A Unified Proof of the Extended, Generalized, and Grand Riemann Hypothesis

1. Introduction

Inspired by Lemma 8 in Ref. [1], we first establish Theorems 1–4 progressively (the proofs do not depend on Lemma 8). Thereafter, we prove the Extended, Generalized, and Grand Riemann Hypotheses within a unified framework built upon Theorem 4. For convenience, Lemma 8 and other related lemmas of Ref. [1] are listed in the Appendix of this paper.

It should be noted that in this article and Ref. [1], ’j’ is used to denote the imaginary unit ($j^2=-1$), while ’i’ serves as a natural number index.

2. Four Theorems

As pointed out in Ref. [2] (on page 57), we can enumerate the non-trivial zeros of the zeta function in order of the increasing absolute value of their imaginary parts, where zeros whose imaginary parts have the same absolute value are arranged arbitrarily. Thus we remove the default ordering of $|{\beta }_i|$, $|{\beta }_1|\le |{\beta }_2|\le \dots$, as condition hereafter for simplicity.

Theorem 1: Given two entire functions represented as absolutely convergent (on the whole complex plane) infinite products of polynomial factors

$$F\left(s\right)=\prod _{i=1}^{\infty }(1+{\displaystyle \frac{{(s-{\alpha }_i)}^2}{{\beta }_i^2}}{)}^{m_i}$$

(1)

and

$$F(k-s)=\prod _{i=1}^{\infty }(1+{\displaystyle \frac{{(k-s-{\alpha }_i)}^2}{{\beta }_i^2}}{)}^{m_i}$$

(2)

where s is a complex variable, ${\rho }_i={\alpha }_i+j{\beta }_i$ and ${\overline{\rho }}_i={\alpha }_i-j{\beta }_i$ are the complex conjugate zeros of $F\left(s\right)$, $0<{\alpha }_i<k$, $k>0$, and ${\beta }_i\ne 0$ are real numbers, $m_i\ge 1$ is the multiplicity of quadruplets of zeros $({\rho }_i,{\overline{\rho }}_i,k-{\rho }_i,k-{\overline{\rho }}_i)$, ${\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}<\infty$.

Then we have

$$F\left(s\right)=F(k-s)\Rightarrow \left\{\begin{array}{cc}& {\alpha }_i={\displaystyle \frac{k}{2}},i=1,2,3,\dots \hfill \\ & 0<|{\beta }_1|<|{\beta }_2|<|{\beta }_3|<\dots \hfill \end{array}\right.$$

(3)

Proof. 

According to the definition of divisibility of entire functions ^[3][4] (or more specifically the definition that a polynomial divides an entire function expressed as infinite product of polynomial factors ^[1]), the functional equation $F\left(s\right)=F(k-s)$ implies that each polynomial factor on either side divides the infinite product on the opposite side, i.e.,

$$F\left(s\right)=F(k-s)\Rightarrow \left\{\begin{array}{cc}& (1+{\displaystyle \frac{{(s-{\alpha }_i)}^2}{{\beta }_i^2}})\mid F(k-s)\hfill \\ \hfill & (1+{\displaystyle \frac{{(k-s-{\alpha }_i)}^2}{{\beta }_i^2}})\mid F\left(s\right)\hfill \\ \hfill & i=1,2,3,\dots \hfill \end{array}\right.$$

(4)

where "∣" is the divisible sign.

Since $1+{\displaystyle \frac{{(s-{\alpha }_i)}^2}{{\beta }_i^2}}$ (with discriminant $\Delta =-4·{\displaystyle \frac{1}{{\beta }_i^2}}<0$) and $1+{\displaystyle \frac{{(k-s-{\alpha }_i)}^2}{{\beta }_i^2}}$ (with discriminant $\Delta =-4·{\displaystyle \frac{1}{{\beta }_i^2}}<0$) are irreducible over the real number field $\mathbb{R}$, then by Lemma 5 (see Appendix) and considering $(1+{\displaystyle \frac{{(s-{\alpha }_i)}^2}{{\beta }_i^2}}{)}^{m_i}=\underset{\underset{m_i\mathrm{times}}{︸}}{(1+{\displaystyle \frac{{(s-{\alpha }_i)}^2}{{\beta }_i^2}})\dots }$, we have from Equation (4):

$$\left\{\begin{array}{cc}& (1+{\displaystyle \frac{{(s-{\alpha }_i)}^2}{{\beta }_i^2}})\mid (1+{\displaystyle \frac{{(k-s-{\alpha }_l)}^2}{{\beta }_l^2}})\hfill \\ \hfill & (1+{\displaystyle \frac{{(k-s-{\alpha }_i)}^2}{{\beta }_i^2}})\mid (1+{\displaystyle \frac{{(s-{\alpha }_l)}^2}{{\beta }_l^2}})\hfill \\ \hfill & i=1,2,3,\dots \hfill \end{array}\right.$$

(5)

where l takes one value from $1,2,3,\dots$.

For the special kind of polynomials in Equation (5), "divisible" means "equal", which can be verified by comparing the like terms in equation

$$(1+{\displaystyle \frac{{(s-{\alpha }_i)}^2}{{\beta }_i^2}})=k^{\prime }(1+{\displaystyle \frac{{(k-s-{\alpha }_l)}^2}{{\beta }_l^2}}),k^{\prime }\ne 0\in \mathbb{R}$$

(6)

to get $k^{\prime }=1$. Further, due to the uniqueness of the multiplicity $m_i$, the only solution to Equation (5) is $i=l$, otherwise, duplicated zeros with ${\alpha }_l=k-{\alpha }_i,k-{\alpha }_l={\alpha }_i,{\beta }_i^2={\beta }_l^2,l\ne i$ would be generated to change $m_i$. Therefore we have from Equation (5):

$$(1+{\displaystyle \frac{{(s-{\alpha }_i)}^2}{{\beta }_i^2}})=(1+{\displaystyle \frac{{(k-s-{\alpha }_i)}^2}{{\beta }_i^2}}),i=1,2,3,\dots$$

(7)

By comparing the like terms in Equation (7), we obtain ${\alpha }_i={\displaystyle \frac{k}{2}}$. Further, to ensure the uniqueness of $m_i$ while ${\alpha }_i={\displaystyle \frac{k}{2}}$, we need limit the ${\beta }_i$ values to be distinct, i.e., $0<|{\beta }_1|<|{\beta }_2|<|{\beta }_3|<\dots$.

That completes the proof of Theorem 1. □

Remark: Theorem 1 extends the critical strip from $0<\Re \left(s\right)<1$ to $0<\Re \left(s\right)<k$ ($k>0$ is a real constant), and Lemma 8 is a special case of Theorem 1 with $k=1$.

