# Simple proof for horizontal zero density estimate for the Riemann-Zeta-function

Let $$N_c(T)$$ denote the cardinality of the set $$\{\rho | \zeta(\rho)=0 \text{ and } \Re \rho > c \text{ and } 0 \leq \Im \rho \leq T\}$$. I am trying to find a prove for the fact $$N_c(T) = o(T)$$ whenever $$c > \tfrac 12$$. This result follows from any (good) horizontal zero-density estimate. For example the bound $$N_c(T) \ll T^{\frac {3(1-\sigma)}{2-\sigma}}\log(T)^{15},$$ which is due to Ingham, readily implies the above. However most of these results are rather hard to prove, and a lot stronger than the above. So I am wondering: Is it possible to show this with simple methods?

Counting zeros of $$L$$-functions is generally a problem that requires some machinery to tackle. To my knowledge, the proofs of estimates like the one you wrote all require a form of Montgomery's zero detection method. Chapter 10 of Iwaniec and Kowalski's Analytic Number Theory begins with a three page proof of the estimate $$N(\sigma,T) \ll T^{4(1-\sigma)}(\log T)^{13}.$$ The proof is relatively easy to follow, but it still relies on producing "zero-detecting" Dirichlet polynomials, i.e. functions $$D(s) = \sum_{n} \frac{a_n}{n^s}$$ that attain unusually large values at zeros of $$\zeta(s)$$. Estimates for $$N(\sigma,T)$$ have implications for the distribution of primes in short intervals, so you should expect any result of the form $$N(\sigma,T) = o(T)$$ to be of comparable difficulty. For instance, the estimate I wrote in the first display implies that every interval of the form $$[x,x+x^{3/4+\epsilon}]$$ contains a prime. Even to prove that the (much longer) intervals $$[x,x+x/\log x]$$ always contain a prime requires the Prime Number Theorem.
With all that being said, I doubt that one can prove $$N(\sigma,T) = o(T)$$ using methods that most would consider "simple."