Should we (the U.S.) increase the number executions?

[Middle-aged Europe]

There's no such thing like a deterrent for murder.

Murder is either premeditated (in which case the culprit didn't plan on getting caught in the first place) or it happens "in the heat of the moment" (in which case the culprit didn't plan on committing a murder).

And I highly doubt that the line of reasoning in cases of premeditated murders is: "Hey, I live in a place without the death penalty. The worst that can happen to me is that I get a life sentence. That doesn't sound so bad. At least I won't have to worry about becoming homeless some day. And it's totally worth it when it means that I can kill that bastard."

[Middle-aged Europe]

Sorry, but you definition is a little defective.  Murder is criminal homicide with malice aforethought.

Doesn't rule out the possibility that actual manslaughterers are charged with murder though... in which case a culprit without malice aforethought could be executed anyway.

Since the justice system is far from being infallible, there are murderers who are convicted of manslaughter and manslaughterers who are convicted of murder. Since the line is that blurred my argument bears relevance.

Finally, execution is a proven specific deterrence.  Executed persons do not break out of prison to kill again.

That's no deterrence though. That's a preemptive measure against prison breaks.

According to your line of thought, the death penalty would make prison breaks more likely though. Actually facing an execution gives you another strong incentive to escape prison.