A Variational Problem In The GPY Sieve Method

The twin prime conjecture is an important topic in the study of the dis-tribution of prime numbers.For hundreds of years,it has attracted attentions of numerous excellent mathematicians.Although this conjecture has not been proved,many results are obtained recently approaching this conjecture.The twin prime conjecture states that there are infinitely many primes p,such that p + 2 is also a prime number.In 1921,Hardy and Littlewood[1],proposed the circle method and gave a more accurate formula for the number of twin primes,namely(?)Many mathematicians have studied this problem.The basic idea to study the conjecture of twin primes is to consider the distance between consecutive primes.A weaker version of the conjecture is that there is a positive con-stant C0,such that pn+1-pn≤C0 holds for infinitely many twin prime pairs(pn,pn+1).Note that C0 = 2 is the twin prime conjecture.In 2005,Goldston,Pintz and Yildirim[2]made a significant breakthrough in the twin prime conjecture.They proposed the GPY sieve method and obtain a thin gap between consecutive primes.However,their results predicted that the distance is of the order of powers of logn.In 2013,Yitang Zhang[3]improved their method and proved that lower limit of the distance between consecutive prime numbers was less than 70 million.That is to say,C0 is less than 70000000.In 2014,Maynard[4]improved the GPY sieve method and improved the results of Zhang to 600.Maynard’s method is different from Zhang’s method.Actually,he successfully reduced the problem to a variational problem and a key parameter in his method is the Mk.He obtained the bound M5>1417255/70816.C0 = 12 follows by assuming the θ of the Elliott-Halberstam conjecture can be taken 1416432/1417255+ ε.In 2014,the POLYMATH[5]group,studied this problem extensively and got the best result so far.That is C0 could take 246.In this thesis,we will continue to study the variational problem proposed by Maynard.First,we give a better lower bound estimate of M5,exactly,M5>1661/830.Using this result,combined with the work of Maynard[4],C0 can be taken as 12 if the θ of Elliott-Halberstam conjecture can be taken as 1660/1661+ε.Moreover,we proved that the corresponding function of the maximum value of the variational problem on the continuous function class is a solution of a linear integral equation.Therefore,the original nonlinear variational problem is transformed into the eigenvalue of a linear operator(see Theorem 1.2).Finally,we considered how many natural numbers n ∈[N,2N)such that the sequencen,n + 2,n + 6,n + 8,n + 12 contains at least two primes.A lower bound estimate of the number is given.