Beethoven Moonlight Sonata?

Okay my question is kind of specific. The piece is played in C# as I understand. I understand the c# scale and how to play it from sheet music (at least I think i do). The beginning of the melody is noted as G, C, E, meaning G#, C#, and E on the C# scale. However, the b is noted as 2 A notes an octave away from each other. In the C# scale it should be played as 2 A notes. But the song is played using two C#s and octave away from each other. Can somebody explain why the key signature is in a different format for the treble and the b?