Theorem 2: Given two entire functions represented as absolutely convergent (on the whole complex plane) infinite products of polynomial factors

$$\begin{array}{cc}\hfill G\left(s\right)& =\prod _{i=1}^{\infty }({\displaystyle \frac{{\beta }_i^2}{{\alpha }_i^2+{\beta }_i^2}}+{\displaystyle \frac{{(s-{\alpha }_i)}^2}{{\alpha }_i^2+{\beta }_i^2}}{)}^{m_i}\hfill \\ \hfill & =\prod _{i=1}^{\infty }(1-{\displaystyle \frac{s}{{\rho }_i}}{)}^{m_i}(1-{\displaystyle \frac{s}{{\overline{\rho }}_i}}{)}^{m_i}\hfill \\ \hfill & =\prod _{i=1}^{\infty }(1-{\displaystyle \frac{s}{{\rho }_i}})\hfill \end{array}$$

(8)

and

$$\begin{array}{cc}\hfill G(k-s)& =\prod _{i=1}^{\infty }({\displaystyle \frac{{\beta }_i^2}{{\alpha }_i^2+{\beta }_i^2}}+{\displaystyle \frac{{(k-s-{\alpha }_i)}^2}{{\alpha }_i^2+{\beta }_i^2}}{)}^{m_i}\hfill \\ \hfill & =\prod _{i=1}^{\infty }(1-{\displaystyle \frac{k-s}{{\rho }_i}}{)}^{m_i}(1-{\displaystyle \frac{k-s}{{\overline{\rho }}_i}}{)}^{m_i}\hfill \\ \hfill & =\prod _{i=1}^{\infty }(1-{\displaystyle \frac{k-s}{{\rho }_i}})\hfill \end{array}$$

(9)

where s is a complex variable, ${\rho }_i={\alpha }_i+j{\beta }_i$ and ${\overline{\rho }}_i={\alpha }_i-j{\beta }_i$ are the complex conjugate zeros of $G\left(s\right)$, $0<{\alpha }_i<k$, $k>0$, and ${\beta }_i\ne 0$ are real numbers, $m_i\ge 1$ is the multiplicity of quadruplets of zeros $({\rho }_i,{\overline{\rho }}_i,k-{\rho }_i,k-{\overline{\rho }}_i)$, ${\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}<\infty$.

Then we have

$$G\left(s\right)=G(k-s)\Rightarrow \left\{\begin{array}{cc}& {\alpha }_i={\displaystyle \frac{k}{2}},i=1,2,3,\dots \hfill \\ & 0<|{\beta }_1|<|{\beta }_2|<|{\beta }_3|<\dots \hfill \end{array}\right.$$

(10)

Proof. 

According to Theorem 1 and Lemma 9 (see Appendix), we have

$$\begin{array}{cc}\hfill & F\left(s\right)=F(k-s)\hfill \\ \iff \hfill \\ \hfill & \prod _{i=1}^{\infty }({\displaystyle \frac{{\beta }_i^2}{{\alpha }_i^2+{\beta }_i^2}}{)}^{m_i}F\left(s\right)=\prod _{i=1}^{\infty }({\displaystyle \frac{{\beta }_i^2}{{\alpha }_i^2+{\beta }_i^2}}{)}^{m_i}F(k-s)\hfill \\ \iff \hfill \\ \hfill & \prod _{i=1}^{\infty }({\displaystyle \frac{{\beta }_i^2}{{\alpha }_i^2+{\beta }_i^2}}+{\displaystyle \frac{{(s-{\alpha }_i)}^2}{{\alpha }_i^2+{\beta }_i^2}}{)}^{m_i}=\prod _{i=1}^{\infty }({\displaystyle \frac{{\beta }_i^2}{{\alpha }_i^2+{\beta }_i^2}}+{\displaystyle \frac{{(k-s-{\alpha }_i)}^2}{{\alpha }_i^2+{\beta }_i^2}}{)}^{m_i}\hfill \\ \iff \hfill \\ \hfill & G\left(s\right)=G(k-s)\hfill \end{array}$$

(11)

Then we know that

$$G\left(s\right)=G(k-s)\Rightarrow \left\{\begin{array}{cc}& {\alpha }_i={\displaystyle \frac{k}{2}},i=1,2,3,\dots \hfill \\ & 0<|{\beta }_1|<|{\beta }_2|<|{\beta }_3|<\dots \hfill \end{array}\right.$$

(12)

That completes the proof of Theorem 2. □

Theorem 3: Given two entire functions represented by their Hadamard products:

$$\mathsf{\Lambda }(\lambda ,s)=e^{A\left(\lambda \right)+B\left(\lambda \right)s}\prod _{i=1}^{\infty }(1-{\displaystyle \frac{s}{{\rho }_i}})e^{{\displaystyle \frac{s}{{\rho }_i}}}$$

(13)

and

$$\mathsf{\Lambda }(\overline{\lambda },k-s)=e^{A\left(\overline{\lambda }\right)+B\left(\overline{\lambda }\right)(k-s)}\prod _{i=1}^{\infty }(1-{\displaystyle \frac{k-s}{{\rho }_i}})e^{{\displaystyle \frac{k-s}{{\rho }_i}}}$$

(14)

where s is a complex variable, $\lambda$ denotes a mathematical object (e.g., Dirichlet character, modular form, automorphic representation), $\overline{\lambda }$ is the dual of $\lambda$, ${\rho }_i={\alpha }_i+j{\beta }_i$ and ${\overline{\rho }}_i={\alpha }_i-j{\beta }_i$ are the complex conjugate zeros of $\mathsf{\Lambda }(\lambda ,s)$, $0<{\alpha }_i<k$, $k>0$, and ${\beta }_i\ne 0$ are real numbers, ${\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}<\infty$.

Then we have

$$\mathsf{\Lambda }(\lambda ,s)=\epsilon \left(\lambda \right)\mathsf{\Lambda }(\overline{\lambda },k-s)\Rightarrow \left\{\begin{array}{cc}& {\alpha }_i={\displaystyle \frac{k}{2}},i=1,2,3,\dots \hfill \\ & 0<|{\beta }_1|<|{\beta }_2|<|{\beta }_3|<\dots \hfill \end{array}\right.$$

(15)

where $\epsilon \left(\lambda \right)$ is a complex number of absolute value 1, called the "root number" of L-function $L(\lambda ,s)$.

Remark: For more details about $\epsilon \left(\lambda \right)$, see Ref. [5] on page 94.

Proof. 

First, we have

$$\begin{array}{cc}\hfill \mathsf{\Lambda }(\lambda ,s)& =e^{A\left(\lambda \right)+B\left(\lambda \right)s}\prod _{i=1}^{\infty }(1-{\displaystyle \frac{s}{{\rho }_i}})e^{{\displaystyle \frac{s}{{\rho }_i}}}\hfill \\ \hfill & =e^{A\left(\lambda \right)+B\left(\lambda \right)s}\prod _{i=1}^{\infty }(1-{\displaystyle \frac{s}{{\rho }_i}})(1-{\displaystyle \frac{s}{{\overline{\rho }}_i}})e^{{\displaystyle \frac{2{\alpha }_{i}s}{{\alpha }_i^2+{\beta }_i^2}}}\hfill \\ \hfill & =e^{A\left(\lambda \right)+B\left(\lambda \right)s}{e}^{s·{\sum }_{i=1}^{\infty }{\displaystyle \frac{2{\alpha }_i}{{\alpha }_i^2+{\beta }_i^2}}}\prod _{i=1}^{\infty }(1-{\displaystyle \frac{s}{{\rho }_i}})(1-{\displaystyle \frac{s}{{\overline{\rho }}_i}})\hfill \end{array}$$

(16)

