# Joint universality for dependent *L*-functions

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## Abstract

We prove that, for arbitrary Dirichlet *L*-functions \(L(s;\chi _1),\ldots ,L(s;\chi _n)\) (including the case when \(\chi _j\) is equivalent to \(\chi _k\) for \(j\ne k\)), suitable shifts of type \(L(s+i\alpha _jt^{a_j}\log ^{b_j}t;\chi _j)\) can simultaneously approximate any given set of analytic functions on a simply connected compact subset of the right open half of the critical strip, provided the pairs \((a_j,b_j)\) are distinct and satisfy certain conditions. Moreover, we consider a discrete analogue of this problem where *t* runs over the set of positive integers.

## Keywords

Joint universality Uniform distribution*L*-functions

## Mathematics Subject Classification

11M41## 1 Introduction

*f*(

*s*) on

*K*, analytic in the interior of

*K*, and every \(\varepsilon >0\), we have

*L*-functions associated with non-equivalent characters can simultaneously and uniformly approximate non-vanishing analytic functions in the above sense. In other words, in order to approximate a collection of non-vanishing continuous functions on some compact subset of \(\mathrm{\{s\in \mathbb {C}{:}\;1/2<\mathrm{Re}(s)<1\}}\) with connected complement, which are analytic in the interior, it is sufficient to take twists of the Riemann zeta function with non-equivalent Dirichlet characters. The requirement that characters are pairwise non-equivalent is necessary, since it is well known that Dirichlet

*L*-functions associated with equivalent characters differ from each other by a finite product and, in consequence, one cannot expect joint universality for them. This idea was extended by Šleževičienė [17] to certain

*L*-functions associated with multiplicative functions, by Laurinčikas and Matsumoto [9] to

*L*-functions associated with newforms twisted by non-equivalent characters, and by Steuding in [18, Sect. 12.3] to a wide class of

*L*-functions with Euler product, which can be compared to the well-known Selberg class. Thus, one possible way to approximate a collection of analytic functions by a given

*L*-function is to consider its twists with sufficiently many non-equivalent characters.

Another possibility to obtain a joint universality theorem by considering only one *L*-function was observed by Kaczorowski et al. [5]. They introduced the Shifts Universality Principle, which says that for every universal *L*-function *L*(*s*), in the Voronin sense, and any distinct real numbers \( \lambda _1,\ldots ,\lambda _n\), the functions \(L(s+i\lambda _1),\ldots ,L(s+i\lambda _n)\) are jointly universal for any compact set \(K\subset \mathrm{\{s\in \mathbb {C}{:}\;1/2<\mathrm{Re}(s)<1\}}\) satisfying \(K_k\cap K_j=\emptyset \) for \(1\le k\ne j\le n\), where \(K_j=\{s+\lambda _j:s\in K\}\).

*L*-function in general, to approximate arbitrary given collection of analytic functions. For example, we might consider an

*L*-function, a compact set \(K\subset \mathrm{\{s\in \mathbb {C}{:}\;1/2<\mathrm{Re}(s)<1\}}\) with connected complement, and non-vanishing continuous functions \(f_1,\ldots ,f_n\) on

*K*, analytic in the interior of

*K*, and ask for functions \(\gamma _1,\ldots ,\gamma _n:\mathbb {R}\rightarrow \mathbb {R}\) satisfying

*K*) answer for the simplest case when \(\gamma _j(\tau ) = \tau + \lambda _j\). The consideration for other linear functions \(\gamma _j(\tau )=a_j\tau +b_j\) might be restricted, without loss of generality, to the case when \(\gamma _j(\tau )=a_j\tau \), which was firstly investigated by Nakamura [11, 12]. He proved that (2) holds, provided \(\gamma _j(\tau )=a_j\tau \) with algebraic real numbers \(a_1,\ldots ,a_n\) linearly independent over \(\mathbb {Q}\). Although Nakamura’s result is the best known result concerning all positive integers

*n*, the case \(n=2\) is already much better understood, and from the work of the author and Nakamura (see [11, 13, 14, 15]), we know that (2) holds if \(\gamma _1(\tau )=a_1\tau \), \(\gamma _2(\tau )=a_2\tau \) with non-zero real \(a_1,a_2\) satisfying \(a_1\ne \pm a_2\).

