### The average least negative Hecke eigenvalue

Linfoot Number Theory Seminar

15th November 2023, 11:00 am – 12:00 pm

Fry Building, 4th Floor Seminar Room

In this talk we discuss the first sign change of Fourier coefficients of newforms, or equivalently Hecke eigenvalues. We will see this to be an analogue of the least quadratic non-residue problem, of which the average was investigated by Erdős in 1961. In fact, we will see that the average least negative prime Hecke eigenvalue holds the same (finite) value as the average least quadratic non-residue, under GRH. This is mainly due to the fact that Hecke eigenvalues are equidistributed with respect to the Sato-Tate measure, a consequence of the Sato-Tate conjecture that was proven in 2011. We further explore the so-called vertical Sato-Tate conjecture to show the average least Hecke eigenvalue has a finite value unconditionally.

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