# Equivalent Statements of Goldbach Conjecture in Terms of the Properties of Riemann Zeta Function?

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 ?

• Why you think that Goldbach's Conjecture had connection with generalized Riemann hypothesis? Sep 26, 2019 at 23:14
• you should specify which Goldbach's verison you talk about (binary or ternary), it is mensioned here : fr.wikipedia.org/wiki/Conjecture_faible_de_Goldbach that G.R.H implies that ternary Goldbach's conjecture holds. Sep 27, 2019 at 0:06
• I don't see why you think the Goldbach conjecture alone should have many equivalent statements. Most conjectures 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 $k = 2$. Sep 27, 2019 at 4:56
• The more structural a conjecture, the more it implies other interesting things, so the more likely one has other interesting statements that are equivalent. The Goldbach Conjecture is a very natural problem to think about, but it turns out that it doesn't really restrict the behavior of the primes much at all in a useful way, so it doesn't directly imply many things at all. To see this, it might help to ask "If I knew Goldbach, what could I say about the distribution of the primes." The answer is not much. For example, even assuming Goldbach, it doesn't help you recover the PNT. Sep 27, 2019 at 13:22
• @JoshuaZ Indeed, the fact that every number is representable as a sum of two primes says barely anything more than that there are at least $\sqrt{n}$ primes below $n$! There are more quantitative versions of Goldbach conjecture (indicating how many representations there are) which I believe gives the right order of magnitude as a lower bound, but falls short of providing a correct constant. Sep 27, 2019 at 15:42