Intereting Posts

How to define $A\uparrow B$ with a universal property as well as $A\oplus B$, $A\times B$, $A^B$ in category theory?
Software to find out adjacency matrix of a graph.
Elements of order 5 in $S_7$, odd permutations of order 4 in $S_4$, and find a specific permutation in $S_7$
Pólya's Enumeration formula and isomers
Prove that $\int_0^1|f''(x)|dx\ge4.$
minimal surface of revolution when endpoints on x-axis?
A new imaginary number? $x^c = -x$
Solving a congruence without Fermat's little theorem
isometry $f:X\to X$ is onto if $X$ is compact
Mathematical meaning of certain integrals in physics
Why does $\int_0^\infty\frac{\ln (1+x)}{\ln^2(x)+\pi^2}\frac{dx}{x^2}$ give the Euler-Mascheroni constant?
Measure on the set of rationals
What is the difference between a variety and a manifold?
When is $\mathrm{Hom}(A,R) \otimes B =\mathrm{Hom}(A,B)$?
Is $\sqrt{x}$ concave?

Legendre’s conjecture is that there exists a prime number between $n^2$ and $(n+1)^2$. This has been shown to be very likely using computers, but this is merely a heuristic. I have read that if this conjecture is true, the biggest gap between two consecutive primes is $O(\sqrt{p})$; the Riemann Hypothesis, on the other hand, implies that this gap is $O(\sqrt{p}\log{p})$, which is a wider gap for sufficiently large inputs. This leaves me asking the following question, which I am aware may be a bit naive, but I want to be sure:

Would a proof of Legendre’s conjecture also be considered a proof of Riemann’s hypothesis?

EDIT: I am aware that the way the asymptotic above were expressed was in terms of prime numbers, and that the Riemann Hypothesis is concerned about everything in between as well; would a proof of this sort need to show this upper bound for all inputs, or would the prime numbers be sufficient?

- Why does zeta have infinitely many zeros in the critical strip?
- Does dividing by zero ever make sense?
- A number-theory question on the deficiency function $2x - \sigma(x)$
- Importance of the zero free region of Riemann zeta function
- Relationship between Dixonian elliptic functions and Borwein cubic theta functions
- Are there any Combinatoric proofs of Bertrand's postulate?

EDIT 2: It seems to me that, if a proof of Legendre’s conjecture *could* result in a proof of the Riemann hypothesis, that this would come from the $\log{x}$ term of the asymptotic, showing that this term is never smaller than what results from the distance between primes. In other words, this would have to be an inductive proof, showing an initial case and, as a result of that initial case and the fact that the gap is $O(\sqrt{p})$ for that case, all future cases must therefore be $O(\sqrt{p} \log{p})$. I hate to add to my question once again, but please tell me if this is completely off.

- Bijection between the set of classes of positive definite quadratic forms and the set of classes of quadratic numbers in the upper half plane
- Counting the Number of Integral Solutions to $x^2+dy^2 = n$
- Verifying Carmichael numbers
- Is this argument valid? (Number Theory)
- Is there a way to determine how many digits a power of 2 will contain?
- lower bound for the prime number function
- If $\gcd( a, b ) = 1$, then is it true to say $\gcd( ac, bc ) = c$?
- Show that lcm$(a,b)= ab$ if and only if gcd$(a,b)=1$
- Determine all primes $p$ for which $5$ is a quadratic residue modulo $p$
- What percentage of numbers is divisible by the set of twin primes?

The Riemann hypothesis (RH) implies that $p_{k+1}-p_k=O(\sqrt{p_k}\log p_k)$, which was shown by Cramer in $1919$. However, this corollary is much weaker than RH. A proof of Legendre’s conjecture would imply that $p_{k+1}-p_k=O(\sqrt{p_k})$, which is indeed better than Cramer’s result. But this does not necessarily mean that it implies RH. It only means that Legendre

is better than a *consequence* of RH.

In fact, it is conjectured that the gap $p_{k+1}-p_k$ is even better than

$O(\sqrt{p_k})$, i.e., $O(\log^2 p_k)$ which is much smaller, and that between $n^2$ and $(n+1)^2$ there lies not only *one* prime but quite a lot of primes.

- What is a number?
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- Prove that there is an irrational number and a rational number between any two distinct real numbers
- Get the adjacency matrix of the dual of a 3-connected $k$-regular $G$ without pen and paper
- Left Adjoint of a Representable Functor
- Proving $\sqrt2$ is irrational
- What is the highest power of $n$ in $(n^r-1)!$
- Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$
- Why is the differentiation of $e^x$ is $e^x$?
- Countability of local maxima on continuous real-valued functions
- Prove that $\frac{ 5^{125}-1}{ 5^{25}-1}$ is a composite number
- How to find the interval $$ on which fixed-point iteration will converge of a given function $f(x)$?
- Context free languages closure property $\{a^n b^n : n\geq 0\} \cup \{a^n b^{2n}: n\geq 0\}$
- Sketch the Solid of Integration
- Proving that two systems of linear equations are equivalent if they have the same solutions