Some Commentary on Atle Selberg’s Mathematics

One way of characterizing Atle Selberg’s mathematical genius is that he had a “golden touch”. In those domains he thought about in depth, he saw further than generations before him, repeatedly uncovering truths lying below the surface. His breakthroughs on long-standing problems were based on imaginative and novel ideas which, once digested, were appreciated as simple and decisive. The impact of his work is far greater than what appears in his collected works [S]. He developed many tools and techniques which are the basis of far-reaching achievements by others in contiguous fields. There are few to whom the term “mathematician’s mathematician” applies so well. It is impossible in a couple of paragraphs to do justice to Selberg’s achievements and his impact. Our hope is that by pointing to some of his major results and placing them in a modern context, we can at least give the reader a taste. Selberg burst into the limelight with his proof that the Riemann zeta-function has a positive proportion of its zeros on the critical line. More important than the result was the technique of “mollification” (as it is referred to today) that he introduced in the proof. The method of weighting averages involving zeta and L-functions by squares of (Dirichlet) polynomials whose coefficients are optimized only at the end of the analysis allowed him to smooth out the large values of zeta on the critical line. This, in turn, enabled him to examine the zeta-function on short intervals and to establish the positive proportion theorem. The mollification method and its many variants remains today as one of the most powerful tools in the study of more general zeta and L-functions on the critical line. Selberg himself, at the age of 80, in a technical tour-de-force, extended these ideas to show that any real linear combination of modular L-functions of a certain type also has a positive proportion (though certainly not all!) of its zeros on the line Re(s) = 1/2. Selberg’s original work from the early 1940’s on the zeta-function led him to his elegant and powerful “lambda-squared” sieve and, from there, to an in-depth analysis of sieve methods and especially their limits. In particular, he identified and clarified various fundamental issues intrinsically associated with sieve methods, such as the parity problem. Selberg’s “fundamental formula”, which lies at the heart of the celebrated elementary proof of the prime number theorem, also arose naturally from this work on the zeta-function and the sieve. During this same period (1940–1950), Selberg developed what is known today as the Rankin–Selberg method, as well as the Selberg Integral. The former was