The main purpose of the paper is to find other example of functions \(\gamma _1,\ldots ,\gamma _n\) such that (2) holds. Our approach is rather general and based on Lemmas 1 and 3, which are stated in the general form. However, we focus our attention only on the case when \(\gamma _j(t)=\alpha _j t^{a_j}(\log t)^{b_j}\). The consideration when \(a_j=a_k\) and \(b_j=b_k\) for some \(j\ne k\) is very similar to the already quoted work of the author and Nakamura for linear functions \(\gamma (t)\) and essentially relies on investigating a kind of independence of \(\alpha _j\) and \(\alpha _k\), so in the sequel we assume that \(a_j\ne a_k\) or \(b_j\ne b_k\) for \(j\ne k\). Moreover, for the sake of simplicity we will restrict ourselves only to Dirichlet *L*-functions, but it should be noted that our approach can be easily generalized to other *L*-functions (as in [18]), at least in the strip where the mean square of a given *L*-function is bounded on vertical lines, namely \(\int _{-T}^{T}|L(\sigma +it)|^2dt\ll T\). On the other hand, we consider any collection of Dirichlet *L*-functions as an input instead of a single *L*-function. Hence, the following theorem gives an easy way to approximate any collection of analytic functions by taking some shifts of any *L*-functions, even equal or dependent.

## Theorem 1

*K*, analytic in the interior of

*K*. Then, for every \(\varepsilon >0\),

*k*is more subtle, since the set \(\{\frac{\alpha _1\log p}{2\pi }:p\in \mathbb {P}\}\cup \{1\}\) is not linearly independent over \(\mathbb {Q}\), which plays a crucial role in the proof. The case \(n\ge 2\) for Dirichlet

*L*-functions associated with non-equivalent characters and \(\gamma _j(k)=\alpha _j k\) was investigated by Dubickas and Laurinčikas in [2], where they proved discrete joint universality under the assumption that

*L*-functions associated with non-equivalent characters are discretely jointly universal, provided that

## Theorem 2

*K*, analytic in the interior of

*K*. Then, for every \(\varepsilon >0\),

*uniformly distributed mod 1*in \(\mathbb {R}^n\) if for every \(\alpha _j,\beta _j\), \(j=1,2,\ldots ,n\), with \(0\le \alpha _j<\beta _j\le 1\), we have

*continuously uniformly distributed mod 1*in \(\mathbb {R}^n\) if for every \(\alpha _j,\beta _j\), \(j=1,2,\ldots ,n\), with \(0\le \alpha _j<\beta _j\le 1\), we have

One can easily notice that Weyl’s criterion (see [7, Theorems 6.2 and 9.2]) shows that (4) and (5) imply that (7) is (continuous) uniformly distributed mod 1. Thus, our approach allows to improve the result of Dubickas and Laurinčikas, and the result due to Laurinčikas, Macaitienė and Šiaučiūnas in the following two aspects. First, we see that the assumption that Dirichlet characters are pairwise non-equivalent is superfluous. Secondly, it shows that one can consider more general functions than \(\gamma _j(t)=\alpha _jt^a\), \(a\in (0,1]\).

## 2 Approximation by finite product

Essentially, we shall follow the original proof of Voronin’s result, which, roughly speaking, might be divided into two parts. The first one relies mainly on uniform distribution mod 1 of the sequence of numbers \(\gamma _j(t)\tfrac{\log p}{2\pi }\) (or a kind of independence of \(p^{i\gamma _j(t)}\)) and deals with the approximation of any analytic function by shifts of a truncated Euler product. The second one deals with an application of the second moment of *L*-functions to approximate a truncated Euler product by a corresponding *L*-function in the mean-square sense.

*M*, and real numbers \(\theta _p\) indexed by primes, we put

## Lemma 1

*K*, which are analytic in the interior of

*K*. Then, for every \(\varepsilon >0\), there is \(v>0\) such that for every \(y>v\) we have

*y*.

Before we give a proof of the above result, let us recall the following crucial result on approximation of any analytic function by a truncated Euler product twisted by a suitable sequence of complex numbers from the unit circle.

We call an open and bounded subset *G* of \(\mathbb {C}\) *admissible*, when for every \(\varepsilon > 0\) the set \(G_\varepsilon = \{s\in \mathbb {C} : |s-w| < \varepsilon \text { for certain }w\in G \}\) has connected complement.

