A Revised Proof of the Riemann Hypothesis via Contradiction

1. Introduction

The Riemann Hypothesis, proposed by Bernhard Riemann in 1859 [1], posits that all non-trivial zeros of the Riemann zeta function $\zeta \left(s\right)$ have real part $\Re \left(s\right)={\displaystyle \frac{1}{2}}$. This conjecture profoundly influences number theory, particularly the distribution of prime numbers [2]. Despite significant progress, including Levinson’s theorem [5] showing at least one-third of zeros lie on the critical line and zero-free regions [6], the hypothesis remains unproven. Our proof assumes a non-trivial zero off the critical line and derives contradictions using classical tools in complex analysis and analytic number theory, refined with rigorous zero-density estimates and Hardy space analysis.

2. Formal Framework

The Riemann zeta function is defined as

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

extended to $\mathbb{C}\setminus \left\{1\right\}$ via analytic continuation [3]. Non-trivial zeros lie in the critical strip $0<\Re \left(s\right)<1$. The Hadamard product is

$$\zeta \left(s\right)=e^{A+Bs}\prod _{\rho }\left(1-{\displaystyle \frac{s}{\rho }}\right)e^{s/\rho },$$

where $\rho$ are non-trivial zeros. The functional equation is

$$\zeta \left(s\right)=2^s{\pi }^{s-1}sin\left({\displaystyle \frac{\pi s}{2}}\right)\mathsf{\Gamma }(1-s)\zeta (1-s).$$

The Chebyshev function is

$$\psi \left(x\right)=\sum _{n\le x}\mathsf{\Lambda }\left(n\right),\psi \left(x\right)=x-\sum _{\rho }{\displaystyle \frac{x^{\rho }}{\rho }}-log2\pi -{\displaystyle \frac{1}{2}}log\left(1-{\displaystyle \frac{1}{x^2}}\right).$$

The Hardy space $H^2\left({\mathbb{C}}_{+}\right)$ consists of analytic functions on $\Re \left(s\right)>0$ with square-integrable boundary values, crucial for analyzing the Laplace transform of $\psi \left(x\right)-x$.

• $s=\sigma +it$: A complex number.
• ${\rho }_0={\sigma }_0+i{t}_0$: A non-trivial zero.
• $\mathsf{\Lambda }\left(n\right)=logp$ if $n=p^k$ for a prime p and integer $k\ge 1$, and 0 otherwise.
• Critical strip: $0<\Re \left(s\right)<1$; critical line: $\Re \left(s\right)={\displaystyle \frac{1}{2}}$.
• $O\left(f\right(t\left)\right)$: A bound $\le Cf\left(t\right)$ for $\left|t\right|>T_0$.

3. Proof of the Riemann Hypothesis

We assume a non-trivial zero ${\rho }_0={\sigma }_0+i{t}_0$, ${\sigma }_0>{\displaystyle \frac{1}{2}}$, exists, implying $\zeta (1-{\rho }_0)=0$ by the functional equation. We derive three independent contradictions.

3.1. Hadamard Product

The Hadamard product yields

$${\displaystyle \frac{{\zeta }^{\prime }}{\zeta }}\left(s\right)=B+\sum _{\rho }{\displaystyle \frac{1}{s-\rho }}.$$

For a zero of multiplicity m, the term is ${\displaystyle \frac{m}{s-\rho }}$. We analyze the behavior near $s=1-{\rho }_0$.

Lemma 1.

For $s=\sigma +i{t}_0$, $\sigma \to {(1-{\sigma }_0)}^{-}$, the sum ${\sum }_{\rho \ne 1-{\rho }_0}{\displaystyle \frac{1}{s-\rho }}$ converges uniformly in $|s-(1-{\sigma }_0+i{t}_0)|<\delta$, contributing $O(log|t_0\left|\right)$.

Proof. 

Let $\rho =\beta +i\gamma$. We have $\left|s-\rho \right|\ge |\gamma -t_0|-\delta$. By Backlund’s theorem [3] and recent zero-density estimates [4], the number of zeros with $|\gamma -t_0|\le 1$ is $O(log|t_0\left|\right)$. Split the sum:

$$\left|\sum _{\rho \ne 1-{\rho }_0}{\displaystyle \frac{1}{s-\rho }}\right|\le \sum _{|\gamma -t_0|\le 1}{\displaystyle \frac{1}{\left|s-\rho \right|}}+\sum _{|\gamma -t_0|>1+\delta }{\displaystyle \frac{1}{|\gamma -t_0|-\delta }}.$$

