with s = 12 + it, and shows that ξ(t) is an even entire function of t whose zeros have imaginary part between −i/2 and i/2. He further states, sketching a proof, that in the range between 0 and T the function ξ(t) has about (T/2π) log(T/2π) − T/2π zeros. Riemann then continues “Man findet nun in der That etwa so viel reelle Wurzeln innerhalb dieser Grenzen, und es ist sehr wahrscheinlich, dass ...

There has been some interest on how the average character degree affects structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, number zeros characters group G as in table divided irreducible $. show that if < 1 then is solvable also \frac{1}{2} supersolvable. characterise abelian groups showing \frac{1}{3} only abelian.

We determine the asymptotics for variance of number zeros random linear combinations orthogonal polynomials degree ≤n in subintervals [a,b] support underlying orthogonality measure μ. show that, as n→∞, this is asymptotic to cn, some explicit constant c>0.

These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this study was its potential application to the tangential Hilbert 16th problem on zeros of complete Abelian integrals. The exposition consists mostly of examples ill...