Remarks on the Connection of the Riemann Hypothesis to Self-Approximation

Abstract

By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function $\zeta \left(s\right)$ by shifts $\zeta (s+i\tau )$. In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating $\zeta \left(s\right)$ with all but at most countably many accuracies $\epsilon >0$. Also, the analogue of an equivalent in terms of positive density in short intervals is discussed.

1. Introduction

Let $s=\sigma +it$ be a complex variable, and $\mathcal{P}$ denote the set of all prime numbers. For $\sigma >1$, the Riemann zeta-function $\zeta \left(s\right)$ is given by

$$\zeta \left(s\right)=\sum _{m=1}^{\infty }{\displaystyle \frac{1}{m^s}}=\prod _{p\in \mathcal{P}}{\left(1-{\displaystyle \frac{1}{p^s}}\right)}^{-1}$$

and is analytically continuable to the whole complex plane, except for the point $s=1$ that is a simple pole with residue 1.

The function $\zeta \left(s\right)$ plays a very important role in analytic number theory, and in mathematics in general. The function $\zeta \left(s\right)$ was already studied by L. Euler; however, he investigated $\zeta \left(s\right)$ only for $s\in \mathbb{R}$. The significance of $\zeta \left(s\right)$ was observed by B. Riemann who began to consider (in 1859) $\zeta \left(s\right)$ as a function of a complex variable. He indicated a method of applying the $\zeta \left(s\right)$ function for the research of the number of prime numbers

$$\pi \left(x\right)=\sum _{p⩽x}1$$

as $x\to \infty$. It turned out that this method is correct, and C.J. de la Valleé Poussin (in 1896) and J. Hadamard (in 1896) independently obtained the asymptotic distribution law of prime numbers using an Riemann’s idea:

$$\pi \left(x\right)\sim {\int }_2^x{(logu)}^{-1}\mathrm{d}u,x\to \infty .$$

(1)

The zeros of the function $\zeta \left(s\right)$ play a crucial role in the proof of the above theorem.

From the functional equation for $\zeta \left(s\right)$ proved by Riemann

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

where $\Gamma \left(s\right)$ denotes the Euler gamma-function, which then gives $\zeta (-2k)=0$, $k\in \mathbb{N}$. These zeros $s=-2k$, $k\in \mathbb{N}$, of $\zeta \left(s\right)$ are called trivial and are not important for applications. Moreover, the function $\zeta \left(s\right)$ has infinitely many of the non-trivial complex zeros located in the region $\{s\in \mathbb{C}:\sigma \in (0,1\left)\right\}$. The asymptotics of the function $\pi \left(x\right)$ as $x\to \infty$ is closely related to the location of non-trivial zeros of $\zeta \left(s\right)$. For example, to prove (1), it is sufficient to show that $\zeta \left(s\right)\ne 0$ on the line $\sigma =1$, while the zero-free region

$$\left\{s\in \mathbb{C}:\sigma >1-{\displaystyle \frac{c}{log\left(\right|t|+2)}},c>0\right\}$$

implies the asymptotic formula with $c_1>0$

$$\pi \left(x\right)={\int }_2^x{(logu)}^{-1}\mathrm{d}u+O(xexp\{-c_1\sqrt{logx}\}),x\to \infty .$$

Riemann also stated a few conjectures for the function $\zeta \left(s\right)$. The most famous of them —the Riemann hypothesis (RH) —says that all non-trivial zeros of $\zeta \left(s\right)$ are on the line $\sigma =1/2$, or, equivalently, $\zeta \left(s\right)\ne 0$, for $\sigma >1/2$. The line $\sigma =1/2$ is called critical. This hypothesis, until present day, remains neither proved not disproved; it is one of the seven millennium problems of mathematics. Therefore, every result concerning RH is important.

