Hardy-Littlewood conjecture

The two Hardy-Littlewood conjectures are unproven mathematical conjectures from the field of number theory . They were drawn up by the two English mathematicians Godfrey Harold Hardy and John Edensor Littlewood and in 1923 in the work " Some Problems of 'Partitio Numerorum." III. On the Expression of a Number as a Sum of Primes. "

In 1974 Ian Richards succeeded in showing that the two Hardy-Littlewood conjectures are incompatible with each other. This means that they cannot both be correct, but at most one.

First Hardy-Littlewood guess

The first Hardy-Littlewood conjecture is also called the k-tuple conjecture or the strong twin prime conjecture. The reason for the latter is that proving the first Hardy-Littlewood conjecture also proves the twin prime conjecture - according to which there are an infinite number of twins prime numbers . It says that there are infinitely many prime number tuples for all correct (and not necessarily the most dense) configurations and gives an explicit function for the density of these. The differences between the tuple elements are described with a configuration of a prime number tuple. For example, one possible correct configuration of a prime 2-tuple (also known as a prime twin) is. In order for a configuration to be considered correct, not all possible residues with respect to every prime number may appear in the tuple (→ prime number tuple ). The densest configurations are also called constellations . ${\displaystyle (0,2)}$ ${\displaystyle \leq k}$

In the following, let be the function that gives the set of all prime numbers less than or equal to this number for any number. Formally: ${\displaystyle P}$

${\displaystyle P:\mathbb {R} \rightarrow {\mathcal {P}}(\mathbb {P} ),x\mapsto [0,x]\cap \mathbb {P} }$

Where the square brackets stand for a closed interval and where stands for the set of all prime numbers . Let be the prime number function , i.e. it indicates the number of prime numbers that are less than or equal to its function argument. This can be easily formalized thanks to the definition of the function : ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle \mathbb {P} }$ ${\displaystyle \pi }$ ${\displaystyle P}$

${\displaystyle \pi :\mathbb {R} \rightarrow \mathbb {N} ,x\mapsto \left|P(x)\right|}$

Now a constant can be introduced for any correct configuration of the quantity , which is defined by the following convergent infinite product : ${\displaystyle (p,p+2m_{1},\dots ,p+2m_{k})}$ ${\displaystyle k+1}$ ${\displaystyle C_{m_{1},\dots ,m_{k}}}$

${\displaystyle C_{m_{1},\dots ,m_{k}}=2^{k}\cdot \prod _{p\in \mathbb {P} \setminus \{2\}}{\frac {1-{\frac {w(p;m_{1},\dots ,m_{k})}{p}}}{\left(1-{\frac {1}{p}}\right)^{k+1}}}}$

Where denotes the number of different residues of with respect to the divider . Formally: ${\displaystyle w(p;m_{1},\dots ,m_{k})}$ ${\displaystyle 0,m_{1},\dots m_{k}}$ ${\displaystyle p}$

${\displaystyle w(p;m_{1},\dots ,m_{k})=\left|\left\{n\in \mathbb {N} \mid \exists i\in \mathbb {N} :n=m_{i}\mod p\right\}\right|}$

The number is also called the twin prime constant. (Follow A005597 in OEIS ) ${\displaystyle C_{2}\approx 0{,}66016\dots }$

${\displaystyle C_{2}=\left(\prod _{p\in \mathbb {P} \setminus \{2\}}{\frac {p\cdot (p-2)}{(p-1)^{2}}}\right)=\left(\prod _{p\in \mathbb {P} \setminus \{2\}}1-{\frac {1}{(p-1)^{2}}}\right)\approx 0{,}66016\dots }$

The following formula exists for the constant for pairs of prime numbers ( ) with any difference : ${\displaystyle k=1}$ ${\displaystyle m}$ ${\displaystyle C_{m}}$

${\displaystyle C_{m}=2\prod _{p\in \mathbb {P} }{\frac {p\cdot (p-2)}{(p-1)^{2}}}\prod _{p\in \left\{n\in \mathbb {P} \mid n\mid m\right\}}{\frac {p-1}{p-2}}}$

Where stands for the divisibility relation . ${\displaystyle n\mid m}$

The above-mentioned value of about 0.66016 has established itself for. A distinction must be made here that with and consequently is twice as large as , which is why there are two different formulas for the assumption of asymptotic behavior. ${\displaystyle C_{2}}$ ${\displaystyle C_{m}}$ ${\displaystyle m=2}$ ${\displaystyle k=1}$ ${\displaystyle C_{2}}$

Interestingly, the constant is not necessarily the same for different configurations of the same size. The smallest counterexample is a constellation of size 8. ${\displaystyle C}$

