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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?

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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."

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