RH has several equivalents; some classical ones can be found in [1]. It is well known that RH is equivalent to the estimate

$$\pi \left(x\right)={\int }_2^x{(logu)}^{-1}\mathrm{d}u+O(x^{1/2}logx),x\to \infty .$$

In this note, we are connected to RH with approximation properties of the function $\zeta \left(s\right)$. The starting point for this is the universality property of $\zeta \left(s\right)$ discovered in [2] by S.M. Voronin. Suppose that $0<r<1/4$, and $f\left(s\right)$ is a continuous non-vanishing function on $\left|s\right|⩽r$ that is analytic for $\left|s\right|<r$. Then, Voronin obtained that, for every $\epsilon >0$, there exists $\tau =\tau \left(\epsilon \right)\in \mathbb{R}$, such that

$$\underset{\left|s\right|⩽r}{max}\left|\zeta \left(s+{\displaystyle \frac{3}{4}}+i\tau \right)-f\left(s\right)\right|<\epsilon .$$

This shows that the function $\zeta \left(s\right)$ is universal in an approximation sense; therefore, Voronin’s theorem is called the universality theorem. The initial form of Voronin’s theorem was improved in [3,4] (see also [5,6]). Set $\mathcal{D}=\{s\in \mathbb{C}:\sigma \in (1/2,1\left)\right\}$, and denote by $\mathcal{K}$ the set of all compact subsets of the strip $\mathcal{D}$ having connected complements. Moreover, let ${\mathcal{H}}_0\left(K\right)$ and $K\in \mathcal{K}$ be the set all of continuous non-vanishing functions on K and the analytic inside of K. Let $\mathcal{L}A$ be the Lebesgue measure of the set $A\subset \mathbb{R}$. Then, the modern version of the universality theorem for $\zeta \left(s\right)$ is the following statement:

Theorem 1. 

Suppose that $K\in \mathcal{K}$ and $f\left(s\right)\in {\mathcal{H}}_0\left(K\right)$. Then, for every $\epsilon >0$,

$$\underset{T\to \infty }{lim\; inf}{\displaystyle \frac{1}{T}}\mathcal{L}\left\{\tau \in [0,T]:\underset{s\in K}{sup}|\zeta (s+i\tau )-f\left(s\right)|<\epsilon \right\}>0.$$

A natural interesting question arises: Is Theorem 1 valid with $f\left(s\right)=\zeta \left(s\right)$, i. e., can $\zeta \left(s\right)$ be approximated by shifts $\zeta (s+i\tau )$? It turned out that this question is closely connected to the one which RH. B. Bagchi proved [7] using the following theorem.

Theorem 2. 

The Riemann hypothesis is equivalent to the assertion that, for every $K\in \mathcal{K}$ and every $\epsilon >0$,

$$\underset{T\to \infty }{lim\; inf}{\displaystyle \frac{1}{T}}\mathcal{L}\left\{\tau \in [0,T]:\underset{s\in K}{sup}|\zeta (s+i\tau )-\zeta \left(s\right)|<\epsilon \right\}>0.$$

Five years latter, Bagchi found a new proof of Theorem 2 by means of topological dynamics [8]. An extension of Theorem 2 for other zeta-functions is given in [6]. The property of $\zeta \left(s\right)$ contained in Theorem 2 is called the strong recurrence.

Theorem 2 inspired a series of works devoted to the positivity of

$$\underset{T\to \infty }{lim\; inf}{\displaystyle \frac{1}{T}}\mathcal{L}\left\{\tau \in [0,T]:\underset{s\in K}{sup}|\zeta (s+i\tau )-\zeta (s+i\tau d)|<\epsilon \right\}$$

for all $K\in \mathcal{K}$ and $\epsilon >0$ with $d\in \mathbb{R}$ (see [9,10,11,12,13]). Obviously, $d=0$ implies RH.

The aim of this note is to establish a new version of Theorem 2 inspired by the following modification of Theorem 1 [14,15].

Theorem 3. 

