论文标题
大型算术进展中三个素数的产物
Product of three primes in large arithmetic progressions
论文作者
论文摘要
For any $ε>0$, there exists $q_0(ε)$ such for any $q\ge q_0(ε)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three primes, all of which are below $q^{\frac{3}{2}+ε}$.如果我们将注意力限制在与1 Mod 4一致的奇数$ Q $上，我们可以在$ q^{\ frac {\ frac {11} {8} {8}+ε} $中找到此类素数。如果我们进一步将模量限制为Prime $ Q $，以至于$（Q-1,4 \ CDOT7 \ CDOT11 \ CDOT17 \ CDOT17 \ CDOT23 \ CDOT29）= 2 $，我们可以在$ Q^{\ frac {6} {6} {5} {5}+ε} $下找到此类数量。最后，对于任何$ε> 0 $，存在$ q_0（ε）$，因此，当$ q \ ge q_0（ε）$时，存在一个与$ a $ a $ a $ modulo $ q $一致的天然数字，这是四个prime的产物，所有这些都是四个prime的产物，所有这些都是低于$ q（\ log q）^6 $。
For any $ε>0$, there exists $q_0(ε)$ such for any $q\ge q_0(ε)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three primes, all of which are below $q^{\frac{3}{2}+ε}$. If we restrict our attention to odd moduli $q$ that do not have prime factors congruent to 1 mod 4, we can find such primes below $q^{\frac{11}{8}+ε}$. If we further restrict our set of moduli to prime $q$ that are such that $(q-1,4\cdot7\cdot11\cdot17\cdot23\cdot29)=2$, we can find such primes below $q^{\frac{6}{5}+ε}$. Finally, for any $ε>0$, there exists $q_0(ε)$ such that when $q\ge q_0(ε)$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly four primes, all of which are below $q(\log q)^6$.
