Nontrivial zero of the Riemann zeta function

A nontrivial zero of the Riemann zeta function lies in the critical strip $0<\Re s<1$. The functional equation of the Riemann zeta function reflects such zeros across the critical line $\Re s=1/2$.

The critical strip for the Riemann zeta function is $0<\Re s<1$.

The critical line is the symmetry line $\Re s=1/2$ inside the critical strip.

The Riemann hypothesis asserts that every Nontrivial zero of the Riemann zeta function lies on the critical line $\Re s=1/2$.

The subpower zeta bound to the right of the critical line gives an arbitrarily small exponent at $1/2+\delta$. The functional equation gives exponent $\delta +\epsilon$ at $1/2-\delta$. The Phragmén–Lindelöf principle gives exponent $\delta /2+\epsilon$ at the midpoint. Letting these fixed positive parameters be arbitrarily small proves the Lindelöf hypothesis.

Under Riemann hypothesis, integrate a smoothed explicit formula from $1/2+\delta$ to two, taking $x=\log |t|$. The prime contribution is $O_{\delta }(x^{1-2\delta })$, while the local zero count bounds the zero contribution by $O_{\delta }(x^{-\delta }\log |t|/\log x)$. Both are $o(\log |t|)$ for fixed $0<\delta <1/4$, yielding the displayed bound; other fixed positive shifts follow by standard strip bounds and the Euler product.

Riemann hypothesis is equivalent to monotonicity of $\sigma \mapsto |\xi (\sigma +it)|$ for every fixed $t$ on $\sigma \ge 1/2$. Under Riemann hypothesis the real logarithmic derivative of Riemann xi is positive for $\sigma >1/2$. Conversely a zero right of that line would force the nonnegative monotone modulus to vanish on an interval, contrary to the identity theorem. Functional symmetry rules out zeros to the left.

The Riemann hypothesis is equivalent tofor every $ϵ>0$. In fact, the hypothesis and the Riemann–von Mangoldt explicit formula give the stronger $O(x^{1/2}(\log x{)}^2)$ bound. Conversely, the stated error continues $-{\zeta }^{\prime }/\zeta -s/(s-1)$ holomorphically to $\Re s>1/2$, excluding zeros there; the functional equation supplies the other half.

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.

If $N(T)$ counts the nontrivial zeta zeros $\rho =\beta +i\gamma$ with $0<\gamma \le T$, including multiplicity, thenIn particular,

List positive ordinates of Nontrivial zeros of the Riemann zeta function in nondecreasing order, including multiplicity. If all sufficiently late gaps were at most $c/\log n$, summing would give ${\gamma }_n\le (c+o(1))n/\log n$, since ${\sum }_{j\le n}1/\log j\sim n/\log n$ with the sum starting at two. The Riemann–von Mangoldt formula would then give $n\le N({\gamma }_n)\le (c/(2\pi )+o(1))n$, a contradiction when $c<2\pi$. The same argument applies when only distinct ordinates are listed, because $n\le N({\gamma }_n)$ still holds.

List positive ordinates of Nontrivial zeros of the Riemann zeta function in nondecreasing order, including multiplicity. The Riemann–von Mangoldt formula gives $N(T)\sim T\log T/(2\pi )$. For each fixed $\epsilon >0$, evaluate this at $T=(1\pm \epsilon )2\pi n/\log n$ to bracket the $n$th ordinate. Letting $\epsilon$ decrease to zero proves the displayed asymptotic without assuming simple zeros or distinct ordinates.

The number of Nontrivial zeros of the Riemann zeta function in a height interval of length one is $O(\log (|t|+3))$, including multiplicity and either endpoint. Subtract the Riemann–von Mangoldt formula at the endpoints and use conjugation at negative heights. Summing these counts with reciprocal-height weights gives ${\sum }_{|\Im \rho |\le T}1/|\rho |\ll {\log }^2(T+3)$.

Put $F(s)=(s-1)\zeta (s)$. The fractional-part continuation formula for the Riemann zeta function gives $|F(s)|\ll (|s|+1{)}^2$ for $\mathrm{Re}s\ge 1/4$. At $2+ij$, the reciprocal Euler product bounds $|\zeta (2+ij)|$ below by $1/\zeta (2)$. Applying the Jensen zero-count bound with outer radius $7/4$ and inner radius $8/5$ gives $O(\log (|j|+2))$ zeros in the latter disk. Those disks cover the half-strip $1/2\le \mathrm{Re}s\le 1$. Reflecting zeros using the functional equation of the Riemann zeta function completes the $O(T\log T)$ bound without requiring left-half-plane growth estimates.

Evaluate the real logarithmic derivative of Riemann xi at real part two, where every summand is positive and comparable to this kernel. The gamma logarithmic derivative and the bounded zeta logarithmic derivative give the logarithmic bound. It implies the local unit-interval count, and conversely such local counts imply this smoothed bound by summing the decaying tails.