Riemann Hypothesis has many equivalent statements.

Many of them are not about prime distribution, instead, are about the properties of Riemann Zeta function, such as the distribution of zeros of Zeta function or the positivity properties of Zeta functions (ex: Li-criterion).

To my knowledge, Goldbach conjecture does not have many equivalent statements.

Question: What are some equivalent statements of Goldbach conjecture in terms of the properties of Zeta function or Dirichlet-L functions ?

Can anyone point out some other references or works in this direction ?

Mostconjectures don't have lots of very different-sounding formulations. Anyway, in the early 20th century, Hardy & Littlewood proved under GRH for Dirichlet $L$-functions (they really use no zeros for all Dirichlet $L$-functions with real part greater than $3/4 - \varepsilon$ for some $\varepsilon > 0$) an asymptotic formula for the number of representations of large even numbers as a sum of $k$ primes for even $k \geq 4$, and they acknowledge that their reasoning falls short when $k = 2$. $\endgroup$