Noticing that ${\sum }_{i=1}^{\infty }{\displaystyle \frac{2{\alpha }_i}{{\alpha }_i^2+{\beta }_i^2}}<2k·{\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}<\infty$, then we have ${\sum }_{i=1}^{\infty }{\displaystyle \frac{2{\alpha }_i}{{\alpha }_i^2+{\beta }_i^2}}=c,c\in \mathbb{R},c\ne 0$. With further consideration of the multiplicity $m_i\ge 1$ of each zero, we obtain

$$\mathsf{\Lambda }(\lambda ,s)=e^{A\left(\lambda \right)+\left[B\right(\lambda )+c]s}\prod _{i=1}^{\infty }(1-{\displaystyle \frac{s}{{\rho }_i}}{)}^{m_i}(1-{\displaystyle \frac{s}{{\overline{\rho }}_i}}{)}^{m_i}$$

(17)

Accordingly

$$\mathsf{\Lambda }(\overline{\lambda },k-s)=e^{A\left(\overline{\lambda }\right)+[B\left(\overline{\lambda }\right)+c](k-s)}\prod _{i=1}^{\infty }(1-{\displaystyle \frac{k-s}{{\rho }_i}}{)}^{m_i}(1-{\displaystyle \frac{k-s}{{\overline{\rho }}_i}}{)}^{m_i}$$

(18)

Then we conclude that Theorem 3 is true according to Theorem 2, considering $e^{A\left(\lambda \right)+\left[B\right(\lambda )+c]s}$ and $\epsilon \left(\lambda \right)e^{A\left(\overline{\lambda }\right)+[B\left(\overline{\lambda }\right)+c](k-s)}$ have no zeros, thus both of them have no effect on the complex zeros related divisibility in the functional equation $\mathsf{\Lambda }(\lambda ,s)=\epsilon \left(\lambda \right)\mathsf{\Lambda }(\overline{\lambda },k-s)$.

That completes the proof of Theorem 3. □

In the following Theorem 4, we make further efforts to lay a foundation for the study of completed L-functions that possess both real and complex zeros, denoted by $\rho \in {\mathcal{Z}}_{\mathrm{real}}$ ($\Im \left(\rho \right)=0$) and $\rho \in {\mathcal{Z}}_{\mathrm{complex}}$ ($\Im \left(\rho \right)\ne 0$), respectively. When these two zero sets have no common elements, we express their disjointness by: ${\mathcal{Z}}_{\mathrm{real}}\cap {\mathcal{Z}}_{\mathrm{complex}}=⌀$.

The reason we need to consider this case is that, so far, we cannot rule out the existence of exceptional zeros (or Landau-Siegel zeros), although their numbers are very limited even if they do exist.

Denote the set of real zeros in the critical strip as

$${\mathcal{Z}}_{\mathrm{real}}=\left\{a_n\in \mathbb{R}\mid 0<a_n<k,n=1,2,\dots ,N\right\}$$

where N is a finite natural number. This finiteness follows from the Identity Theorem, which implies that any non-zero entire function cannot have infinitely many zeros in a bounded region.

Theorem 4: Given two entire functions represented by their Hadamard products:

$$\begin{array}{cc}\hfill \mathsf{\Lambda }(\lambda ,s)& =e^{A\left(\lambda \right)+B\left(\lambda \right)s}\prod _{\rho }(1-{\displaystyle \frac{s}{\rho }})e^{{\displaystyle \frac{s}{\rho }}}\hfill \\ \hfill & =e^{A\left(\lambda \right)+B\left(\lambda \right)s}\prod _{\rho \in {\mathcal{Z}}_{\mathrm{real}}}(1-{\displaystyle \frac{s}{\rho }})e^{{\displaystyle \frac{s}{\rho }}}\prod _{\rho \in {\mathcal{Z}}_{\mathrm{complex}}}(1-{\displaystyle \frac{s}{\rho }})e^{{\displaystyle \frac{s}{\rho }}}\hfill \\ \hfill & =e^{A\left(\lambda \right)+\left[B\right(\lambda )+c]s}\prod _{n=1}^N(1-{\displaystyle \frac{s}{a_n}})e^{{\displaystyle \frac{s}{a_n}}}\prod _{i=1}^{\infty }(1-{\displaystyle \frac{s}{{\rho }_i}})(1-{\displaystyle \frac{s}{{\overline{\rho }}_i}})\hfill \end{array}$$

(19)

and

$$\begin{array}{cc}\hfill \mathsf{\Lambda }(\overline{\lambda },k-s)& =e^{A\left(\overline{\lambda }\right)+B\left(\overline{\lambda }\right)(k-s)}\prod _{\rho }(1-{\displaystyle \frac{k-s}{\rho }})e^{{\displaystyle \frac{k-s}{\rho }}}\hfill \\ \hfill & =e^{A\left(\overline{\lambda }\right)+B\left(\overline{\lambda }\right)(k-s)}\prod _{\rho \in {\mathcal{Z}}_{\mathrm{real}}}(1-{\displaystyle \frac{k-s}{\rho }})e^{{\displaystyle \frac{k-s}{\rho }}}\prod _{\rho \in {\mathcal{Z}}_{\mathrm{complex}}}(1-{\displaystyle \frac{k-s}{\rho }})e^{{\displaystyle \frac{k-s}{\rho }}}\hfill \\ \hfill & =e^{A\left(\overline{\lambda }\right)+[B\left(\overline{\lambda }\right)+c](k-s)}\prod _{n=1}^N(1-{\displaystyle \frac{k-s}{a_n}})e^{{\displaystyle \frac{k-s}{a_n}}}\prod _{i=1}^{\infty }(1-{\displaystyle \frac{k-s}{{\rho }_i}})(1-{\displaystyle \frac{k-s}{{\overline{\rho }}_i}})\hfill \end{array}$$

(20)

where s is a complex variable, $c={\sum }_{i=1}^{\infty }{\displaystyle \frac{2{\alpha }_i}{{\alpha }_i^2+{\beta }_i^2}}$, $\lambda$ denotes a mathematical object, $\overline{\lambda }$ is the dual of $\lambda$, ${\rho }_i={\alpha }_i+j{\beta }_i$ and ${\overline{\rho }}_i={\alpha }_i-j{\beta }_i$ are the complex conjugate zeros of $\mathsf{\Lambda }(\lambda ,s)$, $0<{\alpha }_i<k$, $0<a_n<k$, $k>0$, and ${\beta }_i\ne 0$ are real numbers, ${\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}<\infty$, ${\mathcal{Z}}_{\mathrm{real}}\cap {\mathcal{Z}}_{\mathrm{complex}}=⌀.$

Then we have

$$\mathsf{\Lambda }(\lambda ,s)=\epsilon \left(\lambda \right)\mathsf{\Lambda }(\overline{\lambda },k-s)\Rightarrow \left\{\begin{array}{cc}& {\alpha }_i={\displaystyle \frac{k}{2}},i=1,2,3,\dots \hfill \\ & 0<|{\beta }_1|<|{\beta }_2|<|{\beta }_3|<\dots \hfill \\ \hfill & a_n={\displaystyle \frac{k}{2}},n=1\hfill \end{array}\right.$$

(21)

i.e., all the zeros (both real and complex) of $\mathsf{\Lambda }(\lambda ,s)$ lie on the critical line.

Proof. 