Suppose that $K\in \mathcal{K}$ and $f\left(s\right)\in {\mathcal{H}}_0\left(K\right)$. Then, the limit

$$\underset{T\to \infty }{lim}{\displaystyle \frac{1}{T}}\mathcal{L}\left\{\tau \in [0,T]:\underset{s\in K}{sup}|\zeta (s+i\tau )-f\left(s\right)|<\epsilon \right\}$$

exists and is positive for all but at most countably many $\epsilon >0$.

Thus, we will prove the following.

Theorem 4. 

The Riemann hypothesis is true if and only if, for every $K\in \mathcal{K}$, the limit

$$\underset{T\to \infty }{lim}{\displaystyle \frac{1}{T}}\mathcal{L}\left\{\tau \in [0,T]:\underset{s\in K}{sup}|\zeta (s+i\tau )-\zeta \left(s\right)|<\epsilon \right\}$$

(2)

exists and is positive for all but at most countably many $\epsilon >0$.

Theorem 4 has a more general statement.

Theorem 5. 

Suppose that $T^{1/3}{(logT)}^{26/15}⩽H⩽T$. Then, the Riemann hypothesis is equivalent to the assertion that, for every $K\in \mathcal{K}$, the limit

$$\underset{T\to \infty }{lim}{\displaystyle \frac{1}{H}}\mathcal{L}\left\{\tau \in [T,T+H]:\underset{s\in K}{sup}|\zeta (s+i\tau )-\zeta \left(s\right)|<\epsilon \right\}$$

exists and is positive for all but at most countably many $\epsilon >0$.

2. Proof of Theorems 4 and 5

Proofs of Theorems 4 and 5 are based on limit theorems for probability measures in the space of analytic functions. Therefore, first, we present useful results of such a type.

Denote by $\mathcal{B}\left(\mathbb{X}\right)$ the Borel $\sigma$-field of the space $\mathbb{X}$. Define the set

$$\mathbb{T}=\prod _{p\in \mathcal{P}}{y}_p,$$

where $y_p=\{s\in \mathbb{C}:|s|=1\}$ for all primes p. The infinite-dimensional torus $\mathbb{T}$, with the product topology and pointwise multiplication, is a compact topological Abelian group. Therefore, on $(\mathbb{T},\mathcal{B}(\mathbb{T}\left)\right)$, the probability Haar measure ${\mu }_H$ can be defined. This gives the probability space $(\mathbb{T},\mathcal{B}\left(\mathbb{T}\right),{\mu }_H)$. Denote by $\mathfrak{t}\left(p\right)$ the pth component of an element $\mathfrak{t}\in \mathbb{T}$, by $\mathcal{H}$ the space of analytic functions on $\mathcal{D}$ endowed with the topology of uniform convergence on compacta, and, on $(\mathbb{T},\mathcal{B}\left(\mathbb{T}\right),{\mu }_H)$, define the $\mathcal{H}$-valued random element

$$\zeta (s,\mathfrak{t})=\prod _{p\in \mathcal{P}}{\left(1-{\displaystyle \frac{\mathfrak{t}\left(p\right)}{p^s}}\right)}^{-1}.$$

Note that the infinite product, for almost all $\mathfrak{t}\in \mathbb{T}$, converges uniformly on compact subsets of the strip $\mathcal{D}$. Let $P_{\zeta }$ be the distribution of the random element $\zeta (s,\mathfrak{t})$, that is

$$P_{\zeta }\left(A\right)={\mu }_H\left\{\mathfrak{t}\in \mathbb{T}:\zeta (s,\mathfrak{t})\in A\right\},A\in \mathcal{B}\left(\mathcal{H}\right).$$

For $A\in \mathcal{B}\left(\mathcal{H}\right)$, define

$$P_T\left(A\right)={\displaystyle \frac{1}{T}}\mathcal{L}\left\{\tau \in [0,T]:\zeta (s+i\tau )\in A\right\}.$$

Lemma 1. 

