I have always thought that righteousness was pretty simple. It's me doing the right thing, keeping the rules, being obedient. The checklist includes things like Bible reading, prayer, helping people, keeping promises, being honest, staying in Christian fellowship.
In short, righteousness is about me being good. For the most part, it feels like righteousness is just between me and God.
But thanks to the Bible Project's study of the Sermon, I have a clearer understanding of righteousness. It's not just about me being good. It is about me being in right relationship with others. As Tim Mackie says, righteousness is "doing right by you."
This understanding better captures the meaning of the original words. And we can clearly see that emphasis in other parts of the Sermon. Jesus goes on to say that if there is a problem between you and someone else, you need to chase the person down and make it right. You need to repair the relationship before you even worship. Don't murder, but don't even be angry. Don't commit adultery, but don't even lust. It's all about relationship.
And this tracks well with the second part of the Great Commandment: Love your neighbor as yourself.
So, yes, be righteous. But know that it means making relationships right. It's not just between me and God. We are all in this together. It's the kingdom of heaven.