By Theorem 3, to determine the distribution of the complex zeros of $\mathsf{\Lambda }(\lambda ,s)$, we only need to show that the newly added parts ${\prod }_{\rho \in {\mathcal{Z}}_{\mathrm{real}}}(1-{\displaystyle \frac{s}{\rho }})e^{{\displaystyle \frac{s}{\rho }}}$ and ${\prod }_{\rho \in {\mathcal{Z}}_{\mathrm{real}}}(1-{\displaystyle \frac{k-s}{\rho }})e^{{\displaystyle \frac{k-s}{\rho }}}$ do not affect the complex zeros related divisibility in the functional equation $\mathsf{\Lambda }(\lambda ,s)=\epsilon \left(\lambda \right)\mathsf{\Lambda }(\overline{\lambda },k-s)$, which is an obvious fact since the given condition ${\mathcal{Z}}_{\mathrm{real}}\cap {\mathcal{Z}}_{\mathrm{complex}}=⌀$ means that ${\prod }_{\rho \in {\mathcal{Z}}_{\mathrm{real}}}(1-{\displaystyle \frac{s}{\rho }})e^{{\displaystyle \frac{s}{\rho }}}$ and ${\prod }_{\rho \in {\mathcal{Z}}_{\mathrm{real}}}(1-{\displaystyle \frac{k-s}{\rho }})e^{{\displaystyle \frac{k-s}{\rho }}}$ are co-prime according to Ref. [3] (on pages 174, 208) and Ref. [4] (see its THEOREM 4).

Thus, we conclude by Theorem 3 that

$$\mathsf{\Lambda }(\lambda ,s)=\epsilon \left(\lambda \right)\mathsf{\Lambda }(\overline{\lambda },k-s)\Rightarrow \left\{\begin{array}{cc}& {\alpha }_i={\displaystyle \frac{k}{2}},i=1,2,3,\dots \hfill \\ & 0<|{\beta }_1|<|{\beta }_2|<|{\beta }_3|<\dots \hfill \end{array}\right.$$

(22)

Next, we consider the real zeros of $\mathsf{\Lambda }(\lambda ,s)$.

By canceling the complex non-trivial zeros related polynomial factors on both sides of $\mathsf{\Lambda }(\lambda ,s)=\epsilon \left(\lambda \right)\mathsf{\Lambda }(\overline{\lambda },k-s)$, we have

$$e^{A\left(\lambda \right)+\left[B\right(\lambda )+c]s}\prod _{n=1}^N(1-{\displaystyle \frac{s}{a_n}})e^{{\displaystyle \frac{s}{a_n}}}=\epsilon \left(\lambda \right)e^{A\left(\overline{\lambda }\right)+[B\left(\overline{\lambda }\right)+c](k-s)}\prod _{n=1}^N(1-{\displaystyle \frac{k-s}{a_n}})e^{{\displaystyle \frac{k-s}{a_n}}}$$

(23)

Further Equation (23) is equivalent to

$$e^{A\left(\lambda \right)+[B\left(\lambda \right)+c^{\prime }]s}\prod _{n=1}^N(s-a_n)=\epsilon \left(\lambda \right)e^{A\left(\overline{\lambda }\right)+[B\left(\overline{\lambda }\right)+c^{\prime }](k-s)}\prod _{n=1}^N(s-(k-a_n))$$

(24)

where $c^{\prime }=c+{\sum }_{n=1}^N{\displaystyle \frac{1}{a_n}}$.

Suppose the multiplicity of zero $s=a_n$ is $m_n$ ($m_n\ge 1$) that is finite and unique although unknown. Then Equation (24) becomes

$$e^{A\left(\lambda \right)+[B\left(\lambda \right)+c^{\prime }]s}\prod _{t=1}^T(s-a_t{)}^{m_t}=\epsilon \left(\lambda \right)e^{A\left(\overline{\lambda }\right)+[B\left(\overline{\lambda }\right)+c^{\prime }](k-s)}\prod _{t=1}^T(s-(k-a_t){)}^{m_t}$$

(25)

where ${\sum }_{t=1}^T{m}_t=N$.

Considering $(s-a_t)$ and $(s-(k-a_t))$ are irreducible over $\mathbb{R}$, then by Lemma 3 (see Appendix), Equation (25) means

$$\left\{\begin{array}{cc}& (s-a_t)\mid (s-(k-a_m)\hfill \\ \hfill & (s-(k-a_t))\mid (s-a_m)\hfill \\ \hfill & t=1,2,3,\dots ,T\hfill \end{array}\right.$$

(26)

where m takes one value from $1,2,3,\dots ,T$.

The only solution to Equation (26) is $t=m,a_t={\displaystyle \frac{k}{2}},t=1,\dots ,T$, otherwise the uniqueness of $m_t$ would be violated with $a_t+a_m=k,t\ne m$. To avoid changing the multiplicity of $m_t$ while $a_t={\displaystyle \frac{k}{2}}$, we need to limit $T=1$. Thus we get

$$\mathsf{\Lambda }(\lambda ,s)=\epsilon \left(\lambda \right)\mathsf{\Lambda }(\overline{\lambda },k-s)\Rightarrow a_t={\displaystyle \frac{k}{2}},t=1$$

(27)

where zero $s=a_1={\displaystyle \frac{k}{2}}$ with multiplicity $m_1=N$ .

Putting Equation (22) and Equation (27) together, we proved Equation (21).

That completes the proof of Theorem 4. □

Remark: If the completed L-function has poles at $s=0$ or $s=1$ with multiplicity $m\ge 1$, then the Hadamard product expression can only be applied to $s^m{(k-s)}^m\mathsf{\Lambda }(\lambda ,s)$, not to $\mathsf{\Lambda }(\lambda ,s)$ itself. This adjustment does not affect the results of Theorem 4, since we have $s^m{(k-s)}^m\mathsf{\Lambda }(\lambda ,s)=\epsilon \left(\lambda \right){(k-s)}^m{(k-(k-s))}^m\mathsf{\Lambda }(\overline{\lambda },k-s)\iff \mathsf{\Lambda }(\lambda ,s)=\epsilon \left(\lambda \right)\mathsf{\Lambda }(\overline{\lambda },k-s)$.

Remark: Theorem 4 effectively rules out the existence of Landau-Siegel zeros, thereby proving the Landau-Siegel zeros conjecture.

Remark: As pointed out in Ref. [5] (on page 102), if ${\rho }_i={\alpha }_i+j{\beta }_i$ is a zero of $\mathsf{\Lambda }(\lambda ,s)$, then ${\overline{\rho }}_i={\alpha }_i-j{\beta }_i$ is a zero of $\mathsf{\Lambda }(\overline{\lambda },s)$. Therefore, to use Theorem 4 while $\lambda \ne \overline{\lambda }$, we need to construct a new symmetric functional equation $\mathsf{\Lambda }(\overline{\lambda },s)\mathsf{\Lambda }(\lambda ,s)=\epsilon \left(\lambda \right)\epsilon \left(\overline{\lambda }\right)\mathsf{\Lambda }(\lambda ,k-s)\mathsf{\Lambda }(\overline{\lambda },k-s)$ to ensure that the conjugate zeros appear together in the related Hadamard products. For more details, see the proofs of Theorem 5 and Theorem 7.

3. The Applications of Theorem 4

We will make use of Theorem 4 to prove the Extended Riemann Hypothesis (for Dedekind zeta function), the Generalized Riemann Hypothesis (for Dirichlet L-function), and the Grand Riemann Hypothesis (for modular form L-Function, automorphic L-function, and etc.).

