Attack on the Riemann Hypothesis

1. Introduction

Throughout we write $z=\sigma +it$ with $\sigma ,t\in \mathbb{R}$. Let $\zeta \left(z\right)$ be the Riemann zeta function and

$$\xi \left(z\right)={\displaystyle \frac{z(z-1)}{2}}{\pi }^{-{\displaystyle \frac{z}{2}}}\Gamma \left({\displaystyle \frac{z}{2}}\right)\zeta \left(z\right)$$

the Riemann xi function where $\Gamma \left(z\right)$ is the gamma function. It is well known that the Riemann zeta function has zeros at negative even integers which are called trivial zeros. The Riemann hypothesis asserts that all nontrivial complex zeros satisfy $\sigma =1/2$.

In 1992 Keiper [4] studied the power series

$${\displaystyle \frac{{\xi }^{\prime }(1/z)}{\xi (1/z)}}=\sum _{n=0}^{\infty }{\tau }_n{(1-z)}^n,$$

(1)

$$log\left(2\xi \left(1/z\right)\right)=\sum _{n=0}^{\infty }{\lambda }_n^K{(1-z)}^n.$$

(2)

In 1997 Li [5] defined the numbers

$${\lambda }_n^L={\displaystyle \frac{1}{(n-1)!}}{\left.{\displaystyle \frac{d^n}{d{z}^n}}\left[z^{n-1}log\xi \left(z\right)\right]\right|}_{z=1}$$

and they are also the coefficients of the power series expansion of

$${\displaystyle \frac{{\phi }^{\prime }\left(z\right)}{\phi \left(z\right)}}=\sum _{n=0}^{\infty }{\lambda }_{n+1}^L{z}^n,$$

where

$$\phi \left(z\right)=\xi \left({\displaystyle \frac{1}{1-z}}\right).$$

Li’s ${\lambda }_n^L$ has the expression

$${\lambda }_n^L=\sum _{\rho }\left[1-{\left(1-{\displaystyle \frac{1}{\rho }}\right)}^n\right]$$

(3)

where $\rho$ runs over the nontrivial zeros of the Riemann zeta function.

Remark 1.

The relation between Keiper’s ${\lambda }_n^K$ and Li’s ${\lambda }_n^L$ is ${\lambda }_n^L=n{\lambda }_n^K$. In the following we call both ${\lambda }_n^K$ and ${\lambda }_n^L$ the Keiper-Li coefficients. We will use Keiper’s and set ${\lambda }_n:={\lambda }_n^K$.

Keiper [4] noticed that the Riemann hypothesis implies ${\lambda }_n>0$ for all $n>0$ and Li [5] showed the equivalence, which is now known as Li’s criterion:

$$\mathrm{Riemann}\hspace{0.17em}\mathrm{Hypothesis}⟺{\lambda }_n>0.$$

Given this the Keiper-Li coefficient ${\lambda }_n$ was extensively studied, see for example [1,2,3] and the references therein.

Another observation of Keiper is that the Riemann hypothesis is equivalent to the radius of convergence of the series (1) being 1. In particular as Keiper ([4] p. 769) stated we have

Theorem 2.

If the $|{\tau }_n|$ in (1) are bounded, then the Riemann hypothesis is true.

The purpose of this paper is to show that $|{\tau }_n|$ are bounded and therefore the Riemann hypothesis is true.

Theorem 3.

The $|{\tau }_n|$ in (1) are bounded.

The proof involves asymptotic estimates of modulus of nontrivial zeros of the Riemann zeta function.

2. Proof of Theorem 3

For the coefficients ${\tau }_n$ in (1) Keiper ([4] p. 769) proved that

$${\tau }_{n-1}=-\sum _{\rho }{\displaystyle \frac{1}{{\rho }^2}}{\left({\displaystyle \frac{\rho }{\rho -1}}\right)}^n$$

(4)

and that they are the second central difference of $n{\lambda }_n$:

$${\tau }_n=(n+1){\lambda }_{n+1}-2n{\lambda }_n+(n-1){\lambda }_{n-1}.$$

(5)

Remark 4.

We note that there is also

$${\tau }_{n-1}=-\sum _{\rho }{\displaystyle \frac{1}{{\rho }^2}}{\left(1-{\displaystyle \frac{1}{\rho }}\right)}^{n-2}.$$

(6)

Indeed by Keiper ([4] (30)-(32)) we have

$$\begin{array}{ccc}\hfill {\tau }_{n-1}& =& -\sum _{\rho }{(\rho -1)}^{-n}{\rho }^{n-2}\hfill \\ & =& -\sum _{\rho }{(\rho -1)}^{-(n-2)}{\rho }^{n-2}{(\rho -1)}^{-2}\hfill \\ & =& -\sum _{\rho }{\left({\displaystyle \frac{\rho }{\rho -1}}\right)}^{n-2}·{\displaystyle \frac{1}{{(\rho -1)}^2}}.\hfill \end{array}$$

Now replace $\rho$ by $1-\rho$ we obtain (6).

