ON GOOD UNIVERSALITY AND THE RIEMANN HYPOTHESIS

HAL Id: hal-02860568

https://hal.archives-ouvertes.fr/hal-02860568v2

Submitted on 5 Apr 2021

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ON GOOD UNIVERSALITY AND THE RIEMANN HYPOTHESIS

Jean-Louis Verger-Gaugry, Radhakrishnan Nair, Michel Weber

To cite this version:

Jean-Louis Verger-Gaugry, Radhakrishnan Nair, Michel Weber. ON GOOD UNIVERSAL- ITY AND THE RIEMANN HYPOTHESIS. Advances in Mathematics, Elsevier, In press,

�10.1016/j.aim.2021.107762�. �hal-02860568v2�

ON GOOD UNIVERSALITY AND THE RIEMANN HYPOTHESIS

RADHAKRISHNAN NAIR^‡, JEAN-LOUIS VERGER-GAUGRY^✸, AND MICHEL WEBER^†

ABSTRACT. We use subsequence and moving average ergodic theorems applied to Boole’s transformation and its variants and their invariant measures on the real line to give new characterisations of the Lindel¨of Hypothesis and the Riemann Hypothesis. These ideas are then used to study the value distribution of DirichletL-functions, and the zeta functions of Dedekind, Hurwitz and Riemann and their derivatives. This builds on earlier work of R. L.

Adler and B. Weiss, M. Lifshits and M. Weber, J. Steuding, J. Lee and A. I. Suriajaya using Birkhoff’s ergodic theorem and probability theory.

Keywords: Ergodic Averages, Boole’s Transformation, Dedekind Zeta Function, Dirichlet L-function, L-Series, Hurwitz Zeta Function, Riemann Zeta Function, The Lindel¨of Hy- pothesis, The Riemann Hypothesis.

2020 Mathematics Subject Classification: 11M06, 28D05, 37A44, 11M26, 11M35, 30D35.

CONTENTS

1. Introduction 2

2. Applications of Theorem 1.1 6

2.1. The Riemann zeta function 6

2.2. DirichletL-functions 10

2.3. The Dedekind zeta function of a number field 11

2.4. The Hurwitz zeta function 12

3. Proof of Theorem 1.1 12

4. Proof of Theorems 1.2 to 1.5 14

5. Comparing dynamical and probabilistic models 21

6. Hartman uniformly distributed and good universal sequences 24

7. Moving averages 26

7.1. Stoltz sequences and Theorem 7.2 26

7.2. Applications and moving average ergodic Theorems 27

8. Sublinearity and the Riemann zeta function 28

Acknowledgements 29

References 29

1

1. INTRODUCTION

Equivalent statements of the Riemann Hypothesis are well-known and can be found e.g. in Titchmarsh [T] or more recently e.g. in Mazur and Stein [MSt]. The Lindel¨of Hypothesis asserts that for everyε >0 we have

(1.1) |ζ(1/2+it)|=O(|t|^ε) as|t| →∞,

whereζ(z)is the Riemann zeta funtion, and Landau’s notation “O” means that there exists a neighbourhoodV of∞and a constantc≥0 such that:|ζ(1/2+it)| ≤c|t|^ε for allt∈V. Before stating the main Theorems 1.1–1.5, and Theorem 7.2 in the context of moving average ergodic theorems, let us make precise the context in probability theory.

Let(Xi)_i≥1be a sequence of independent Cauchy random variables, with characteristic functionφ(t) =e^|t|and consider the partial sumsS_n=X₁+. . .+X_n(n=1,2, . . .).

M. Lifshits and M. Weber [LW3] studied the value distribution of the Riemann zeta functionζ(s)sampled along the random walk(Sn)n≥1showing, forb>2, that

(1.2) lim

N→∞

1 N

N n=1

∑

ζ 1

2+iS_n

=1+o

(logN)^b N¹²

.

They also showed that (1.3)

sup

n≥1

|∑ⁿ_q=1ζ(¹₂+iSq)−n|

n¹²(logn)^b

2

<∞.

Here of course f(x) = o(g(x)) means lim_x→∞g(x)^f(x) =0. This result was extended to L- functions and Hurwitz zeta functions by T. Srichan [Sr]. In [St2] J. Steuding replaced (Sn)n≥1in [LW3] by(Tⁿx)n≥1for almost allxonRwith respect the Lebesgue measure, for the Boolean dynamical system byT x:=x−¹_x. This result has its roots in the observation, due to G. Boole [Bo], subsequently developed by J.W.L. Glashier [G1] [G2], that if f is integrable on the real numbers, then

Z _∞

−∞f(x)dx= Z _∞

−∞f x−1

x dx.