## Lemma 2

*G*such that \(\overline{G}~\subset \{s\in \mathbb {C}\;:\;\frac{1}{2}<\mathrm{Re}(s)<1\}\), every analytic and non-vanishing function

*f*on the closure \(\overline{G}\), and every finite set of primes \(\mathcal {P}\), there exist \(\theta _p\in \mathbb {R}\) indexed by primes and a sequence of finite sets \(M_1\subset M_2\subset \ldots \) of primes such that \(\bigcup _{k=1}^\infty M_k=\mathbb {P}\setminus \mathcal {P}\) and, as \(k\rightarrow \infty \),

## Proof

This is Lemma 7 in [4].

## Proof of Lemma 1

*G*such that \(K\subset G\subset \overline{G}\subset \{s\in \mathbb {C}:1/2<\sigma <1\}\) and each \(f_j\) is analytic non-vanishing on \(\overline{G}\). Therefore, by Lemma 2 with \(\mathcal {P}=\emptyset \), there exist real numbers \(\theta _{jp}\) for \(p\in \mathbb {P}\), \(1\le j\le n\) such that, for any \(z>0\) and \(\varepsilon >0\), there are finite sets of primes \(M_1,\ldots ,M_n\) such that \(\{p:p\le z\}\subset M_j\) for every \(j=1,2,\dots ,r\) and

*A*of real \(\tau \ge 2\) satisfying

*z*, we have

*G*and \(\mathrm{dist}(A,B) = \inf \{|a-b|:a\in A,\ b\in B\}\).

*f*and

*s*lying in the interior of

*G*(see [3, Chap. III, Lemma 1.1]), we observe that the measure of the set of \(\tau \in A_T\) satisfying

## 3 Application of the second moment

As we described in Sect. 2, in order to complete the proof of universality, we need to show how to approximate shifts of a truncated Euler product by shifts of a corresponding *L*-function. In general, a given *L*-function is not well approximated by a corresponding truncated Euler product in the critical strip with respect to the supremum norm. Nevertheless, it is well known that the situation is much easier if we consider the \(L^2\)-norm, which we use to prove the following result.

## Lemma 3

*b*are real numbers, and \(\gamma (t)=\alpha t^a(\log t)^b\). Then, for every \(\varepsilon >0\) and sufficiently large integer

*y*, we have

## Proof

*T*and

*y*we have

*X*we have

*T*, so, by Carlson’s theorem (see, for example, Theorem A.2.10 in [6]), we obtain

*n*are less than

*y*, and \(c_n=1\) otherwise. Hence, the second factor on the right-hand side of (12) is

*X*and

*y*, which gives (11), and the proof is complete. \(\square \)

Now we are in the position to prove Theorem 1.

## Proof of Theorem 1

Without loss of generality, we can assume that for every *j* there is at least one \(p\in M_j\) such that \(h_{jp}\ne 0\). Therefore, \(c_j:= \sum _{p\in M_j}h_{jp}\frac{\log p}{2\pi }\ne 0\) for every \(1\le j\le n\), and again, by Weyl’s criterion, it suffices to show that \(g(t) = \sum _{j=1}^n c_j\gamma _j(t)\) is continuously uniformly distributed mod 1 in \(\mathbb {R}\). In order to prove it, we shall use [7, Theorem 9.6] and show that for almost all \(t\in [0,1]\) the sequence \((g(nt))_{n\in \mathbb {N}}\) is uniformly distributed mod 1 in \(\mathbb {R}\) for any real \(c_j\ne 0\).

Let \(a=\max _{1\le j\le n} a_j\), \(b=\max \{b_j: 1\le j\le n,\ a_j=a\}\) and \(j_0\) be an index satisfying \((a_{j_0},b_{j_0}) = (a,b)\). First, let us assume that \(a_{j_0}\not \in \mathbb {Z}\), \(b_{j_0}\in \mathbb {R}\) or \(a_{j_0}\in \mathbb {Z}\), \(b_{j_0}<0\). Then it is clear that for every \(t\in (0,1)\) the function \(g_t(x) = \sum _{j=1}^n c_j\gamma _j(x)\) is \(\lceil a\rceil \) times differentiable and \(g_t^{(\lceil a\rceil )}(x)\asymp x^{a-\lceil a\rceil }\log ^b x\). Hence, \(g_t^{(\lceil a\rceil )}(x)\) tends monotonically to 0 as \(x\rightarrow \infty \) and \(x\bigg |g_t^{(\lceil a\rceil )}(x)\bigg |\rightarrow \infty \) as \(x\rightarrow \infty \), so, by [7, Theorem 3.5], the sequence \((g_t(n))=(g(nt))\), \(n=1,2,\ldots \), is uniformly distributed mod 1.

