I actually wrote this awhile ago but I pulled it up for another reason. Fixed it up a little bit for this venue.
In "Permission to Believe" Rabbi Lawrence Keleman breaks it down into four basic arguments.
1. Arguing against the alternative: Atheism is irrational because it would necessarily imply that one knows all that exists. As long as you don't know everything, it is possible God exists.
2. Teleological: There must be a God because the Universe is way too complex for there not to be.
3. Moralistic: We all have morality, what can it be based on except for an almighty moral being who created us?
4. Historical: Jews have an amazing history and beat all the odds, thus God must exist because he must have helped them.
My problems against number one is that there is no reason to assume there is a God anymore than we should assume an invisible dragon is living on the Moon. Atheism is logical as long as you take in all the information obtained as of yet. It's true that we cannot rule out conclusively the existence of a transcendent superbeing, but that doesn't justify the belief that one does exist. Atheism is not necessarily saying that God definitely does not exist but that the individual is believing in accordance with the available evidence. We cannot speak about what we don't know. Maybe there really is an invisible dragon on the Moon.
Number two - Science has discovered more and more about nature and more and more has lost deification. We no longer assume the weather is controlled directly by supernatural beings, but that cascading competitive temperatures are the direct cause. The evidence for evolution is mountainous and now the realm which God touches directly is confined to what science has not yet been able to explain. It is a weak argument based on what we don't have information about. God of the gaps is a weak thesis indeed. Complexity in the universe is being explained more and more by science without positing God. When the French physicist Pierre Simon de Laplace explained his theory of the universe to Napoleon, Napoleon is said to have asked, "Where does God fit into your theory?" to which Laplace replied, "I have no need of that hypothesis."
Three - Morality is a curious thing, but I don't see why it necessitates a God. Fundamentally, morality is based on the Golden Rule. Something any society would understand as basic for civilized living. Indeed, any society which did not have a basic standard for living would be self-destructive. Why would people create anything if theft was acceptable? Who would feel safe walking around if they could be murdered freely? Societies encourage the formation of moralities for their collective well being.
Four - Jewish history is remarkable, true. But I don't think one needs a God to be involved to explain it. The Jewish ability to retain unity can be explained in terms of strong common faith, communal resistance to outside forces that try to force Jews apart (Jewish identity is strongest in the most antisemtitic places), strong extensive dictates on lifestyle (read: social controls from things like kashrut which prevent "intermingling") and "approved" reading material, etc. The restoration of a Jewish state is the reaction of a self-fulfilling prophecy. You can follow the origins of Zionism and history all the way to the culmination of the formation of the State of Israel. But nowhere in it do you need to posit divine intervention.