The prime number function can now also be extended by the index , so that the number of all prime number tuples denotes that are of the form and whose components are not larger than the function argument. An example is given because up to 9 there are the prime twins and . ${\displaystyle \pi }$ ${\displaystyle (m_{1},\dots ,m_{k})}$ ${\displaystyle \pi _{m_{1},\dots ,m_{k}}}$ ${\displaystyle (p,p+2m_{1},\dots ,p+2m_{k})}$ ${\displaystyle \pi _{2}(9)=2}$ ${\displaystyle (3,5)}$ ${\displaystyle (5,7)}$

With the first Hardy-Littlewood conjecture, it is now asserted that the asymptotic behavior applies

${\displaystyle \pi _{2}(n)\sim 2C_{2}\int _{2}^{n}{\frac {\mathrm {d} t}{(\ln t)^{2}}}}$

which can also be formalized as a limit value as follows :

${\displaystyle \lim _{n\rightarrow \infty }{\frac {\pi _{2}(n)}{2C_{2}\int _{2}^{n}{dt \over (\ln t)^{2}}}}=1}$

Generalized to any configuration is the assumption

${\displaystyle \pi _{m_{1},\dots ,m_{k}}(n)\sim C_{m_{1},\dots ,m_{k}}\int _{2}^{n}{\frac {\mathrm {d} t}{\mathrm {d} \ln ^{k+1}t}}}$

what can be transformed into a limit value in an analogous way.

Since the number of prime numbers is asymptotically equivalent to - according to the prime number theorem - the assumption seems plausible, and the asymptotic form can also be confirmed numerically, which, however, is not sufficient for a proof. ${\displaystyle \pi }$ ${\displaystyle {\frac {x}{\ln x}}}$ ${\displaystyle \pi (x)\sim {\frac {x}{\ln x}}}$

Second Hardy-Littlewood conjecture

The second Hardy-Littlewood conjecture makes the statement about the number of prime numbers in an interval. More precisely, it is about the following inequality :

${\displaystyle \forall x,y\in \left\{n\in \mathbb {N} \mid n\geq 2\right\}:\pi (x+y)\leq \pi (x)+\pi (y)}$

Where is again the prime number function, i.e. indicates the number of prime numbers. ${\displaystyle \pi }$

In general, it is assumed that this guess is wrong because - as mentioned at the beginning - it is incompatible with the more plausible first Hardy-Littlewood conjecture.

The case for is trivial. The prime number function grows more slowly than linearly , so formally it can be said that it holds where the identical mapping is. See Landau symbols for the o notation. Consequently, so the inequality needs to apply. ${\displaystyle x=y}$ ${\displaystyle \pi \in {\hbox{o}}(\operatorname {id} _{\mathbb {R} })}$ ${\displaystyle \operatorname {id} _{\mathbb {R} }}$ ${\displaystyle \pi (2\cdot x)\leq 2\cdot \pi (x)}$ ${\displaystyle x\geq 2}$

Examples of values ​​for which the equation applies are specifically mentioned . In general, all pairs or the equation in which the smaller element is a prime number twin pair satisfy . are related prime twins. ${\displaystyle x,y\in \left\{n\in \mathbb {N} \mid n\geq 2\right\}}$ ${\displaystyle \pi (x+y)=\pi (x)+\pi (y)}$ ${\displaystyle (2,3)}$ ${\displaystyle (x,y)=(2,p)}$ ${\displaystyle (p,2)}$ ${\displaystyle p}$ ${\displaystyle (3,5)}$

Similarly, the inequality applies to all or those in which is not the smaller element of a pair of prime numbers. An example is , because is not a prime twin pair because 9 is not prime. ${\displaystyle \pi (x+y)<\pi (x)+\pi (y)}$ ${\displaystyle (2,p)}$ ${\displaystyle (p,2)}$ ${\displaystyle p}$ ${\displaystyle p=7}$ ${\displaystyle (7,9)}$

Web links

• Tamar Ziegler : Linear equations in primes and dynamics of nilmanifolds , Arxiv 2014 (pdf), (overview of current developments on the first Hardy-Littlewood conjecture, especially from the ergodic theory )

Individual evidence

1. ^ ^{A b} " Hardy-Littlewood Conjectures - from Wolfram MathWorld ". Retrieved June 12, 2014.
2. ↑ " On the incompatibility of two conjectures Concerning primes; a discussion of the use of computers in attacking a theoretical problem- ". Retrieved June 12, 2014.
3. ^ " K-Tuple Conjecture - from Wolfram MathWorld ". Retrieved June 12, 2014.
4. ^ " The Prime Glossary: ​​prime k-tuple conjecture ". Retrieved June 12, 2014.
5. ^ " Hardy-Littlewood constants ". Retrieved June 12, 2014.