The inputs that give the output zero are called zeros of the zeta function . Deligne 's proof of the Riemann hypothesis over finite fields used the zeta functions of product varieties , whose zeros and poles correspond to sums of zeros and poles of the original zeta function , in order to bound the real parts of the zeros of the original zeta function . By analogy , introduced multiple zeta functions whose zeros and poles correspond to sums of zeros and poles of the Riemann zeta function .