To facilitate the subsequent discussion, we give the details of the Extended Riemann Hypothesis, the Generalized Riemann Hypothesis, and the Grand Riemann Hypothesis. In the following contents, the critical line means: $\mathfrak{R}\left(s\right)={\displaystyle \frac{1}{2}}$, or more generally, $\mathfrak{R}\left(s\right)={\displaystyle \frac{k}{2}}$, $k>0$ is a constant real number.

The Generalized Riemann Hypothesis: The non-trivial zeros of Dirichlet L-functions lie on the critical line.

The Extended Riemann Hypothesis: The non-trivial zeros of Dedekind zeta functions lie on the critical line.

The Grand Riemann Hypothesis: The non-trivial zeros of all L-functions lie on the critical line.

Another version of Grand Riemann Hypothesis: The non-trivial zeros of all automorphic L-functions lie on the critical line.

To begin with, we provide a general property of L-functions, which was labeled Lemma 5.5 in Ref. [5] on page 101.

Lemma 5.5^[5]: Let $L(f,s)$ be an L-function. All zeros $\rho$ of $\mathsf{\Lambda }(f,s)$ are in the critical strip $0\le \sigma \le 1$. For any $ϵ>0$, we have

$$\sum _{\rho \ne 0,1}{\left|\rho \right|}^{-1-ϵ}<+\infty .$$

where, $\mathsf{\Lambda }(f,s)$ is the completed L-function corresponding to $L(f,s)$, $\sigma$ is the real part of $\rho$, and f is identical to $\lambda$ in this paper as a symbol representing a mathematical object (e.g., Dirichlet character, modular form, automorphic representation).

Another general property of L-functions is as follows.

The zeros of $\mathsf{\Lambda }(\lambda ,s)$ are precisely the non-trivial zeros of $L(\lambda ,s)$, as the trivial zeros of $L(\lambda ,s)$ are canceled by the poles of the Gamma factors in the completion process (see Ref. [5] on page 96 for more details, with f therein replaced by $\lambda$).

Thus, we can discuss the non trivial zeros of L-functions based on the zeros of the corresponding completed L-functions.

3.1. Dirichlet L-Function

Definition: The Dirichlet L-function associated with a Dirichlet character $\chi$ modulo q is defined for $\Re \left(s\right)>1$ by the series:

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

(28)

For the principal (or trivial) character ${\chi }_0$ (where ${\chi }_0\left(n\right)=1$ if $gcd(n,q)=1$ and ${\chi }_0\left(n\right)=0$ otherwise), the L-function is related to the Riemann zeta function by:

$$L({\chi }_0,s)=\zeta \left(s\right)\prod _{p|q}\left(1-{\displaystyle \frac{1}{p^s}}\right)$$

(29)

Completed L-function: The completed Dirichlet L-function is defined as:

$$\mathsf{\Lambda }(\chi ,s)={\left({\displaystyle \frac{q}{\pi }}\right)}^{{\displaystyle \frac{s+a}{2}}}\mathsf{\Gamma }\left({\displaystyle \frac{s+a}{2}}\right)L(\chi ,s)$$

(30)

where $a=0$ if $\chi (-1)=1$ (even character) and $a=1$ if $\chi (-1)=-1$ (odd character).

Functional Equation: The completed Dirichlet L-function satisfies the functional equation:

$$\mathsf{\Lambda }(\chi ,s)=W\left(\chi \right)\mathsf{\Lambda }(\overline{\chi },1-s)$$

(31)

where $W\left(\chi \right)$ is the Gauss sum:

$$W\left(\chi \right)={\displaystyle \frac{\tau \left(\chi \right)}{i^a\sqrt{q}}}$$

(32)

and $\tau \left(\chi \right)={\sum }_{n=1}^q\chi \left(n\right)e^{2\pi in/q}$ is the Gauss sum associated with $\chi$.

Hadamard Product: For non-principal characters $\chi$, the completed L-function $\mathsf{\Lambda }(\chi ,s)$ is an entire function and has the Hadamard product:

$$\mathsf{\Lambda }(\chi ,s)=e^{A\left(\chi \right)+B\left(\chi \right)s}\prod _{\rho }\left(1-{\displaystyle \frac{s}{\rho }}\right)e^{s/\rho }$$

(33)

where the product is over all zeros $\rho$ of $\mathsf{\Lambda }(\chi ,s)$, and $A\left(\chi \right)$ and $B\left(\chi \right)$ are constants depending on $\chi$.

Next we prove the Generalized Riemann Hypothesis.

Theorem 5: The non-trivial zeros of the above-described Dirichlet L-functions lie on the critical line.

Remark: We only need to prove that all the zeros of the completed Dirichlet L-function $\mathsf{\Lambda }(\chi ,s)$ have real part ${\displaystyle \frac{1}{2}}$, i.e., all the zeros of $\mathsf{\Lambda }(\chi ,s)$ lie on the critical line.

Proof. 

We conduct the proof in two cases.

CASE 1: $\chi =\overline{\chi }$ (self-dual)

It suffices to verify that the properties of $\mathsf{\Lambda }(\chi ,s)$ with $\chi =\overline{\chi }$ match the conditions of Theorem 4 with $\lambda =\chi$, $\epsilon \left(\lambda \right)=W\left(\chi \right)$, $k=1$. Equation (33) is equivalent to Equation (19) in Theorem 4 by separating all zeros into two sets ${\mathcal{Z}}_{\mathrm{real}}$ and ${\mathcal{Z}}_{\mathrm{complex}}$. Actually, to restrict $\chi =\overline{\chi }$ is to guarantee that the complex conjugate zeros of $\mathsf{\Lambda }(\chi ,s)$ appear in pairs, and then the quadruplets of zeros $({\rho }_i,{\overline{\rho }}_i,1-{\rho }_i,1-{\overline{\rho }}_i)$ appear together according to Equation (31). The condition ${\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}<\infty$ can be assured by Lemma 5.5, considering that ${\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}$ is a subseries of ${\sum }_{\rho \ne 0,1}{\displaystyle \frac{1}{{\left|\rho \right|}^2}}$; The condition ${\mathcal{Z}}_{\mathrm{real}}\cap {\mathcal{Z}}_{\mathrm{complex}}=⌀$ holds because ${\mathcal{Z}}_{\mathrm{real}}$ and ${\mathcal{Z}}_{\mathrm{complex}}$ are mutually exclusive sets, i.e., if $\rho \in {\mathcal{Z}}_{\mathrm{real}}$, then $\Im \left(\rho \right)=0$; if $\rho \in {\mathcal{Z}}_{\mathrm{complex}}$, then $\Im \left(\rho \right)\ne 0$.

Therefore, by Theorem 4 with $\lambda =\chi$, $\epsilon \left(\lambda \right)=W\left(\chi \right)$, $k=1$, we know that both the real (if exists) and the complex zeros of $\mathsf{\Lambda }(\chi ,s)$ with $\chi =\overline{\chi }$ lie on the critical line.

CASE 2: $\chi \ne \overline{\chi }$

In this case, the complex conjugate zeros do not appear together in Equation (33), because if $\rho$ is a zero of $\mathsf{\Lambda }(\chi ,s)$, then $\overline{\rho }$ is a zero of $\mathsf{\Lambda }(\overline{\chi },s)$.

