Zero-free region of the Riemann zeta function

The classical zero-free region asserts that for some $c>0$,Together with bounds for the logarithmic derivative, it permits contour arguments with exponentially small errors in $\sqrt{\log x}$.

For sufficiently large $|t|$, the Riemann zeta function has no zeros in the displayed region, for a fixed positive $c$. To derive it from the Richert bound for the Riemann zeta function, take $\eta \asymp (\log \log t/\log t{)}^{2/3}$. The logarithm of the maximum on the two discs in the Landau zero-free-region theorem is $O(\log \log t)$, as is $\log (1/\eta )$. Dividing $\eta$ by this logarithmic factor gives the displayed width.

For $t\ge 2$ and $0<\eta \le 1/4$, suppose $|\zeta |\le M$, $M\ge 2$, on radius-$\eta$ discs centred at $1+\eta /4+it$ and $1+\eta /4+2it$. Every zero $\beta +it$ has the displayed gap. The reciprocal Euler product bounds both centre values below by a constant times $\eta$. The local logarithmic-derivative lemma gives errors $B/\eta$, $B=1+\log (M/\eta )$. The three-four-one zero-free-region argument then implies $4/(\sigma -\beta )-3/(\sigma -1)\ll B/\eta$. For a zero close to one, choose $\sigma =1+6(1-\beta )$, making the left side $1/(14(1-\beta ))$. Zeros farther away already satisfy the bound. Thus a logarithmic upper bound for zeta translates into a quantitative zero-free width.

The logarithm of the Euler product and $3+4\cos \theta +\cos 2\theta \ge 0$ give $\zeta (\sigma {)}^3|\zeta (\sigma +it){|}^4|\zeta (\sigma +2it)|\ge 1$. At $\sigma =1+a/{\log }^9|t|$, upper bounds $\zeta (\sigma )\ll {\log }^9|t|/a$ and $|\zeta (\sigma +2it)|\ll \log |t|$ imply the lower bound $|\zeta |\gg a^{3/4}{\log }^{-7}|t|$. The Cauchy estimate for derivatives gives $|{\zeta }^{\prime }|\ll {\log }^2|t|$ nearby, allowing a sufficiently small leftward displacement of order ${\log }^{-9}|t|$. This elementary region is weaker than the classical logarithmic region but gives an explicit reciprocal bound by simple estimates.

For $F=-{\zeta }^{\prime }/\zeta$ and $\sigma >1$, positivity ofterm by term in the Euler product givesComparing this with the pole of $F$ at one and the local poles of ${\zeta }^{\prime }/\zeta$ at zeros proves the classical zero-free region.

After shrinking the zero-free-region constant, one hasin the half-width region $\sigma >1-c/(2\log |t|)$ for $|t|$ large. Compare the local partial-fraction expansion at $s$ with one at a point just to the right of one, where the Euler product controls the logarithmic derivative.