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| THE GOLDBACH-VINOGRDOV THEOREM WITH THREE PRIMES IN ATHIN SUBSET |
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Citation： |
Liu Jianya.THE GOLDBACH-VINOGRDOV THEOREM WITH THREE PRIMES IN ATHIN SUBSET[J].Chinese Annals of Mathematics B,1998,19(4):479~488 |
| Page view： 1188
Net amount： 754 |
Authors： |
Liu Jianya; |
Foundation： |
the National Natural Science Foundation of China |
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| Abstract： |
It is proved constructively that there exists a thin subset $S$ of primes, satisfying $$|S\cap[1,x]|\ll x^{\frac{9}{10}}\log^c x$$ 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. |
Keywords： |
Goldbach-Vinogradov theorem, Exponential sum, Primes,
Arithmetic progression, Mean-value theorem |
Classification： |
11P32, 11L07 |
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