Thus, we need to extend Equation (31) to another form, i.e.,

$$\mathsf{\Lambda }(\overline{\chi },s)=W\left(\overline{\chi }\right)\mathsf{\Lambda }(\chi ,1-s)$$

(34)

Combining (34) with (31), we get a new functional equation

$$\mathsf{\Lambda }(\overline{\chi },s)\mathsf{\Lambda }(\chi ,s)=W\left(\overline{\chi }\right)W\left(\chi \right)\mathsf{\Lambda }(\chi ,1-s)\mathsf{\Lambda }(\overline{\chi },1-s)$$

(35)

Both sides of Equation (35) are the products of entire functions, thus they are still entire functions. And that the complex conjugate zeros of $\mathsf{\Lambda }(\overline{\chi },s)\mathsf{\Lambda }(\chi ,s)$ appear in pairs, then the quadruplets of zeros $({\rho }_i,{\overline{\rho }}_i,1-{\rho }_i,1-{\overline{\rho }}_i)$ appear together according to Equation (35). Further, based on Equation (33), we have

$$\begin{array}{cc}\hfill \mathsf{\Lambda }(\chi ,s)\mathsf{\Lambda }(\overline{\chi },s)& =e^{A\left(\chi \right)+A\left(\overline{\chi }\right)+[B\left(\chi \right)+B\left(\overline{\chi }\right)+c]s}\prod _{\rho \in {\mathcal{Z}}_{\mathrm{real}}}(1-{\displaystyle \frac{s}{\rho }})e^{{\displaystyle \frac{s}{\rho }}}\prod _{i=1}^{\infty }(1-{\displaystyle \frac{s}{{\rho }_i}})(1-{\displaystyle \frac{s}{{\overline{\rho }}_i}})\hfill \end{array}$$

(36)

where $c={\sum }_{i=1}^{\infty }{\displaystyle \frac{2{\alpha }_i}{{\alpha }_i^2+{\beta }_i^2}}$.

The condition ${\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}<\infty$ and condition ${\mathcal{Z}}_{\mathrm{real}}\cap {\mathcal{Z}}_{\mathrm{complex}}=⌀$ hold for the same reasons as in CASE 1.

Therefore, by Theorem 4, we know that both the real (if exists) and the complex zeros of $\mathsf{\Lambda }(\chi ,s)\mathsf{\Lambda }(\overline{\chi },s)$ (thus of $\mathsf{\Lambda }(\chi ,s)$) with $\chi \ne \overline{\chi }$ lie on the critical line.

Combining CASE 1 and CASE 2, we conclude that Theorem 5 holds as a specific case of Theorem 4 with $k=1$ and $\lambda =\chi$. □

3.2. Dedekind Zeta Function

Definition: For a number field K with ring of integers ${\mathcal{O}}_K$, the Dedekind zeta function is defined for $\Re \left(s\right)>1$ by:

$${\zeta }_K\left(s\right)=\sum _{\mathfrak{a}}{\displaystyle \frac{1}{N{\left(\mathfrak{a}\right)}^s}}$$

(37)

where the sum is over all non-zero ideals $\mathfrak{a}$ of ${\mathcal{O}}_K$, and $N\left(\mathfrak{a}\right)$ is the norm of the ideal.

Completed Zeta Function: The completed Dedekind zeta function is defined as:

$${\mathsf{\Lambda }}_K\left(s\right)={\left|D_K\right|}^{s/2}{\left({\pi }^{-s/2}\mathsf{\Gamma }\left({\displaystyle \frac{s}{2}}\right)\right)}^{r_1}{\left({\left(2\pi \right)}^{-s}\mathsf{\Gamma }\left(s\right)\right)}^{r_2}{\zeta }_K\left(s\right)$$

(38)

where $D_K$ is the discriminant of K, $r_1$ is the number of real embeddings of K, $r_2$ is the number of pairs of complex embeddings of K.

Functional Equation: The completed Dedekind zeta function satisfies:

$${\mathsf{\Lambda }}_K\left(s\right)=\epsilon \left(K\right){\mathsf{\Lambda }}_K(1-s)$$

(39)

where $\epsilon \left(K\right)=1$ for all number fields K, showing the symmetry of the functional equation.

Hadamard Product: The completed Dedekind zeta function has a simple pole at $s=1$ with residue ${\displaystyle \frac{2^{r_1}{\left(2\pi \right)}^{r_2}{h}_K{R}_K}{w_K\sqrt{|D_K|}}}$, where $h_K$ is the class number, $R_K$ is the regulator, and $w_K$ is the number of roots of unity in K. The function $s(s-1){\mathsf{\Lambda }}_K\left(s\right)$ is entire and has the Hadamard product:

$$s(s-1){\mathsf{\Lambda }}_K\left(s\right)=e^{A_K+B_{K}s}\prod _{\rho \ne 0,1}\left(1-{\displaystyle \frac{s}{\rho }}\right)e^{s/\rho }$$

(40)

where the product is over all zeros $\rho$ of ${\mathsf{\Lambda }}_K\left(s\right)$ except $\rho =0$ and $\rho =1$, and $A_K$ and $B_K$ are constants depending on K.

For more details of the completed Dedekind zeta function, please be referred to Ref. [5] (Chapter 5.10) and Ref. [6] (Section 10.5.1).

Theorem 6: The non-trivial zeros of the above-described Dedekind zeta function lie on the critical line.

Remark: We only need to prove that all the zeros of ${\mathsf{\Lambda }}_K\left(s\right)$ have real part ${\displaystyle \frac{1}{2}}$, i.e., all the zeros of ${\mathsf{\Lambda }}_K\left(s\right)$ lie on the critical line.

Proof. 

It suffices to show that the properties of ${\mathsf{\Lambda }}_K\left(s\right)$ match the conditions of Theorem 4 with $\lambda =K$, $\epsilon \left(\lambda \right)=1$, $k=1$.

Actually, Equation (39) and Equation (40) guarantee that

$$e^{A_K+B_{K}s}\prod _{\rho \ne 0,1}\left(1-{\displaystyle \frac{s}{\rho }}\right)e^{s/\rho }=e^{A_K+B_K(1-s)}\prod _{\rho \ne 0,1}\left(1-{\displaystyle \frac{1-s}{\rho }}\right)e^{1-s/\rho }$$

(41)

where $\prod _{\rho \ne 0,1}\left(1-{\displaystyle \frac{s}{\rho }}\right)e^{s/\rho }=\prod _{\rho \in {\mathcal{Z}}_{\mathrm{real}}\setminus \{0,1\}}\left(1-{\displaystyle \frac{s}{\rho }}\right)e^{s/\rho }\prod _{\rho \in {\mathcal{Z}}_{\mathrm{complex}}}\left(1-{\displaystyle \frac{s}{\rho }}\right)e^{s/\rho }$.

And that the complex conjugate zeros of ${\mathsf{\Lambda }}_k\left(s\right)$ appear in pairs, then the quadruplets of zeros $({\rho }_i,{\overline{\rho }}_i,1-{\rho }_i,1-{\overline{\rho }}_i)$ appear together according to Equation (39). The condition ${\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}<\infty$ and condition ${\mathcal{Z}}_{\mathrm{real}}\cap {\mathcal{Z}}_{\mathrm{complex}}=⌀$ hold for the same reasons as in CASE 1 in the proof of Theorem 5.

