Maier's matrix method is a technique in analytic number theory due to

Helmut Maier Helmut Maier (born 17 October 1953) is a German mathematician and professor at the University of Ulm, Germany. He is known for his contributions in analytic number theory and mathematical analysis and particularly for the so-called Maier's matr ...

that is used to demonstrate the existence of intervals of natural numbers within which the prime numbers are distributed with a certain property. In particular it has been used to prove

Maier's theorem In number theory, Maier's theorem is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer. The theorem states that if π is the prime-counting function and λ is greater t ...

and also the existence of chains of large gaps between consecutive primes . The method uses estimates for the distribution of prime numbers in arithmetic progressions to prove the existence of a large set of intervals where the number of primes in the set is well understood and hence that at least one of the intervals contains primes in the required distribution.

The method

The method first selects a

primorial In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...

and then constructs an interval in which the distribution of integers coprime to the primorial is well understood. By looking at copies of the interval translated by multiples of the primorial an array (or matrix) of integers is formed where the rows are the translated intervals and the columns are

arithmetic progressions An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...

where the difference is the primorial. By

Dirichlet's theorem on arithmetic progressions In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is al ...

the columns will contain many primes if and only if the integer in the original interval was coprime to the primorial. Good estimates for the number of small primes in these progressions due to allows the estimation of the primes in the matrix which guarantees the existence of at least one row or interval with at least a certain number of primes.

References

* * * {{Citation , last1=Gallagher , first1=Patrick , author-link1=Patrick X. Gallagher , title=A large sieve density estimate near σ=1 , year=1970, journal=Inventiones Mathematicae , volume=11, pages=329–339 , doi=10.1007/BF01403187 , issue=4 Analytic number theory