The Riemann--von Mangoldt zero count

The Hadamard product The Riemann xi function has its genus-one Hadamard product over the nontrivial zeros of zeta gives ${\xi }^{\mathrm{\prime }}\mathrm{/}\xi (s)=B+{\sum }_{\rho }(1\mathrm{/}(s-\rho )+1\mathrm{/}\rho )$. Write $\rho =\beta +i\gamma$, with $0<\beta <1$ by The only zeros of zeta on the nonpositive real axis are the negative even integers, and every other zero lies in the open critical strip. Since ${\sum }_{\rho }\mathrm{\mid }\rho {\mathrm{\mid }}^{-2}<\mathrm{\infty }$, the sum of $\mathrm{\Re }(1\mathrm{/}\rho )=\beta \mathrm{/}\mathrm{\mid }\rho {\mathrm{\mid }}^2$ converges. At $s_0=2+iT$, the xi identity, Stirling's formula and The logarithmic derivative of the zeta Dirichlet series is the Dirichlet series of the von Mangoldt function on Re s greater than 1 give ${\xi }^{\mathrm{\prime }}\mathrm{/}\xi (s_0)=O(\log T)$: the zeta term is bounded by ${\sum }_{n\ge 2}(\log n)n^{-2}$, and the Gamma term is $O(\log T)$. Differentiating Stirling here is justified by Cauchy's estimate for its analytic remainder on disks of radius proportional to $\mathrm{\mid }{s}_0\mathrm{\mid }$ in a larger sector. Taking real parts of the product formula, all variable summands are positive and ${\sum }_{\rho }\frac{1}{4+(T-\gamma {)}^2}\le {\sum }_{\rho }\frac{2-\beta }{(2-\beta {)}^2+(T-\gamma {)}^2}=O(\log T)\mathrm{.}$ In particular there are $O(\log T)$ zeros with $\mathrm{\mid }T-\gamma \mathrm{\mid }<1$, without using the present theorem or its unit-interval corollary.

On the horizontal segment $s=\sigma +iT$, $1\mathrm{/}2\le \sigma \le 2$, subtract the product formula at $s_0$. For $\mathrm{\mid }T-\gamma \mathrm{\mid }\ge 1$, $\mid \frac{1}{s-\rho }-\frac{1}{s_0-\rho }\mid \le \frac{3\mathrm{/}2}{(T-\gamma {)}^2}\le \frac{15\mathrm{/}2}{4+(T-\gamma {)}^2}\mathrm{.}$ Thus step 1.1 bounds the nonlocal difference sum by $O(\log T)$, uniformly in $\sigma$. The local subtracted terms $1\mathrm{/}(s_0-\rho )$ also total $O(\log T)$. The integral of the imaginary part of each remaining local term $1\mathrm{/}(\sigma +iT-\rho )$ is the argument change of a horizontal segment missing zero, of absolute value at most $\pi$. Their number is $O(\log T)$. Including the reference value ${\xi }^{\mathrm{\prime }}\mathrm{/}\xi (s_0)$, the total argument change of $\xi$ on the top segment from $2+iT$ to $1\mathrm{/}2+iT$ is therefore $O(\log T)$.

Fix a height $t_0\in (0,2)$ not equal to any zero ordinate. On the vertical segment from $2+i{t}_0$ to $2+iT$, the factors $s$ and $s-1$ in $\xi (s)=\frac{1}{2}s(s-1){\pi }^{-s\mathrm{/}2}\mathrm{\Gamma }(s\mathrm{/}2)\zeta (s)$ have bounded argument changes. The zeta factor also has bounded argument change: $\mathrm{\mid }\zeta (2+it)-1\mathrm{\mid }\le {\sum }_{n\ge 2}{n}^{-2}<1$, so it remains in a fixed right half-plane. Stirling with a continuous logarithm in the right half-plane gives $\mathrm{\Im }\log \mathrm{\Gamma }(1+iT\mathrm{/}2)=\frac{T}{2}\log \frac{T}{2}-\frac{T}{2}+O(1)\mathrm{.}$ Consequently the argument change along this vertical segment is $\frac{T}{2}\log \frac{T}{2\pi }-\frac{T}{2}+O(1)\mathrm{.}$

By [L1], $\xi (s)=\xi (1-s)$, and conjugation symmetry gives $\xi (1-\bar{s})=\overline{\xi (s)}$. Apply the argument principle to the rectangle with real sides $-1,2$ and heights $t_0,T$. Its zeros are precisely the nontrivial zeta zeros in that height range. Reflection in $\mathrm{\Re }s=1\mathrm{/}2$ pairs the two vertical edges and the two halves of each horizontal edge, doubling the argument change on the right vertical edge followed by the right half of the top; the bottom contributes a constant independent of $T$. Thus $N(T)=\frac{1}{\pi }({\mathrm{\Delta }}_{2+i{t}_0\to 2+iT}\arg \xi +{\mathrm{\Delta }}_{2+iT\to 1\mathrm{/}2+iT}\arg \xi )+O(1)\mathrm{.}$ Steps 2.1 and 3.1 yield the claimed main term and $O(\log T)$ error.

If $T$ is a zero ordinate, take nonzero-ordinate heights decreasing to $T$. Discreteness of zeros in the bounded strip makes their counts eventually equal to the convention $0<\gamma \le T$, with full multiplicities. The preceding error constant is independent of the distance to zero ordinates, so passage to the limit preserves the estimate. Finally compact heights $2\le T\le T_0$ are covered by enlarging the constant.