Robert C. Vaughan

Zeros of Dirichlet series


April 7, 2015, 1:55 — 2:45pm at LIT 368.


We are concerned here with Dirichlet series

\(f(s) = 1 +sum_{n=2}^{infty} frac{c(n)}{n^s} \)

which satisfy a function equation similar to that of the Riemann zeta function, typically of the form

\(f(s) = 2^s q^{1/2-s} pi^{s-1} Gamma(1-s) big(sintextstylefrac{pi}{2}(s+kappa)big) f(1-s), \)

but for which the Riemann hypothesis is false. We establish a plethora of such functions and show that the zeros of such functions are ubiquitous in the complex plane.