Therefore, by Theorem 4 we know that both the real (if exists) and the complex zeros of ${\mathsf{\Lambda }}_K\left(s\right)$ lie on the critical line, i.e., Theorem 6 holds as a specific case of Theorem 4 with $\lambda =K$, $\epsilon \left(\lambda \right)=1$, $k=1$. □

3.3. Modular Form L-Function

Definition: For a modular form $f\left(z\right)={\sum }_{n=1}^{\infty }{a}_n{e}^{2\pi inz}$ of weight k for a congruence subgroup $\mathsf{\Gamma }$, the associated L-function is defined for $\Re \left(s\right)>{\displaystyle \frac{k+1}{2}}$ by:

$$L(f,s)=\sum _{n=1}^{\infty }{\displaystyle \frac{a_n}{n^s}}$$

(42)

Completed L-function: For a cusp form f of weight k for ${\mathsf{\Gamma }}_0\left(N\right)$ (level N is any positive integer) with Nebentypus character $\chi$, the completed modular form L-function is defined as:

$$\mathsf{\Lambda }(f,s)=N^{s/2}{\left(2\pi \right)}^{-s}\mathsf{\Gamma }\left(s\right)L(f,s)$$

(43)

Functional Equation: For a normalized Hecke eigenform f of weight k for ${\mathsf{\Gamma }}_0\left(N\right)$ with Nebentypus character $\chi$, the completed modular form L-function satisfies:

$$\mathsf{\Lambda }(f,s)=\epsilon \left(f\right)\mathsf{\Lambda }(\overline{f},k-s)$$

(44)

where $\epsilon \left(f\right)=\pm 1$ is the epsilon factor, which is the eigenvalue of f under the Atkin-Lehner involution, and $\overline{f}$ is the modular form with Fourier coefficients ${\overline{a}}_n$.

Hadamard Product: For a cusp form f, the completed L-function $\mathsf{\Lambda }(f,s)$ is an entire function of order 1 and has the Hadamard product:

$$\mathsf{\Lambda }(f,s)=e^{A\left(f\right)+B\left(f\right)s}\prod _{\rho }\left(1-{\displaystyle \frac{s}{\rho }}\right)e^{s/\rho }$$

(45)

where the product is over all zeros $\rho$ of $\mathsf{\Lambda }(f,s)$, and $A\left(f\right)$ and $B\left(f\right)$ are constants depending on f.

For more details of the completed modular form L-function, please be referred to Refs. [5,7].

We have the following result about the non-trivial zero distribution of modular form L-Functions.

Theorem 7: The non-trivial zeros of the above-described modular form L-Functions lie on the critical line.

Remark: We only need to prove that all the zeros of $\mathsf{\Lambda }(f,s)$ have real part ${\displaystyle \frac{k}{2}}$, i.e., all the zeros of $\mathsf{\Lambda }(f,s)$ lie on the critical line.

Proof. 

We conduct the proof in two cases.

CASE 1: $f=\overline{f}$ (self-dual)

It suffices to show that the properties of $\mathsf{\Lambda }(f,s)$ with $f=\overline{f}$ match the conditions of Theorem 4 with $\lambda =f$. Equation (45) is equivalent to Equation (19) in Theorem 4 by separating all zeros into two sets ${\mathcal{Z}}_{\mathrm{real}}$ and ${\mathcal{Z}}_{\mathrm{complex}}$. Actually, to restrict $f=\overline{f}$ is to guarantee that the complex conjugate zeros of $\mathsf{\Lambda }(f,s))$ appear in pairs. Then the quadruplets of zeros $({\rho }_i,{\overline{\rho }}_i,k-{\rho }_i,k-{\overline{\rho }}_i)$ appear together according to Equation (44). The condition ${\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}<\infty$ and condition ${\mathcal{Z}}_{\mathrm{real}}\cap {\mathcal{Z}}_{\mathrm{complex}}=⌀$ hold for the same reasons as in CASE 1 in the proof of Theorem 5.

Therefore, by Theorem 4, we know that both the real (if exists) and the complex zeros of $\mathsf{\Lambda }(f,s)$ with $f=\overline{f}$ lie on the critical line.

CASE 2: $f\ne \overline{f}$

To deal with this case $f\ne \overline{f}$, we need first to extend Equation (44) to another form, i.e.,

$$\mathsf{\Lambda }(\overline{f},s)=\epsilon \left(\overline{f}\right)\mathsf{\Lambda }(f,k-s)$$

(46)

Combining Equation (46) with Equation (44), we get a new functional equation

$$\mathsf{\Lambda }(\overline{f},s)\mathsf{\Lambda }(f,s)=\epsilon \left(f\right)\epsilon \left(\overline{f}\right)\mathsf{\Lambda }(f,k-s)\mathsf{\Lambda }(\overline{f},k-s)$$

(47)

Obviously, both sides of Equation (47) are the products of entire functions, thus they are still entire functions. And that the complex conjugate zeros of $\mathsf{\Lambda }(\overline{f},s)\mathsf{\Lambda }(f,s)$ appear in pairs, then the quadruplets of zeros $({\rho }_i,{\overline{\rho }}_i,k-{\rho }_i,k-{\overline{\rho }}_i)$ appear together according to Equation (47). Further, based on Equation (45), we have

$$\mathsf{\Lambda }(f,s)\mathsf{\Lambda }(\overline{f},s)=e^{A\left(f\right)+A\left(\overline{f}\right)+[B\left(f\right)+B\left(\overline{f}\right)+c]s}\prod _{\rho \in {\mathcal{Z}}_{\mathrm{real}}}(1-{\displaystyle \frac{s}{\rho }})e^{{\displaystyle \frac{s}{\rho }}}\prod _{i=1}^{\infty }\left(1-{\displaystyle \frac{s}{{\rho }_i}}\right)\left(1-{\displaystyle \frac{s}{{\overline{\rho }}_i}}\right)$$

(48)

where $c={\sum }_{i=1}^{\infty }{\displaystyle \frac{2{\alpha }_i}{{\alpha }_i^2+{\beta }_i^2}}$.

The condition ${\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}<\infty$ and condition ${\mathcal{Z}}_{\mathrm{real}}\cap {\mathcal{Z}}_{\mathrm{complex}}=⌀$ hold for the same reasons as in CASE 1 in the proof of Theorem 5.

Therefore, by Theorem 4, we know that both the real (if exists) and the complex zeros of $\mathsf{\Lambda }(f,s)\mathsf{\Lambda }(\overline{f},s)$ (thus of $\mathsf{\Lambda }(f,s)$) with $f\ne \overline{f}$ lie on the critical line.

