Gergely Harcos 1:30 pm, May 18, 1999 Topics: analytic number theory, algebraic number theory Committee: Sarnak (chair), Trotter, Luo This is the true story of my general exam. Sarnak asked me what I wanted to start with. I said complex analysis. Trotter asked the following. If you have an analytic function in C-{1,2i} how many power (Laurent) series does it have around zero and how do I find them. Then Sarnak took over. Draw three concentric circles. Suppose you have an analytic function in the annulus bounded by the innermost and outermost circles, and you have an upper bound for its absolute value on the innermost circle and another bound on the outermost circle. How would you bound the function on the middle circle. Let's say both bounds are 5. Then the bound is 5 by the maximum principle, I said. What if one bound is 5 the other one is 7? I said, I would apply the three circle theorem. What does it say? I told them. Ok, let's see if you know what lies behind that theorem. Suppose you have a domain bounded by a Jordan curve and you cut out two regions from that domain bounded by Jordan curves inside the domain. We have an analytic function in the remaining triply connected domain and have various bounds - 3,5,7 - on the boundary components. How would you bound the function in the domain? I said I would construct the harmonic measures corresponding to those components and then the linear combination with coefficients 3,5,7 will give a pointwise bound inside the domain. Let's go back to the annulus and suppose that we have a bounded subharmonic function inside it and we know bounds 5 and 7 on the boundary circles with the exception of finitely many points. I said the same answer applies by the generalized maximum principle. Sarnak then asked me to give the definition of subharmonic functions and explain the proof of the generalized maximum principle. So we discussed barriers a bit. I was asked to construct the harmonic measures for a strip and also to find explicitly the harmonic function inside a semidisc (bounded by a line segment and a semicircle) which has boundary values 0 in the interior of the line segment and 1 in the interior of the semicircle (just use the Thales theorem from elementary geometry). Then he gave the following problem. Let's assume we have a bounded subharmonic function u on the whole complex plane. Can you conclude it is constant? I said yes. Why? My first idea to write u as log|f| with f analytic was silly - as I realized quickly. Then Sarnak told me that if I had not thought about this problem before I probably shouldn't try to do it there. But I said I was sure I could do it quickly using ideas as in the proof of the generalized maximum principle. I got the right intuition that instead of u one should examine v=u+elog|1/z| where e is any positive constant. But I got into trouble with the details, so Sarnak helped me a bit. Here is the proof. Let M be the supremum of u and assume that u is not constant. By upper semicontinuity, u attains its supremum on the unit circle is, call it m. Then m1. The same holds for |z|<1 as well. By letting e tend to 0 we get that u<=m on the whole plane which is a contradiction to m