good of an approximation Li(x) is to the number of primes below x; specifically, it is equivalent to the statement that the difference between these two is never more than some constant times [math]\ln(x^{x^{1/2}})[/math]. So the approximation error, measured as that difference, can grow as x increases, but, the Riemann hypothesis says, it can't grow too fast. (The exponent 1/2 there relates to the claim that 1/2 is the largest real part of a zero of the Riemann zeta function).
Disclaimer: Not actually my area. Please correct any mistakes I've made.
It's well known that, picking an integer at uniform random between 1 and x, it is prime with probability "approximately" [math]\frac{1}{\ln(x)}[/math] if x is large. Specifically, this is in the sense that the ratio between the probability and the approximation gets arbitrarily close to 1 as x gets arbitrarily large. (This is called the Prime Number Theorem). Therefore, in some shaky sense, an arbitrary number n has probability [math]\frac{1}{\ln(n)}[/math] of being prime, and so, in some sense, we should expect the number of primes below x to be [math]\frac{1}{\ln(2)} + \frac{1}{\ln(3)} + ... + \frac{1}{\ln(x)}[/math]; even better, we could replace the discrete sum with a continuous integral [math]\int_{2}^{x} \frac{1}{\ln(n)} \mathrm{d}n[/math], which we give the name Li(x).
The Riemann hypothesis concerns how good of an approximation Li(x) is to the number of primes below x; specifically, it is equivalent to the statement that the difference between these two is never more than some constant times [math]\ln(x^{x^{1/2}})[/math]. So the approximation error, measured as that difference, can grow as x increases, but, the Riemann hypothesis says, it can't grow too fast. (The exponent 1/2 there relates to the claim that 1/2 is the largest real part of a zero of the Riemann zeta function).
Disclaimer: Not actually my area. Please correct any mistakes I've made.