See [AdW] for a proof of the ergodicity of this dynamical system. In Theorem 3.1 in [LS], the maps

(1.4) T_α,β(x) =

( α 2

_x+β

α −_x−β^α

ifx 6=β, β ifx =β,

forα >0 and realβ, are shown to be measure preserving and ergodic with respect to the probability measure

(1.5) µ_α,β(A) = α

π Z

A

dt α²+ (t−β)², for any Lebesgue measureable subsetAof the real numbers.

As noted in [LS], ifλ denotes Lebesgue measure thenµ_α,β(A)≤_απ¹ λ(A)for allA∈B, where B denotes the Lebesgueσ-algebra. Also ifφ_α,β(x) =αx+β, withT =T_1,0 and

µ =µ1,0we have

(1.6) T_α,β =φ_α,β◦T◦φ_α⁻¹_,β,

which implies the µ_α,β-measure preservation and ergodicity of T_α,β. The referee of this paper has brought to the authors’ attention the paper [MSr], which considers a family of rational maps more general than that in [LS], which we consider in this paper. The suggestion is that our methods in this paper will adapt to the more general framework in [MSr] – at least to a degree. We will not however explore this possibility in this paper, as it would be too much of a digression.

We consider a number of general definitions.

We say(kn)n≥1∈NisL^p-good universalif for each dynamical system(X,β,µ,T)and for eachg∈L^p(X,β,µ)the limit

(1.7) ℓT,g(x) = lim

N→∞

1 N

N−1 n=0

∑

g(T^knx) existsµ almost everywhere.

Also throughout the rest of this paperBwill be the Lebesgueσ- algebra onRand when aσ-algebra onRis not explicitly mentioned it is in factB.

We say that a sequencex1, . . .,xN, . . .isuniformly distributed modulo oneif

(1.8) lim

N→∞

1

N#{1≤n≤N:{xn} ∈I}=|I|

for every intervalI ⊆[0,1). For a real numberywe have used{y}to denote its fractional part and let[y] =y− {y}denote its integer part. We say a sequence of integers isuniformly distributed onZif it is uniformly distributed among the residue classes modulom, for each natural numberm>1. We say a sequence of natural numbers(kn)n≥1isHartman uniformly distributed (onZ)if it is uniformly distributed onZ, and for each irrational numberα, the sequence({knα})n≥1is uniformly distributed modulo one . This condition coincides with (kn)n≥1 being uniformly distributed on the Bohr compactification of Z. Some basics on Hartman uniform distribution can be found in the book of Kuipers and Niederreiter [KN].

Note that if(kn)_n≥1is Hartman uniformly distributed onZ, and if forzwith|z|=1 we set

(1.9) F(N,z):= 1

N

N−1 n=0

∑

z^kn (N=1,2,· · ·) then

(1.10) lim

N→∞F(N,z) =

1 ifz=1, 0 otherwise.

For ac∈Rwe useH_cto denote the half plane{z∈C:ℜ(z)>c}and useL_cto denote the line{z∈C:ℜ(z) =c}. We establish the following theorem.

Theorem 1.1. Let f be a meromorphic function onH_csatisfying the following conditions:

(1) there exists a K >0and a c^′>c such that for any t ∈R, we have (1.11) |f({σ+it|σ >c^′})| ≤K;

(2) there exists a non-increasingν :(c,∞)→Rsuch that ifσ is sufficiently near c then ν(σ)≤1+c−σ and that for anyε >0, we have f(σ+it)≪f,ε |t|^ν(σ)+ε as|t| →∞;

(3) f has at most one pole of order m in H_c at s₀=σ0+it₀, that is, we can write its Laurent expansion near s=s₀as

a_−m

(s−s₀)^m+ a_−(m−1)

(s−s₀)^(m−1) +. . . a₋₁

(s−s₀)⁻¹+a0+

∞ n=1

∑

an(s−s0)ⁿ for m≥0where we set m=0if f has no pole inH_c.

Then if (kn)n≥1 is L^p-good universal and Hartman uniformly distributed, for any s ∈ H_c\L_σ0, we have

(1.12) lim

N→∞

1 N

N−1 n=0

∑

f(s+iT_α,β^kn (x)) = α π

Z

R

f(s+iτ) α²+ (τ−β)²dτ, for almost all x inR.

In the casekn=n(n=1,2, . . .)Theorem 1.1 appears in [LS], where it is shown, using (1), (2) and (3) and contour integration, for

l_α,β(s):= α π

Z

R

f(s+iτ) α²+ (τ−β)²dτ, that if f has a pole ats₀=σ0+it₀,

(1.13) l_α,β(s) =









f(s+α+iβ) +Bf_m(so), ifc<ℜ(s)<σo, s6=s_o−α−iβ,

m n=0

∑

a_−n

(−2α)ⁿ, ifc<ℜ(s)<σ_o, s=s_o−α−iβ, f(s+α+iβ), ifℜ(s)>σo,

orl_α,β(s) = f(s−α+iβ)ifs∈H_c and f has no pole. Hereℜ(s)denotes the real part of sand

Bfm(s0) =

m n=1

∑

n a_−n

iⁿ(β+iα−i(s−s₀))ⁿ− a_−n

iⁿ(β−iα−i(s−s₀))ⁿ o.