The first sum is bounded by $O(log|t_0\left|\right)·{\displaystyle \frac{1}{\delta }}$. For the second, use the zero counting function $N\left(u\right)\sim {\displaystyle \frac{ulogu}{2\pi }}$:

$$\sum _{|\gamma -t_0|>1+\delta }{\displaystyle \frac{1}{|\gamma -t_0|-\delta }}\le {\int }_{1+\delta }^{|t_0|}{\displaystyle \frac{N^{\prime }\left(u\right)}{u-\delta }}du.$$

Substitute $v=u-\delta$:

$${\displaystyle \frac{1}{2\pi }}{\int }_1^{|t_0|-\delta }{\displaystyle \frac{log(v+\delta )}{v}}dv\le {\displaystyle \frac{1}{2\pi }}{\left[logvlog(v+\delta )\right]}_1^{|t_0|-\delta }+{\int }_1^{|t_0|-\delta }{\displaystyle \frac{logv}{v+\delta }}dv.$$

The first term is $log|t_0|log(|t_0|+\delta )\sim (log|t_0{\left|\right)}^2$. The second integral is:

$${\int }_1^{|t_0|-\delta }{\displaystyle \frac{logv}{v+\delta }}dv\le {\int }_1^{|t_0|}{\displaystyle \frac{logv}{v}}dv={\displaystyle \frac{1}{2}}(log|t_0{\left|\right)}^2.$$

For $\delta <{\displaystyle \frac{1}{log|t_0|}}$, the total is $O(log|t_0\left|\right)$. Uniform convergence follows from the Weierstrass M-test. □

Lemma 2.

For ${\rho }_0={\sigma }_0+i{t}_0$, ${\sigma }_0>{\displaystyle \frac{1}{2}}$, $|t_0|\gg 1$, and $s=\sigma +i{t}_0$, ${\displaystyle \frac{1}{2}}\le \sigma <{\sigma }_0$,

$$\underset{\sigma \to {(1-{\sigma }_0)}^{-}}{lim}\left|{\displaystyle \frac{{\zeta }^{\prime }}{\zeta }}\left(s\right)\right|=\infty ,$$

contradicting $|{\displaystyle \frac{{\zeta }^{\prime }}{\zeta }}(\sigma +it){|\le C(log\left|t\right|)}^2$.

Proof. 

From the Hadamard product,

$${\displaystyle \frac{{\zeta }^{\prime }}{\zeta }}\left(s\right)=B+{\displaystyle \frac{1}{s-(1-{\rho }_0)}}+\sum _{\rho \ne 1-{\rho }_0}{\displaystyle \frac{1}{s-\rho }}.$$

As $\sigma \to {(1-{\sigma }_0)}^{-}$, ${\displaystyle \frac{1}{s-(1-{\rho }_0)}}\sim {\displaystyle \frac{1}{\sigma -(1-{\sigma }_0)}}\to \infty$. By Lemma 1, the remaining sum is $O(log|t_0\left|\right)$. The logarithmic derivative is:

$${\displaystyle \frac{{\zeta }^{\prime }}{\zeta }}\left(s\right)=\sum _{\rho \mathrm{near}s}{\displaystyle \frac{1}{s-\rho }}+B\left(s\right),$$

where $B\left(s\right)=O(log|t\left|\right)$ away from zeros [3]. The term ${\displaystyle \frac{1}{s-(1-{\rho }_0)}}$ dominates. The Dirichlet series gives:

$$|{\displaystyle \frac{{\zeta }^{\prime }}{\zeta }}\left(s\right)|\le \sum _{n=1}^{\infty }{\displaystyle \frac{\mathsf{\Lambda }\left(n\right)}{n^{\sigma }}}\le C{(log|t\left|\right)}^2,$$

for ${\displaystyle \frac{1}{2}}\le \sigma \le 1$ away from zeros [3], Chapter 5. The divergence contradicts this bound. □

Lemma 3.

For ${\rho }_0={\displaystyle \frac{1}{2}}+ϵ+i{t}_0$, $ϵ>0$, as $ϵ\to 0^{+}$, the contradiction persists.

Proof. 