The formula (6) was also showed by the author in ([7] Theorem 3.2) in terms of Li’s ${\lambda }_n^L$.

Let $N\left(T\right)$ denote the number of zeros of $\zeta \left(z\right)$ in the region $0<\sigma <1,0<t<T$. Then it is well known ([6] Theorem 9.4) that as $T\to \infty$,

$$N\left(T\right)={\displaystyle \frac{1}{2\pi }}TlogT-{\displaystyle \frac{log2\pi e}{2\pi }}T+o\left(T\right).$$

(7)

As a consequence of (7) we have the following estimate.

Lemma 1

([6] p. 214). Let the complex zeros $\beta +i\gamma$ of $\zeta \left(z\right)$ with $\gamma >0$ be arranged in a sequence ${\rho }_n={\beta }_n+i{\gamma }_n$ so that ${\gamma }_{n+1}\ge {\gamma }_n$, then as $n\to \infty$,

$$|{\rho }_n|\sim {\gamma }_n\sim {\displaystyle \frac{2\pi n}{logn}}.$$

(8)

Proof. 

We follow the proof in ([6] p. 214). By (7) we have

$$N\left(T\right)\sim {\displaystyle \frac{1}{2\pi }}TlogT$$

and thus

$$2\pi N({\gamma }_n\pm 1)\sim ({\gamma }_n\pm 1)log({\gamma }_n\pm 1)\sim {\gamma }_{n}log{\gamma }_n.$$

(9)

Also

$$N({\gamma }_n-1)\le n\le N({\gamma }_n+1)$$

(10)

and therefore

$$2\pi n\sim {\gamma }_{n}log{\gamma }_n,$$

(11)

and then taking logarithms on both sides of (11) we have

$$logn\sim log{\gamma }_n.$$

(12)

Combining (11) and (12) gives

$${\gamma }_n\sim {\displaystyle \frac{2\pi n}{logn}}.$$

(13)

That $|{\rho }_n|\sim {\gamma }_n$ is obvious. □

Corollary 5.

Let the complex zeros $\beta +i\gamma$ of $\zeta \left(z\right)$ with $\gamma >0$ be arranged in a sequence ${\rho }_n={\beta }_n+i{\gamma }_n$ so that ${\gamma }_{n+1}\ge {\gamma }_n$. Then for every fixed $\epsilon >0$ there is a positive integer $N\left(\epsilon \right)$ such that when $n>N\left(\epsilon \right)$,

$$|{\rho }_n|>(1-\epsilon ){\displaystyle \frac{2\pi n}{logn}}.$$

(14)

We now prove Theorem 3.

Proof of Theorem 3.

By (6) and $|z_1-z_2|\le |z_1|+|z_2|$ we have

$$|{\tau }_n|\le \sum _{\rho }{\displaystyle \frac{1}{{\left|\rho \right|}^2}}{\left(1+{\displaystyle \frac{1}{\left|\rho \right|}}\right)}^{n-1}.$$

(15)

Together with (14) we take a fixed small $\epsilon >0$,

$$\sum _{\rho }{\displaystyle \frac{1}{{\left|\rho \right|}^2}}{\left(1+{\displaystyle \frac{1}{\left|\rho \right|}}\right)}^{n-1}\ll \sum _{n=1}^{\infty }{\displaystyle \frac{{log}^{2}n}{{(1-\epsilon )}^2{\left(2\pi n\right)}^2}}{\left(1+{\displaystyle \frac{logn}{2\pi n(1-\epsilon )}}\right)}^{n-1}.$$

(16)

Since

$$(n-1)log\left(1+{\displaystyle \frac{logn}{2\pi n(1-\epsilon )}}\right)\le n·{\displaystyle \frac{logn}{2\pi n(1-\epsilon )}}={\displaystyle \frac{logn}{2\pi (1-\epsilon )}},$$

(17)

and thus we have

$${\left(1+{\displaystyle \frac{logn}{2\pi n(1-\epsilon )}}\right)}^{n-1}\le n^{{\displaystyle \frac{1}{2\pi (1-\epsilon )}}}$$

(18)

and

$$\sum _{\rho }{\displaystyle \frac{1}{{\left|\rho \right|}^2}}{\left(1+{\displaystyle \frac{1}{\left|\rho \right|}}\right)}^{n-1}\ll \sum _{n=1}^{\infty }{\displaystyle \frac{{log}^{2}n}{n^{2-1/2\pi (1-\epsilon )}}}={\zeta }^{″}\left(2-{\displaystyle \frac{1}{2\pi (1-\epsilon )}}\right)=O\left(1\right),$$

(19)

provided we take $\epsilon$ such that

$$2-{\displaystyle \frac{1}{2\pi (1-\epsilon )}}>1⟺\epsilon <1-{\displaystyle \frac{1}{2\pi }}.$$

(20)

Now combining (15) and (19) gives Theorem 3. □

3. Conclusion

In this paper we find a proof of the Riemann hypothesis that Keiper missed. In a forthcoming paper we will generalize the method to the study of the extended Riemann hypothesis for general number fields.