$P_T$ converges weakly to $P_{\zeta }$ as $T\to \infty$.

Limit theorems for the Dirichlet series, including the function $\zeta \left(s\right)$, in the space $\mathcal{H}$ introduced by B. Bagchi in his thesis [3]. A proof of the lemma can also be found in [5,6].

For $A\in \mathcal{B}\left(\mathcal{H}\right)$, define

$$P_{T,H}={\displaystyle \frac{1}{H}}\mathcal{L}\left\{\tau \in [T,T+H]:\zeta (s+i\tau )\in A\right\}.$$

Lemma 2. 

Suppose that $T^{1/3}{(logT)}^{26/15}⩽H⩽T$. Then, $P_{T,H}$ converges weakly to $P_{\zeta }$ as $T\to \infty$.

The lemma is Theorem 9 from [16].

In the proofs of Theorems 4 and 5, we will use the support of the measure $P_{\zeta }$. Recall that the support of $P_{\zeta }$ is a minimal closed set $S_{\zeta }\subset \mathcal{H}$ such that $P_{\zeta }\left(S_{\zeta }\right)=1$. The set $S_{\zeta }$ consists of all $g\in \mathcal{H}$ such that, for every open neighbourhood G of g, the inequality $P_{\zeta }\left(G\right)>0$ is satisfied.

Lemma 3. 

The support of $P_{\zeta }$ is the set

$$S_{\zeta }=\{h\in \mathcal{H}:h\left(s\right)\ne 0orh\left(s\right)\equiv 0\}.$$

Proof of the lemma can be found in [5], Lemma 6.5.5.

Let P be a probability measure on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$, and $\partial A$ denote the boundary of the set $A\in \mathcal{B}\left(\mathbb{X}\right)$. If $P(\partial A)=0$, then the set A is called a continuity set of the measure P. For convenience, we state one equivalent of the weak convergence of probability measures in terms of continuity sets.

Lemma 4. 

Let $P_n$, $n\in \mathbb{N}$, and P be probability measures on the space $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$. Then, $P_n$ converges weakly to P as $n\to \infty$ if and only if

$$\underset{n\to \infty }{lim}{P}_n\left(A\right)=P\left(A\right)$$

for every continuity set A of the measure P.

The proof can be found in [17], Theorem 2.1.

We continue with a metric in the space $\mathcal{H}$. It is well known that there is a sequence of embedded compact subsets with connected complements $\{K_l:l\in \mathbb{N}\}$ of the strip $\mathcal{D}$ such that

$$\bigcup _{l=1}^{\infty }{K}_l=\mathcal{D},$$

and if $K\subset \mathcal{D}$ is a compact set, then K lies in some $K_l$. For example, we can take closed rectangles $K_l$. For $h_1,h_2\in \mathcal{H}$, define

$$\rho (h_1,h_2)=\sum _{l=1}^{\infty }{2}^{-l}{\displaystyle \frac{{sup}_{s\in K}|h_1\left(s\right)-h_2\left(s\right)|}{1+{sup}_{s\in K}|h_1\left(s\right)-h_2\left(s\right)|}}.$$

Then, $\rho$ is a desired metric in $\mathcal{H}$ inducing the topology of uniform convergence on compacta.

Proof of Theorem 4. 

