THE GOLDBACH-VINOGRDOV THEOREM WITH THREE PRIMES IN A THIN SUBSET

1.IntroductionandStatementofResultsIn1937,Vinogradovi7]provedthatJ(N),thenumberofrepresefltationsofanilltegerNassumsofthreeprimes,satisfiesthefollowingasymptoticformulawherea(N)isthesingularseries,andu(N)>>1foroddN.Itthereforefollowsthateverysufficientlylargeoddintegeristhesumofthreeprimes.ThissettledtheternaryGoldbachproblem,andtheresultisreferredtoastheGoldbach-Vinogradovtheorein.ManyauthorshaveconsideredthecorrespondingproblemswithrestrictedconditionsposedonthethreeprimesintheGoldbach…

THE GOLDBACH-VINOGRDOV THEOREM WITH THREE PRIMES IN A THIN SUBSET

It is proved constructively that there exists a thin subset $S$ of primes, satisfying $$|S\cap1,x]|\ll x^{\frac{9}{10}}\log^c x$$ for some absolute constant $c>0,$ such that every sufficiently large odd integer $N$ can be represented as $$\left\{ \aligned N&=p_1+p_2+p_3,\\ p_j&\in S, \,\,\,j=1,2,3. \endaligned \right.$$ Let $r$ be prime, and $b_j$ positive integers with $(b_j, r)=1, j=1,2,3.$ It is also proved that,for almost all prime moduli $r\leq N^{\frac{3}{20}}\log^{-c} N,$ every sufficiently large odd integer $N\equiv b_1+b_2+b_3 (\bmod r)$ can be represented as $$ \left\{\aligned N&=p_1+p_2+p_3, \\ p_j&\equiv b_j (\bmod r), \,\,\,j=1,2,3, \endaligned \right. $$ where $c>0$ is an absolute constant.