TY - JOUR

T1 - An elementary approach to the twin primes problem

AU - Baier, S.

PY - 2004/12/1

Y1 - 2004/12/1

N2 - Hardy-Littlewood [4] conjectured an asymptotic formula for the number of prime pairs (twin primes) (p, p + 2d) with p + 2d = y, where d ? N is fixed and y ? 8. Up to now, no one has been able to prove this conjecture, but employing Hardy-Littlewood's circle method, Lavrik [5] showed that in a certain sense this formula holds true for almost-all d = y/2. In the present paper, we use a completely different method to prove Lavrik's almost-all result. Our method is based on an elementary approach developed by Pan Chengdong [7] to the twin primes problem. By a slight modification of our method, we get a corresponding almost-all result for the binary Goldbach problem. From this, according to [3], we derive Vinogradov's [8] well-known Three-Primes-Theorem.

AB - Hardy-Littlewood [4] conjectured an asymptotic formula for the number of prime pairs (twin primes) (p, p + 2d) with p + 2d = y, where d ? N is fixed and y ? 8. Up to now, no one has been able to prove this conjecture, but employing Hardy-Littlewood's circle method, Lavrik [5] showed that in a certain sense this formula holds true for almost-all d = y/2. In the present paper, we use a completely different method to prove Lavrik's almost-all result. Our method is based on an elementary approach developed by Pan Chengdong [7] to the twin primes problem. By a slight modification of our method, we get a corresponding almost-all result for the binary Goldbach problem. From this, according to [3], we derive Vinogradov's [8] well-known Three-Primes-Theorem.

UR - http://www.scopus.com/inward/record.url?scp=11944275637&partnerID=8YFLogxK

U2 - 10.1007/s00605-004-0270-3

DO - 10.1007/s00605-004-0270-3

M3 - Article

AN - SCOPUS:11944275637

VL - 143

SP - 269

EP - 283

JO - Monatshefte für Mathematik

JF - Monatshefte für Mathematik

SN - 0026-9255

IS - 4

ER -