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I am investigating whether or not there exist $\epsilon > 0$ such that $\zeta(s) \neq 0$ on the strip $1-\epsilon < \Re(s) \leq 1$.

Suppose not. Then given $\delta > 0$ there exists a zero of zeta $\rho$ such that $1 -\delta < \Re(\rho) < 1 $. Hence, there exists a sequence of zeros $\{ z_n \}_{n=1}^\infty$ with increasing imaginary parts that have real part getting closer and closer to the line $\Re(z)=1$.

Can we somehow estimate the density of this infinite set of "fake zeros." The idea is to compare the density of this set with the best known estimate for the density of zeros that are on the critical line.

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    $\begingroup$ Please use a high-level tag like "nt.number-theory". I added this tag now. $\endgroup$
    – GH from MO
    Commented Dec 31, 2021 at 22:28
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    $\begingroup$ What you are investigating about would be a consequence of some tightening of the upper bound of the error term in an average form of Goldbach's conjecture as shown by Bhowmik. $\endgroup$ Commented Dec 31, 2021 at 22:53

2 Answers 2

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There are, provably, very few zeros with real part close to $1$ (or bigger than $0.51$ for that matter). These theorems go under the name of "zero density estimates", and they have a vast literature. See Chapter 11 in Ivić: The Riemann zeta function (1985). The book was reprinted in 2013 by Dover Publications.

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    $\begingroup$ It's perhaps worth recalling Iwaniec's description of such results as "estimates for the cardinality of the empty set" (as also noted earlier on MO: mathoverflow.net/questions/50186 and mathoverflow.net/questions/52576). $\endgroup$
    – KConrad
    Commented Jan 1, 2022 at 2:27
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    $\begingroup$ @KConrad Another famous saying of his is: "There is life beyond the Riemann Hypothesis." $\endgroup$
    – GH from MO
    Commented Jan 1, 2022 at 7:43
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To add to GH from MO's answer, Chapter 10 of Iwaniec and Kowalski's "Analytic Number Theory" is another good reference. A few of the classical papers are

  1. Ingham's "On the Estimation of $N(\sigma,T)$,"
  2. Montgomery's "Mean and Large Values of Dirichlet Polynomials" and "Zeros of $L$-functions," and
  3. Huxley's "On the Difference Between Consecutive Primes."

Define $$ N(\sigma,T) = |\left\{\rho=\beta+i\gamma: \zeta(\rho)=0,\ \beta\geq \sigma,\ 0\leq \gamma \leq T\right\}|, $$ so $N(\sigma,T)$ counts the zeros of $\zeta$ with real part at least $\sigma$ and imaginary part at most $T$. In general, zero density estimates for the Riemann zeta function take the form $$ \tag{1} N(\sigma,T) \ll T^{\lambda(\sigma)(1-\sigma)}(\log T)^B $$ for some relatively unimportant positive constant $B$ and some function $\lambda(\sigma)$. Note that the bound shrinks as $\sigma \to 1$. Ingham shows that (1) holds with $$ \lambda(\sigma) = \frac{3}{2-\sigma}. $$ Montgomery proves an analogous result for all Dirichlet $L$-functions and also proves that (1) holds with $$ \lambda(\sigma) = \frac{5}{2}, $$ which Huxley improves to $$ \lambda(\sigma) = \frac{12}{5}. $$ Both Montgomery and Huxley prove results slightly stronger than what I've stated. The density conjecture states that one can take $\lambda(\sigma)=2$, but this is only known in limited ranges of $\sigma$. For instance, $\sigma \geq \frac{5}{6}$ follows from Huxley's work. The full range would follow from the Riemann Hypothesis.

Zero density estimates are connected to the distribution of primes in short intervals. Ingham, in a paper with the same name as Huxley's, shows that any estimate of the form (1) with $\lambda$ constant implies that any interval $[x,x+x^\theta]$ with $\theta > 1-\lambda^{-1}$ and $x$ sufficiently large contains a prime.

The general strategy for proving zero density estimates is to construct "zero detecting polynomials," which are functions that take unusually large values at the zeros of $\zeta$. One then uses "mean and large value estimates" to show that these functions cannot be large too often.

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  • $\begingroup$ I gave Ivić's chapter, because it is more comprehensive (for the Riemann zeta). $\endgroup$
    – GH from MO
    Commented Jan 1, 2022 at 7:45

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