For $s={\displaystyle \frac{1}{2}}+i{t}_0$, $1-{\rho }_0={\displaystyle \frac{1}{2}}-ϵ+i{t}_0$,

$${\displaystyle \frac{{\zeta }^{\prime }}{\zeta }}\left(s\right)\sim {\displaystyle \frac{1}{s-(1-{\rho }_0)}}={\displaystyle \frac{1}{ϵ}}\to \infty \mathrm{as}ϵ\to 0^{+}.$$

By Lemma 1, the sum ${\sum }_{\rho \ne 1-{\rho }_0}{\displaystyle \frac{1}{s-\rho }}=O(log|t_0\left|\right)$, contradicting the bound $C(log|t_0{\left|\right)}^2$. □

3.1.1. Multiplicity of Zeros

For a zero ${\rho }_0$ with multiplicity $m\ge 1$,

$${\displaystyle \frac{{\zeta }^{\prime }}{\zeta }}\left(s\right)=B+{\displaystyle \frac{m}{s-(1-{\rho }_0)}}+\sum _{\rho \ne 1-{\rho }_0}{\displaystyle \frac{1}{s-\rho }}.$$

As $\sigma \to {(1-{\sigma }_0)}^{-}$, ${\displaystyle \frac{m}{s-(1-{\rho }_0)}}\to \infty$, amplifying the contradiction. Similarly, for the functional equation, $\zeta \left(s\right)\sim \chi \left(s\right)·{\displaystyle \frac{cm}{{({\sigma }_0-\sigma )}^m}}$, which diverges faster, strengthening the contradiction in Section 3.2. Multiple zeros are unlikely [3], but the proof holds for all $m\ge 1$.

3.2. Functional Equation

The functional equation is

$$\zeta \left(s\right)=\chi \left(s\right)\zeta (1-s),\chi \left(s\right)=2^s{\pi }^{s-1}sin\left({\displaystyle \frac{\pi s}{2}}\right)\mathsf{\Gamma }(1-s).$$

Assume $\zeta (1-{\rho }_0)=0$. For $s=\sigma +i{t}_0$, ${\displaystyle \frac{1}{2}}\le \sigma <{\sigma }_0$, as $\sigma \to {(1-{\sigma }_0)}^{-}$,

$$\zeta (1-s)\sim {\displaystyle \frac{c}{(1-s)-(1-{\rho }_0)}}={\displaystyle \frac{c}{{\sigma }_0-\sigma }},c\ne 0.$$

Using Stirling’s approximation for $s=\sigma +i{t}_0$, $|t_0|\gg 1$:

$$\left|\mathsf{\Gamma }(1-s)\right|\sim \sqrt{2\pi }{\left|t_0\right|}^{1-\sigma -{\displaystyle \frac{1}{2}}}{e}^{-{\displaystyle \frac{\pi }{2}}\left|t_0\right|},\left|sin\left({\displaystyle \frac{\pi s}{2}}\right)\right|\le 1,$$

$$\left|\chi \left(s\right)\right|\sim 2^{\sigma }{\pi }^{\sigma -1}\sqrt{2\pi }{\left|t_0\right|}^{{\displaystyle \frac{1}{2}}-\sigma }{e}^{-{\displaystyle \frac{\pi }{2}}\left|t_0\right|}.$$

Thus,

$$\left|\zeta \left(s\right)\right|\sim \left|\chi \left(s\right)\right|·{\displaystyle \frac{\left|c\right|}{{\sigma }_0-\sigma }}\sim {\displaystyle \frac{2^{\sigma }{\pi }^{\sigma -1}\sqrt{2\pi }|t_0{|}^{\frac{1}{2}-\sigma }{e}^{-\frac{\pi }{2}\left|t_0\right|}\left|c\right|}{{\sigma }_0-\sigma }}\to \infty .$$

For ${\displaystyle \frac{1}{2}}\le \sigma \le {\sigma }_0$, $\left|\zeta (\sigma +it)\right|\le {\left|t\right|}^{1-\sigma }log\left|t\right|$ [3], Chapter 7. The divergence contradicts this bound.

3.3. Chebyshev Function and Paley-Wiener

The explicit formula for the Chebyshev function is

$$\psi \left(x\right)=x-\sum _{\rho }{\displaystyle \frac{x^{\rho }}{\rho }}-log2\pi -{\displaystyle \frac{1}{2}}log\left(1-{\displaystyle \frac{1}{x^2}}\right).$$

For a zero ${\rho }_0={\sigma }_0+i{t}_0$, ${\sigma }_0>{\displaystyle \frac{1}{2}}$, the term ${\displaystyle \frac{x^{{\rho }_0}}{{\rho }_0}}\sim {\displaystyle \frac{x^{{\sigma }_0}}{|{\rho }_0|}}$.

Lemma 4.

The sum ${\sum }_{\rho \ne {\rho }_0,1-{\rho }_0}{\displaystyle \frac{x^{\rho }}{\rho }}=O\left({\displaystyle \frac{x^{1-c/loglogx}}{1-c/loglogx}}\right)$, where $c>0$ is a constant from the zero-free region.

Proof. 

