Finite Dirichlet series with prescribed zeroes

In 2011 Yuri Matiyasevich began examining finite Diriclet series having the same initial zeroes as Riemann's zeta function. Surprisingly, even with a small numbers of terms such series gave extremely good approximations to the values of the zeta function inside the critical strip. Moreover, the coefficients of these series turned out to have number-theoretical significance, in particular, they produced new connections of the zeta function with prime numbers.

Later these investigations were extended by him and other researchers to divese finite Diriclet series having common zeroes with other functions definied by infinite Diriclet series. This site is devoted to the ongoing study of arising phenomena. You are welcome to share your discoveries here.