Another Disproof of the Riemann Hypothesis

1. Introduction

The infinite series

$$\zeta \left(s\right)=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{n^s}}$$

where $s=\sigma +it$ is a complex number, converges for $\sigma >1$. The Riemann zeta function is its meromorphic continuation to the whole complex plane. The Riemann hypothesis asserts that all zeros in the strip $0<\sigma <1$ satisfy $\sigma =1/2$. For the basic theory of the Riemann zeta function one may refer to [1,4,5,8].

In a recent preprint [9], the author proposed a disproof of the Riemann hypothesis by considering an integral equivalence of Hu [3]. In this short note we give another disproof by Littlewood’s oscillatory theorem and a result of Fujii [2].

Theorem 1.

The Riemann hypothesis is false.

2. Proof of Theorem 1

Let $\mathsf{\Lambda }\left(n\right)$ be the von Mangoldt function and let $\mathsf{\Omega }$ denote the big Omega notation. The following is Littlewood’s oscillatory theorem.

Theorem 2

([7]). As $x\to \infty$,

$$\sum _{n\le x}\mathsf{\Lambda }\left(n\right)=x+\mathsf{\Omega }\left(\sqrt{x}logloglogx\right).$$

(1)

Let

$$r_2\left(n\right)=\sum _{\ell +m=n}\mathsf{\Lambda }(\ell )\mathsf{\Lambda }\left(m\right).$$

In his study of the Goldbach conjecture Fujii proved the following result.

Theorem 3

([2]). Suppose the Riemann hypothesis is true, then

$$\sum _{n\le x}{r}_2\left(n\right)={\displaystyle \frac{1}{2}}{x}^2+O\left(x^{3/2}\right).$$

(2)

We now return to the proof of Theorem 1.

Proof of Theorem 1.

It follows readily from (1) that as $x\to 1^{-}$,

$$f\left(x\right):=\sum _{n=1}^{\infty }\mathsf{\Lambda }\left(n\right)x^n={\displaystyle \frac{1}{1-x}}+\mathsf{\Omega }\left({\displaystyle \frac{1}{\sqrt{1-x}}}logloglog{\displaystyle \frac{1}{1-x}}\right),$$

(3)

and thus as $x\to 1^{-}$,

$$\begin{array}{ccc}\hfill f{\left(x\right)}^2& =& \sum _{n=1}^{\infty }\left(\sum _{\ell +m=n}\mathsf{\Lambda }(\ell )\mathsf{\Lambda }\left(m\right)\right)x^n\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& =& \sum _{n=1}^{\infty }{r}_2\left(n\right)x^n\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{{(1-x)}^2}}+\mathsf{\Omega }\left({\displaystyle \frac{1}{{(1-x)}^{3/2}}}logloglog{\displaystyle \frac{1}{1-x}}\right).\hfill \end{array}$$

(6)

Then by the well known Hardy-Littlewood tauberian theorem [6] we deduce that

$$\sum _{n\le x}{r}_2\left(n\right)={\displaystyle \frac{1}{2}}{x}^2+\mathsf{\Omega }(x^{3/2}logloglogx),$$

(7)

which contradicts (2). □