The case \(a_{j_0}\in \mathbb {N}\) and \(b_{j_0}>1\) is very similar, since \(g_t^{(\lceil a\rceil +1)}(x) \asymp \frac{\log ^{b-1} x}{x}\).

Finally, if \(a_{j_0}\in \mathbb {N}\) and \(b_{j_0}=0\), we see that \(\lim _{x\rightarrow \infty } g_t^{(a)}(x)\rightarrow t^a a! c_{j_0}\alpha _{j_0}\), which is irrational for almost all \(t\in [0,1]\). Therefore, [7, Chap. 1, Sect. 3] (in particular, see [7, Exercise 3.7, p. 31]) shows that the sequence \((g_t(n))=(g(nt))\), \(n=1,2,\ldots \), is uniformly distributed mod 1 for almost all \(t\in [0,1]\), and the proof is complete. \(\square \)

## 4 Discrete version

In this section, we deal with a discrete version of Theorem 1. Let us start with the following discrete analogue of Lemma 1.

## Lemma 4

*K*, which are analytic in the interior of

*K*. Then, for every \(\varepsilon >0\) and every finite set \(\mathcal {A}_j\) with \( \mathcal {P}_j\subset \mathcal {A}_j\subset \mathbb {P}\), there is \(v>0\) such that for every \(y>v\) we have

*y*.

## Proof

The proof closely follows the proof of Lemma 1, and therefore we will be rather sketchy.

*G*, real numbers \(\theta _{jp}\) for \(p\in \mathbb {P}\setminus \mathcal {A}_j\), \(1\le j\le n\), and finite sets of primes \(M_j\), \(1\le j\le n\), containing \(\{p\in \mathbb {P}\setminus \mathcal {A}_j:p\le z\}\) and satisfying

*A*of positive integers

*k*satisfying

*z*, we obtain from [7, Theorem 6.1]) that

The next proposition is a discrete version of Lemma 3 and its proof relies on Carlson’s theorem and the following Gallagher’s lemma.

## Lemma 5

**(Gallagher)**Let \(T_0\) and \(T\ge \delta >0\) be real numbers and

*A*be a finite subset of \([T_0+\delta /2,T+T_0-\delta /2]\). Define \(N_\delta (x) = \sum _{t\in A,\ |t-x|<\delta } 1\) and assume that

*f*(

*x*) is a complex continuous function on \([T_0,T+T_0]\) continuously differentiable on \((T_0,T+T_0)\). Then

## Proof

This is Lemma 1.4 in [10]. \(\square \)

## Proposition 1

*b*are real numbers, and \(\gamma (t)=\alpha t^a(\log t)^b\). Then, for every \(\varepsilon >0\) and sufficiently large integer

*y*, we have

## Proof

*N*and

*y*, and the proof is complete. \(\square \)

## Proof of Theorem 2

First, we shall use Lemma 4, so let us define the sets \(\mathcal {A}_j\) and \(\mathcal {P}_j\) for \(j=1,2,\ldots ,n\). If \(\gamma _j(t)=\alpha _jt^{a_j}\log ^{b_j}t\) with \(a_j\notin \mathbb {Z}\) or \(b_j\ne 0\), then the proof is essentially the same as in the continuous case, so we just take \(\mathcal {P}_j=\mathcal {A}_j=\emptyset \).

*m*,

*l*such that \(\exp (2\pi m/\alpha _j)=\prod _{p\in M_j}p^{lh_{jp}}\in \mathbb {Q}\), which, by the definition of \(m^*_j\), is a power of \(\prod _{p\in \mathcal {A}_j}p^{k_{jp}}\), and we get a contradiction.

*j*satisfying \(a_j\in \mathbb {Z}\) and \(b_j=0\). If \(a_j\not \in \mathbb {Z}\) or \(b_j\ne 0\) for all \(j=1,2,\ldots ,n\), then \(q^*=1\).

*cN*. The second inequality together with (14) gives that

*j*satisfying \(a_j\in \mathbb {Z}\), \(b_j=0\),

*K*, and hence

*k*, one can easily observe that the number of integers \(k\in [2,N]\) satisfying

## Notes

### Acknowledgements

The author would like to cordially thank Professor Kohji Matsumoto for his valuable comments and suggestions.

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