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Interval with primes

Interval with primes under RH

Interval with primes in a.p.

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Short intervals containing primes

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The story seems to start in 1845 when Bertrand conjectured after numerical trials that the interval $]n,2n-3]$ contains a prime as soon as $n\ge4$. This was proved by \v Ceby\v sev in 1852 in a famous work where he got the first good quantitative estimates for the number of primes less than a given bound, say $x$. By now, analytical means combined with sieve methods (see ) ensures us that each of the intervals $[x,x+x^{0.525}]$ for $x \geq x_0$ contains at least one prime. This statement concerns only for the (very) large integers. It falls very close to what we can get under the assumption of the Riemann Hypothesis: the interval $[x-K\sqrt{x}\log x,x]$ contains a prime, where $K$ is an effective large constant and $x$ is sufficiently large (cf for an account on this subject). A theorem of Schoenfeld also tells us that the interval \begin{equation*} [x-\sqrt{x}\log^2x/(4\pi),x] \end{equation*} contains a prime for $x\geq 599$ under the Riemann Hypothesis. These results are still far from the conjecture in on probabilistic grounds: the interval $[x-K\log^2x,x]$ contains a prime for any $K > 1$ and $x\geq x_0(K)$. Note that this statement has been proved for almost all intervals in a quadratic average sense in assuming the Riemann Hypothesis and replacing $K$ by a function $K(x)$ tending arbitrarily slowly to infinity. proved the following.
Theorem (1976)

Let $x$ be a real number larger than $2\,010\,760$. Then the interval $$ \Bigl] x \Bigl(1-\frac1{16\,597}\Bigr),x \Bigr] $$ contains at least one prime.

The main ingredient is the explicit formula and a numerical verification of the Riemann hypothesis. From a numerical point of view, the Riemann Hypothesis is known to hold up to a very large height (and larger than in 1976). and the Zeta grid project verified this hypothesis till height $T_0=2.41\cdot 10^{11}$ and till height $T_0=2.44 \cdot 10^{12}$ thus extending the work who had conducted such a verification in 1986 till height $5.45\times10^8$. This latter computations has appeared in a refereed journal, but this is not the case so far concerning the other computations; section 4 of the paper casts some doubts on whether all the zeros where checked. Discussions in 2012 with Dave Platt from the university of Bristol led me to believe that the results of can be replicated in a very rigorous setting, but that it may be difficult to do so with the results of with the hardware at our disposal. In , we used the value $T_0=3.3 \cdot 10^{9}$ and obtained the following.
Theorem (2002)

Let $x$ be a real number larger than $10\,726\,905\,041$. Then the interval $$ \Bigl] x \Bigl(1-\frac1{28\,314\,000}\Bigr),x \Bigr] $$ contains at least one prime.

If one is interested in somewhat larger value, the paper also contains the following.
Theorem (2002)

Let $x$ be a real number larger than $\exp(53)$. Then the interval $$ \Bigl] x \Bigl(1-\frac1{204\,879\,661}\Bigr),x \Bigr] $$ contains at least one prime.

Increasing the lower bound in $x$ only improves the constant by less than 5 percent. If we rely on , we can prove that
Theorem (2004, conditional)

Let $x$ be a real number larger than $\exp(60)$. Then the interval $$ \Bigl] x \Bigl(1-\frac1{14\,500\,755\,538}\Bigr),x \Bigr] $$ contains at least one prime.

Note that all prime gaps have been computed up to $10^{15}$ in , extending a result of . In , we find
Theorem (2016)

Let $x$ be a real number larger than $2\,898\,242$. The interval $$ \Bigl[ x, x \Bigl(1+\frac1{111(\log x)^2}\Bigr) \Bigr] $$ contains at least one prime.

In , we find
Theorem (2016)

Let $x$ be a real number larger than $468\,991\,632$. The interval $$ \Bigl[ x, x \Bigl(1+\frac1{5000(\log x)^2}\Bigr),x \Bigr] $$ contains at least one prime.
Let $x$ be a real number larger than $89\,693$. The interval $$ \Bigl[ x, x \Bigl(1+\frac1{\log^3 x}\Bigr) \Bigr] $$ contains at least one prime.

The proof of these latter results has an asymptotical part, for $x\ge 10^{20}$ where we used the numerical verification of the Riemann hypothesis together with two other arguments: a (very strong) smoothing argument and a use of the Brun-Titchmarsh inequality. The second part is of algorithmic nature and covers the range $10^{10} \le x \le 10^{20}$ and uses prime generation techniques : we only look at families of numbers whose primality can be established with one or two Fermat-like or Pocklington's congruences. This kind of technique has been already used in a quite similar problem in . The generation technique we use relies on a theorem proven in and enables us to generate dense enough families for the upper part of the range to be investigated. For the remaining (smaller) range, we use theorems of that yield a fast primality test (for this limited range).
Let us recall here that a second line of approach following the original work of \v Ceby\v sev is still under examination though it does not give results as good as analytical means (see for the latest result).
For very large numbers proved the following.
Theorem (2014)

The interval $(x,x+3 x^{2/3}]$ contains a prime for $x\ge \exp(\exp(34.32))$.

This is improved in as follows.
Theorem (2021)

The interval $(x,x+3 x^{2/3}]$ contains a prime for $x\ge \exp(\exp(33.99))$.

Theorem (2002)

Under the Riemann Hypothesis, the interval $\bigl]x-\tfrac85\sqrt{x}\log x,x\bigr]$ contains a prime for $x\ge2$.

This is improved upon in into:
Theorem (2015)

Under the Riemann Hypothesis, the interval $\bigl]x-\tfrac4{\pi}\sqrt{x}\log x,x\bigr]$ contains a prime for $x\ge2$.

In , the authors go one step further and prove the next result.
Theorem (2019)

Under the Riemann Hypothesis, the interval $\bigl]x-\tfrac{22}{25}\sqrt{x}\log x,x\bigr]$ contains a prime for $x\ge4$.

Collecting references: , , .

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