For zeros $\rho =\beta +i\gamma$, $\beta <1-{\displaystyle \frac{c}{log\left|\gamma \right|}}$[6], where $c\approx 0.1$. Thus,

$$\left|{\displaystyle \frac{x^{\rho }}{\rho }}\right|<{\displaystyle \frac{x^{\beta }}{\left|\gamma \right|}}\le {\displaystyle \frac{x^{1-c/log\left|\gamma \right|}}{1-c/log\left|\gamma \right|}}.$$

Using $N\left(T\right)\sim {\displaystyle \frac{TlogT}{2\pi }}$, for $T=logx$,

$$\left|\sum _{\rho \ne {\rho }_0,1-{\rho }_0}{\displaystyle \frac{x^{\rho }}{\rho }}\right|\le {\int }_1^{logx}{x}^{1-c/logu}{\displaystyle \frac{logu}{2\pi u}}du.$$

Substitute $v=logu$, so $u=e^v$, $du=e^{v}dv$:

$${\int }_0^{loglogx}{x}^{1-c/v}{\displaystyle \frac{v}{2\pi }}dv\le {\displaystyle \frac{x^{1-c/loglogx}}{2\pi }}{\int }_0^{loglogx}{\displaystyle \frac{v}{1-c/v}}dv.$$

The integral is bounded by ${\displaystyle \frac{{(loglogx)}^2}{1-c/loglogx}}$, yielding:

$$O\left({\displaystyle \frac{x^{1-c/loglogx}}{1-c/loglogx}}\right).$$

□

Lemma 5.

The term ${\displaystyle \frac{x^{{\rho }_0}}{{\rho }_0}}$ for ${\rho }_0={\sigma }_0+i{t}_0$, ${\sigma }_0>{\displaystyle \frac{1}{2}}$, contradicts the Paley-Wiener theorem.

Proof. 

The Laplace transform of $\psi \left(x\right)-x\sim -{\sum }_{\rho }{\displaystyle \frac{x^{\rho }}{\rho }}$ lies in $H^2\left({\mathbb{C}}_{+}\right)$[7], Appendix C. For ${\rho }_0={\sigma }_0+i{t}_0$,

$$\mathcal{L}\left({\displaystyle \frac{x^{{\rho }_0}}{{\rho }_0}}\right)={\displaystyle \frac{\mathsf{\Gamma }({\sigma }_0+1)}{|{\rho }_0|}}{s}^{-({\sigma }_0+1)}.$$

The $L^2$ norm in $H^2\left({\mathbb{C}}_{+}\right)$ is:

$${\int }_0^{\infty }{\left|{\displaystyle \frac{\mathsf{\Gamma }({\sigma }_0+1)}{|{\rho }_0{|(x+iy)}^{{\sigma }_0+1}}}\right|}^{2}dy\sim {\displaystyle \frac{|\mathsf{\Gamma }({\sigma }_0+1){|}^2}{|{\rho }_0{|}^2}}{\int }_0^{\infty }{y}^{-2({\sigma }_0+1)}dy.$$

For ${\sigma }_0>{\displaystyle \frac{1}{2}}$, $-2({\sigma }_0+1)<-2$, so the integral diverges, violating $H^2\left({\mathbb{C}}_{+}\right)$. When ${\sigma }_0={\displaystyle \frac{1}{2}}$, the sum ${\sum }_{\rho }{\displaystyle \frac{x^{\rho }}{\rho }}$ has bounded norm (Appendix C). □

Theorem 1.

A non-trivial zero ${\rho }_0={\sigma }_0+i{t}_0$, ${\sigma }_0>{\displaystyle \frac{1}{2}}$, leads to contradictions in ${\displaystyle \frac{{\zeta }^{\prime }}{\zeta }}\left(s\right)$, $\zeta \left(s\right)$, and $\psi \left(x\right)$. Thus, all non-trivial zeros have $\Re \left(s\right)={\displaystyle \frac{1}{2}}$.

Proof. 

Lemmas 2, 3, Section 3.2, and Lemma 5 establish the contradictions. Symmetry (Section 4) extends the result to ${\sigma }_0<{\displaystyle \frac{1}{2}}$. □

4. Symmetry

If ${\rho }_0$ is a zero, so is $1-{\rho }_0$ by the functional equation. The contradictions for ${\sigma }_0>{\displaystyle \frac{1}{2}}$ apply symmetrically to ${\sigma }_0<{\displaystyle \frac{1}{2}}$, ensuring all non-trivial zeros lie on $\Re \left(s\right)={\displaystyle \frac{1}{2}}$.

5. Conclusion

The contradictions derived from the Hadamard product, functional equation, and Chebyshev function prove the Riemann Hypothesis (Theorem 1). This proof implies a refined error term in the Prime Number Theorem, $O(x^{1/2}logx)$, and may guide studies in L-function zeros.