Events

I will start from the situation of a cuspidal Hecke eigenform f of real quadratic character, congruent to its complex conjugate modulo a prime P ramified in the coefficient field.

In this talk I present recent results obtained with Markus Neuhauser towards the non-vanishing of the coefficients of the Dedekind eta function in the spirit of G.-C. Rota. This includes Serre’s table, pentagonal numbers, results of Kostant in the context of simple affine Lie algebras and the Lehmer conjecture. In the second part I will talk about partition numbers and the Bessenrodt-Ono inequality.

Let $V$ be the set of the values of the Euler's $\varphi$ function. It is known that the natural density of $V$, namely the limit of $V(x)/x$, as $x$ tends to infinity, is $0$.

In a paper to appear, M. K. Das, P. Eyyunni and B. R. Patil claim that this also applies to a local density of $V$, known as the uniform upper density, or the Banach density, defined by
$$
\delta^{*(}V) = \lim_{H \rightarrow \infty} \limsup_{x \rightarrow \infty} \frac{1}{H}\left(V(x+H) -V(x)\right).
$$

I will talk about lower bounds for the discrete negative 2k-th moment of the derivative of the Riemann zeta function for all
fractional k > 0. The bounds are in line with a conjecture of Gonek and Hejhal. This is a joint work with Winston Heap and
Junxian Li.