Suppose that RH is true. Then, obviously, inequality (2) follows from Theorem 3; however, for fullness, we give a short proof without using Mergelyan’s theorem that asserts that every function continuous on a compact set $K\subset \mathbb{C}$ with connected complement and analytic inside of K can be approximated uniformly on K by a polynomial [18]. Usually, Mergelyan’s theorem is a useful ingredient for proofs of universality theorems for zeta-functions. Since RH is true, in view of Lemma 3, the function $\zeta \left(s\right)\in S_{\zeta }$; therefore, for $K\in \mathcal{K}$ and $\epsilon >0$, the set

$${\mathcal{G}}_{\epsilon ,K}=\left\{h\in \mathcal{H}:\underset{s\in K}{sup}|h\left(s\right)-\zeta \left(s\right)|<\epsilon \right\}$$

is an open neighbourhood of an element $\zeta \left(s\right)$, which lies in the support of the measure $P_{\zeta }$. Hence,

$$P_{\zeta }\left({\mathcal{G}}_{\epsilon ,K}\right)>0.$$

(3)

For fixed K, the boundary $\partial {\mathcal{G}}_{\epsilon ,K}$ of ${\mathcal{G}}_{\epsilon ,K}$ belongs to the set

$$\left\{h\in \mathcal{H}:\underset{s\in K}{sup}|h\left(s\right)-\zeta \left(s\right)|=\epsilon \right\};$$

therefore, the boundaries $\partial {\mathcal{G}}_{{\epsilon }_1,K}$ and $\partial {\mathcal{G}}_{{\epsilon }_2,K}$ cannot intersect for positive ${\epsilon }_1\ne {\epsilon }_2$. Hence, $P_{\zeta }(\partial {\mathcal{G}}_{\epsilon ,K})>0$ for at most countably many values of $\epsilon >0$ because, for every $m\in \mathbb{N}\setminus \left\{1\right\}$, there are at most $m-1$ sets of $\partial {\mathcal{G}}_{\epsilon ,K}$ satisfying

$$P_{\zeta }(\partial {\mathcal{G}}_{\epsilon ,K})>{\displaystyle \frac{1}{m}}.$$

This gives that the set ${\mathcal{G}}_{\epsilon ,K}$ is a continuity set of the measure $P_{\zeta }$ for all but at most countably many $\epsilon >0$. Hence, in view of Lemmas 1 and 4,

$$\underset{T\to \infty }{lim}{P}_T\left({\mathcal{G}}_{\epsilon ,K}\right)=P_{\zeta }\left({\mathcal{G}}_{\epsilon ,K}\right)$$

(4)

for all but at most countably many $\epsilon >0$. This, (3) and the definitions of $P_T$ and ${\mathcal{G}}_{\epsilon ,K}$ prove that (2) is positive.

It remains to prove that the positivity of (2) implies RH. Suppose, on the contrary, that RH is not valid. Then, there exists zeros of the function $\zeta \left(s\right)$ lying in $\mathcal{D}$, thus, $\zeta \left(s\right)\notin S_{\zeta }$, and by Lemma 3, $\zeta \left(s\right)$ is not in the support of the probability measure $P_{\zeta }$. Hence, there exists an open neighbourhood $\mathcal{G}$ of the function $\zeta \left(s\right)$ such that $P_{\zeta }\left(\mathcal{G}\right)=0$. Next, there exists an open ball ${\mathcal{G}}_{\delta }=\{h\in \mathcal{H}:\rho (h,\zeta )<\delta \}$ lying in $\mathcal{G}$. We will show that there exist $K\in \mathcal{K}$ and values of $\epsilon$ such that the set ${\mathcal{G}}_{\epsilon ,K}$ is contained in ${\mathcal{G}}_{\delta }$. Fix $l_0\in \mathbb{N}$, satisfying

$$\sum _{l=l_0+1}^{\infty }{2}^{-l}<{\displaystyle \frac{\delta }{3}}.$$

(5)

Consider the set ${\mathcal{G}}_{\delta /3,K_{l_0}}$, where $K_{l_0}$ is from the definition of the metric $\rho$. Then, by properties of the sequence $\left\{K_l\right\}$, the inclusion $K_l\subset K_{l_0}$ is valid for all $l=1,\cdots ,l_0$. Therefore, for all $l=1,\cdots ,l_0$,

$$\underset{s\in K_l}{sup}\left|h\left(s\right)-\zeta \left(s\right)\right|⩽\underset{s\in K_{l_0}}{sup}\left|h\left(s\right)-\zeta \left(s\right)\right|.$$