Combining CASE 1 and CASE 2, we conclude that Theorem 7 holds as a specific case of Theorem 4 with $\lambda =f$. □

3.4. Automorphic L-Function

Definition: For an automorphic representation $\pi$ of ${\mathrm{GL}}_n\left({\mathbb{A}}_{\mathbb{Q}}\right)$ (${\mathbb{A}}_{\mathbb{Q}}$ is the adele ring over the field of rational numbers $\mathbb{Q}$, see Ref. [7] on page 5 for more details), the associated L-function is defined for $\Re \left(s\right)>1$ by:

$$L(\pi ,s)=\prod _p{L}_p({\pi }_p,s)$$

(49)

where $L_p({\pi }_p,s)$ is the local L-factor at the prime p. For unramified ${\pi }_p$ with Satake parameters $\{{\alpha }_{1,p},\dots ,{\alpha }_{n,p}\}$,

$$L_p({\pi }_p,s)=\prod _{i=1}^n{\left(1-{\displaystyle \frac{{\alpha }_{i,p}}{p^s}}\right)}^{-1}$$

(50)

Completed L-function: The completed automorphic L-function is defined as:

$$\mathsf{\Lambda }(\pi ,s)=Q_{\pi }^{s/2}\prod _{i=1}^n{\mathsf{\Gamma }}_{*}^{\left(i\right)}(s+{\mu }_{i,\pi })·L(\pi ,s)$$

(51)

where $Q_{\pi }$ is the conductor of $\pi$, ${\mu }_{i,\pi }$ are complex numbers determined by the i-th local component of ${\pi }_{\infty }$, and

$${\mathsf{\Gamma }}_{*}^{\left(i\right)}\left(s\right)=\left\{\begin{array}{cc}& {\mathsf{\Gamma }}_{\mathbb{R}}\left(s\right)={\pi }^{-{\displaystyle \frac{s}{2}}}\mathsf{\Gamma }\left({\displaystyle \frac{s}{2}}\right),\mathrm{for}\mathrm{real}\mathrm{representations}.\hfill \\ \hfill & {\mathsf{\Gamma }}_{\mathbb{C}}\left(s\right)={\left(2\pi \right)}^{-s}\mathsf{\Gamma }\left(s\right),\mathrm{for}\mathrm{complex}\mathrm{representations}.\hfill \end{array}\right.$$

(52)

Functional Equation: The completed automorphic L-function satisfies:

$$\mathsf{\Lambda }(\pi ,s)=\epsilon \left(\pi \right)\mathsf{\Lambda }(\tilde{\pi },1-s)$$

(53)

where $\tilde{\pi }$ is the contragredient representation of $\pi$ and $\epsilon \left(\pi \right)$ is the epsilon factor, a complex number of absolute value 1.

Hadamard Product: For a cuspidal automorphic representation $\pi$, the completed L-function $\mathsf{\Lambda }(\pi ,s)$ is an entire function of order 1 and has the Hadamard product:

$$\mathsf{\Lambda }(\pi ,s)=e^{A\left(\pi \right)+B\left(\pi \right)s}\prod _{\rho }\left(1-{\displaystyle \frac{s}{\rho }}\right)e^{s/\rho }$$

(54)

where the product is over all zeros $\rho$ of $\mathsf{\Lambda }(\pi ,s)$, and $A\left(\pi \right)$ and $B\left(\pi \right)$ are constants depending on $\pi$.

For more details of the completed automorphic L-function, please be referred to Refs. [5,7].

We have the following result about the non-trivial zero distribution of automorphic L-Functions.

Theorem 8: The non-trivial zeros of the above-described automorphic L-Functions lie on the critical line.

Remark: We only need to prove that all the zeros of $\mathsf{\Lambda }(\pi ,s)$ have real part ${\displaystyle \frac{1}{2}}$, i.e., all the zeros of $\mathsf{\Lambda }(\pi ,s)$ lie on the critical line.

Proof. 

The proof procedure of Theorem 8 is similar to that of Theorem 7 with $k=1$, f replaced by $\pi$, and $\overline{f}$ replaced by $\tilde{\pi }$. Thus, the proof details are omitted for simplicity. □

Actually, from the above proofs of Theorem 5, Theorem 6, and Theorem 7, we can note that each proof does not depend on the specific definition of the L-function $L(\lambda ,s)$, but rather relies on the following general properties of $\mathsf{\Lambda }(\lambda ,s)$ and $L(\lambda ,s)$:

P1: Symmetric functional equation between $\mathsf{\Lambda }(\lambda ,s)$ and $\mathsf{\Lambda }(\overline{\lambda },k-s)$: $\mathsf{\Lambda }(\lambda ,s)=\epsilon \left(\lambda \right)\mathsf{\Lambda }(\overline{\lambda },k-s)$;

P2: Hadamard product expression of entire function $\mathsf{\Lambda }(\lambda ,s)$ or $s^m{(k-s)}^m\mathsf{\Lambda }(\lambda ,s),m\ge 1$, i.e., $\mathsf{\Lambda }(\lambda ,s)=e^{A\left(\lambda \right)+B\left(\lambda \right)s}{\prod }_{\rho }(1-{\displaystyle \frac{s}{\rho }})e^{{\displaystyle \frac{s}{\rho }}}$ or $s^m{(k-s)}^m\mathsf{\Lambda }(\lambda ,s)=e^{A\left(\lambda \right)+B\left(\lambda \right)s}{\prod }_{\rho }(1-{\displaystyle \frac{s}{\rho }})e^{{\displaystyle \frac{s}{\rho }}},m\ge 1$;

P3: The concurrence of complex conjugate zeros in the Hadamard product expression of entire functions $\mathsf{\Lambda }(\lambda ,s)$ and $\mathsf{\Lambda }(\overline{\lambda },k-s)$, or their variants $\mathsf{\Lambda }(\lambda ,s)\mathsf{\Lambda }(\overline{\lambda },s)$ and $\mathsf{\Lambda }(\overline{\lambda },k-s)\mathsf{\Lambda }(\lambda ,k-s)$;

P4: The property stated in Lemma 5.5 in Ref. [5], which guarantees all zeros $\rho$ in the Hadamard product expression (see P2) lie in the critical strip and satisfy the condition $\sum {\displaystyle \frac{1}{{\left|\rho \right|}^2}}<\infty$ or equivalently ${\sum }_{i=1}^{\infty }{\displaystyle \frac{1}{|{\rho }_i{|}^2}}<\infty$;

P5: The disjointness of real and complex non-trivial zero sets;

P6: The zeros of $\mathsf{\Lambda }(\lambda ,s)$ are precisely the non-trivial zeros of $L(\lambda ,s)$.

Therefore, we have the following result on the Grand Riemann Hypothesis.

Theorem 9: The non-trivial zeros of all L-functions lie on the critical line if only the properties P1, P2, P3, P4, P5, and P6 are satisfied.

Proof. 

It is not difficult to see that P1, P2, P3, P4, and P5 cover all the conditions in Theorem 4. Thus, we know by Theorem4 that all the zeros, both real (if exist) and complex, of $\mathsf{\Lambda }(\lambda ,s)$ lie on the critical line. Further, according to P6, we conclude that all the non-trivial zeros, both real (if exist) and complex, of $L(\lambda ,s)$ lie on the critical line.

That completes the proof of Theorem 9. □

Remark: Condition P5 is always satisfied as we define ${\mathcal{Z}}_{\mathrm{real}}$ ($\Im \left(\rho \right)=0$) and ${\mathcal{Z}}_{\mathrm{complex}}$ ($\Im \left(\rho \right)\ne 0$); P6 does not appear in the conditions of Theorem 4; its role is solely to bridge between the zeros of $\mathsf{\Lambda }(\lambda ,s)$ and the non-trivial zeros of $L(\lambda ,s)$. Therefore, to verify a given L-function or to construct new zeta- or L-type functions satisfying Grand Riemann Hypothesis, we need only focus on verifying P1, P2, P3, and P4 in Theorem 4 plus P6 outside Theorem 4.