Now we know that as much as the Law says it speaks to those in the Law, so that each mouth will be stopped and all the cosmos will become liable to trial by God. For this reason, from works of the Law no one of all flesh will be declared righteous in the sight of Him, for by means of the Law is full knowledge of sin. (Romans 3:19, 20)
Now we know that as much as the Law says...Paul brings his argument to a close with a statement directed to the Jews, followed by a universal pronouncement of accountability, followed by another statement to the Jews.
Those who were "in the Law," were the Jews, and the Law shouted to them that they were sinners (even if some refused to listen).
So that each mouth will be stopped...Every man, whether Jew or Gentile, will appear before the tribunal of God, and when it comes time to present his defense to the Judge there will be a deafening silence, for he will have nothing worth saying. He will simply bow his head and walk around behind the woodshed.
For this reason, from works of the Law...Since everybody has sinned and every Jew has broken the Law, it should be obvious that no one will be declared righteous for his Law-keeping.
For by means of the Law...Now the apostle introduces a theme that he will develop in various ways later: the purpose of the Law was to amplify sin. From the bird’s-eye view of redemptive-history, God did not give the Law to the children of Israel in order to show them the way of blessing, rather He gave it to show them their wickedness, and thereby, their need of a Redeemer.
(Summary of Romans 3:9-20)
In terms of their righteous standing before God, the Jews are no better off than the Gentiles; and both groups are in deep trouble because of sin. Jews and Gentiles are alike in being unrighteous, deficient in kindness, straying from the truth, quick to speak and act wickedly, in a word—they do not respect God and His commands. Furthermore, all Jews and Gentiles will be judged by God, and they will have no excuses to offer for their disobedience. Contrary to Jewish thinking, the Law was not intended to be a means of blessing, but rather it was to be a means of conviction.
(Extended Paraphrase)
So, in the final analysis, are we Jews in a superior position to the Gentiles? Not at all! Because, we made the accusation earlier that all Jews and all Gentiles are sinners. The Scriptures teach this saying, "There is not a single righteous man. There are no men or women who understand or who carefully enquire about the character and ways of God. Everybody has turned away from the straight way, together they have become morally worthless; there is not a man or woman who acts with kindness, not even for a time. Their throat is ready to swallow men whole, their tongues are continually betraying others, the venom of asps lies behind their lips. Their mouths are filled with curses and bitterness. They run toward an opportunity to kill. Their paths are ruin and wretchedness, and they do not know the way to peace. They do not fear God." Now we know that when the Law has something to say, it speaks to those who are subject to the Law, that is, the Jews. The goal is to render every Jew and every Gentile liable to trial and without excuse at the tribunal of God. For this reason, out of all the people of the world, not one of them will be declared righteous before God by keeping the Law, because the Law's purpose was to bring to light that the Jews were sinners.