Hence, for $h\in {\mathcal{G}}_{\delta /3,K_{l_0}}$,

$$\begin{array}{cc}\hfill \rho (h,\zeta )& =\left(\sum _{l=1}^{l_0}+\sum _{l=l_0+1}^{\infty }\right)2^{-l}{\displaystyle \frac{{sup}_{s\in K_l}|h\left(s\right)-\zeta \left(s\right)|}{1+{sup}_{s\in K_l}|h\left(s\right)-\zeta \left(s\right)|}}\hfill \\ & <\underset{s\in K_{l_0}}{sup}\left|h\left(s\right)-\zeta \left(s\right)\right|\sum _{m=1}^{l_0}{2}^{-l}+\sum _{l=l_0+1}^{\infty }{2}^{-l}<{\displaystyle \frac{\delta }{3}}\sum _{l=1}^{\infty }{2}^{-l}+{\displaystyle \frac{\delta }{3}}={\displaystyle \frac{2\delta }{3}}\hfill \end{array}$$

in view of (5) because, by the definition of ${\mathcal{G}}_{\delta /3,K_{l_0}}$,

$$\underset{s\in K_0}{sup}\left|h\left(s\right)-\zeta \left(s\right)\right|<{\displaystyle \frac{\delta }{3}}.$$

Therefore, ${\mathcal{G}}_{\epsilon ,K_{l_0}}$ lies in ${\mathcal{G}}_{\delta }$, and, thus, in $\mathcal{G}$, for all $0<\epsilon <\delta /3$. Hence, $P_{\zeta }\left({\mathcal{G}}_{\epsilon ,K_{l_0}}\right)=0$ for all $0<\epsilon <\delta /3$. Since the segment $0<\epsilon <\delta /3$ contains the continuum of the values of $\epsilon$, the equality $P_{\zeta }\left({\mathcal{G}}_{\epsilon ,K_{l_0}}\right)=0$ and (4) contradict the positivity of (2). The proof is complete. □

Proof of Theorem 5. 

We follow the proof of Theorem 4 by replacing Lemma 1 with Lemma 2. □

The abovementioned Bagchi theorem and Theorems 4 and 5 show that the Riemann hypothesis is closely related to the self-approximation of $\zeta \left(s\right)$ by shifts $\zeta (s+i\tau )$.

3. Conclusions

The Riemann hypothesis (RH) states that all non-trivial zeros of the Riemann zeta-function $\zeta \left(s\right)$, $s=\sigma +it$, lie on the line $\sigma =1/2$. In this paper, we obtained a new equivalent of RH, namely, RH is equivalent to the statement that, for every compact set of the strip $\{s\in \mathbb{C}:1/2<\sigma <1\}$ with a connected complement, the limit

$$\underset{T\to \infty }{lim}{\displaystyle \frac{1}{T}}\mathcal{L}\left\{t\in [0,T]:\underset{s\in K}{sup}|\zeta (s+i\tau )-\zeta \left(s\right)|<\epsilon \right\}$$

exists and is positive for all but at most countably many $\epsilon >0$.

A similar equivalent is valid in terms of self-approximation in short intervals.

In the future, we plan to obtain RH equivalents by using generalized shifts. Also, we expect that a similar equivalence of RH can be described by using discrete shifts. Maybe, in this case, computer calculations can be useful.

It is well known that all large computer calculations support RH. For example, it was obtained in [19] that the ${10}^{13}$ first zeros of $\zeta \left(s\right)$ lie on the line $\sigma =1/2$. In [20], it was found that the 10²²nd zero is also on the critical line. However, this way cannot prove RH. By calculations, RH can only be disproven if some zeroes in the strip $\{s:1/2<\sigma <1\}$ will be detected.