# On the asymptotic bound for $\arg\zeta(s)$ on the critical line

I am currently trying to prove

$$N(T)={T\over2\pi}\log{T\over2\pi}-{T\over2\pi}+\mathcal O(\log T)$$

in which $$N(T)$$ denotes the number of $$\zeta$$'s nontrivial zeros with imaginary part between $$(0,T]$$. Currently, using symmetric properties of $$\xi(s)$$, I am able to obtain

$$N(T)={T\over2\pi}\log{T\over2\pi}-{T\over2\pi}+\frac78+\frac1\pi\arg\zeta\left(\frac12+iT\right)+\mathcal O\left(\frac1T\right)$$

Apparently, the remaining job is to show that the argument of $$\zeta$$ on the critical line is of logarithmic growth, and I become stuck on interpreting the meaning of $$\arg\zeta$$. According to H. M. Edwards' Riemann's zeta function, this argument is bounded by the number of zeros of $$\Re\zeta(s)$$ on a certain curve (section 6.7 of his book), and I wonder if anybody could provide a more intuitive and clear explanation on that. Thank you!

• I suggest to look up the survey paper of karatsuba and korolev, the argument of the zeta function (free pdf top google search) as it is best reference imho – Conrad Jan 17 at 17:24
• @Conrad Although my problem is not solved yet, thank you for providing that source since it convinces me that approximating exponential sums are inevitable to learn :D – TravorLZH Feb 3 at 11:59

## Contour integral over three line segments

It can be shown that

$$\frac1\pi\arg\zeta\left(\frac12+iT\right)={1\over2\pi i}\int_\mathcal L{\zeta'\over\zeta}(s)\mathrm ds$$

in which $$\mathcal L$$ represents line segments $$1/2-iT\to2-iT\to2+iT\to1/2+iT$$. Since $$|\log\zeta(2+it)|\le\log\zeta(2)$$ for all $$t\in\mathbb R$$, we see that the integral on the line segment $$2-iT\to2+iT$$ is bounded. Consequently, all we need is to estimate the upper bound for integrals on lines segments $$1/2-iT\to2-iT$$ and $$2+iT\to1/2+iT$$. Because the procedures are so similar, we here only present the proof for line segment $$2+iT\to1/2+iT$$.

## Meromorphic expansion of $$\zeta'/\zeta$$

Since we know little information about the location of $$\zeta$$'s nontrivial zeros in the critical strip, we consider meromorphically expanding $$\zeta'/\zeta$$ near $$2+iT$$. Let $$S=\{\rho\in\mathbb C:\zeta(\rho)=0\wedge|\rho-2-iT|\le2\}$$, then define

$$g(z)={\zeta(z+2+iT)\over\zeta(2+iT)}\prod_{\rho\in S}\left(1-{z\over\rho-2-iT}\right)^{-1}$$

Because $$|1-z/(\rho-2-iT)|\ge1$$ on $$|z|=4$$, it follows from maximum modulus principle that if we set

$$M=\max_{|z|\le4}\left|\zeta(z+2+iT)\over\zeta(2+iT)\right|$$

then $$\log|g(z)|\le\log M$$. By Borel-Caretheodory lemma, we see that

$${g'\over g}(z)\ll\log M$$

for all $$|z|\le3/2$$. Now, all we need is to find an explicit estimate of $$M$$:

## Growth of $$\zeta$$ and $$\zeta'/\zeta$$

According to The theory of Riemann zeta-function by E.C. Titchmarsh, we know that there exists a fixed positive constant such that $$\zeta(\sigma+it)\ll|t|^A$$ for large $$|t|$$ for all $$\sigma\in\mathbb R$$. This indicates

$$\log|\zeta(\sigma+it)|\ll\log|t|$$

Moreover because

$$|\zeta^3(\sigma)\zeta^4(\sigma+it)\zeta(\sigma+2it)|\ge1$$

we see that

$$-\log|\zeta(\sigma+it)|\ll\log|t|$$

for all $$\sigma>1$$. Consequently we have $$\log M\ll\log T$$. This, combining with the definition of $$g$$, indicates that

$${\zeta'\over\zeta}=\sum_{\rho\in S}{1\over s-\rho}+\mathcal O(\log T)$$

when $$s$$ is on the line segment $$[1/2+iT,2+iT]$$.

## Final shot: the integral is $$\mathcal O(\log T)$$

\begin{aligned} \int_{2+iT}^{1/2+iT}{\zeta'\over\zeta}(s)\mathrm ds &=\sum_{\rho\in S}\int_{2+iT}^{1/2+iT}{\mathrm ds\over s-\rho}+\mathcal O(\log T) \\ &\ll\sum_{\rho\in S}1+\mathcal O(\log T) \end{aligned}

To estimate the sum, we apply Jensen's inequality:

$$\sum_{\rho\in S}1\le{\log M\over4/2}\ll\log T$$

Therefore, the expansion of Riemann-von Mangoldt formula becomes

$$N(T)={T\over2\pi}\log{T\over2\pi}-{T\over2\pi}+\mathcal O(\log T)$$