Riemann hypothesis

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.