Moreover, whenm=1, we can extend Theorem 1.1 to the lineL_σ0 by setting l_α,β(σ0+it) = f(σ0+α+i(t+β))− a₋₁α

α²+ (t0−t−β)². Applications will be given in the next section.

As noted earlier Steuding [St2] showed that ifζ denotes the Riemann zeta function

(1.14) lim

N→∞

1 N

N−1

∑

n=0

ζ(s+iTⁿx) = 1 π

Z

R

ζ(s+iτ) 1+τ^{2 d}τ

existsµ almost everywhere (forx). We can measure the stability of these averages another way. HenceforthC, possibly with subscripts, will denote a positive constant, though non- necessarily the same on each occasion.

Theorem 1.2. Suppose s=σ+it with ¹₂<σ <1. Let Y_N(σ) =Y_N(σ,x):= 1

N

N−1 n=0

∑

ζ(s+iTⁿx). (N=1,2, . . .) Suppose(Nk)_k≥1is a strictly increasing sequence of positive integers. Then there exists an absolute C>0such that

(1.15)

∞ k=1

∑

k sup

Nk≤N<N_k+1

|YN(s)−Y_Nk(s)| k²₂≤ C πσ

ζ(2σ)

2 +ζ(2σ−1) 2σ−1

.

There are other ergodic theoretic measures of the stability of the Riemann zeta function.

In this direction we consider the frequencies associated to the sequences ζ(s+iTⁿx)

n

n≥1. Evidently

n→∞lim

ζ(s+iTⁿx)

n =0,

for almost all x. It is possible to make this more precise as follows. Suppose 0<p<∞.

For a sequence of real numbersx={xn:n≥1}set

(1.16) kxkp,∞:= sup

t>0

{t^p#{n:|xn|>t}}¹_p .

Theorem 1.3. (i) Suppose ¹₂ <σ <1. Then

(1.17) lim

m→∞

#{n: ^|ζ(σ+iT_n ^nx)| ≥_m¹}

m =ζ(σ+1)− 2

σ(2−σ), for almost all x∈R. and in L¹(µ)norm.

(ii) Further (1.18)

|ζ(σ+iTⁿx)|

n :n≥1_1,∞

1

<∞.

Moreover,

Theorem 1.4. Suppose s=σ+it with ¹₂ <σ <1. Suppose κ = (kn)n≥1 is L^p- good universal for p>2and let

(1.19) R_N(s,x,κ):= 1 N

N−1 n=0

∑

ζ(s+iT^knx). (N=1,2, . . .).

Then

(i) there exists a constant C>0such that (1.20)

∞ N=1

∑

R_N+1(s, .,κ)−R_N(s, .,κ)²₂≤ C πσ

ζ(2σ)

2 +ζ(2σ−1) 2σ−1

, (ii) there exists C>0

(1.21) µ_α,βⁿx:

∞ N=1

∑

|RN+1(s,x,κ)−R_N(s,x,κ)|²≥λ^o≤ C λ²

ζ(2σ)

2 +ζ(2σ−1) 2σ−1

.

We will provide additional information about various special families of sequences in Section 4 and Section 8. On the other hand if we consider theL¹-norm we have the follow- ing theorem in the opposite direction.

Theorem 1.5. Suppose κ = (kn)_n≥1 is Hartman uniformly distributed and L^p- good uni- versal for fixed p≥ 1andσ∈(¹₂,1). Then

(1.22)

∑

N≥1

R_N+1(σ,x,κ)−R_N(σ,x,κ) = +∞,

almost everywhere in x with respect to Lebesgue measure.

It would be interesting to get good bounds almost everywhere inxfor

J N=1

∑

R_N+1(σ,x,κ)−RN(σ,x,κ), (J=1,2. . .)

even for specific cases ofκ.

The paper is organized as follows: in Section 2 new characterisations of the (extended) Lindel¨of Hypothesis are obtained for the Riemann zeta function, and for Dirichlet L- functions, Dedekind zeta functions of number fields, Hurwitz zeta functions, as applica- tions of Theorem 1.1. The sequences here are assumed to be Hartman uniformly distributed and L^p-good universal. Replacing this assumption by being Stoltz leads to similar limit theorems, which are reported in Section 7. The role played by sublinear sequences hav- ing controlled growth is investigated in Section 8 for the Riemann zeta function. Section 6 contains some examples of L^p-good universal sequences and Hartman uniformly dis- tributed sequences. Comparison between the dynamical and probabilistic models of Weber [W], resp. Srichan [Sr], is done in Section 5.

2. APPLICATIONS OFTHEOREM 1.1

From now, we assume that(kn)n≥1satisfies the hypothesis in Theorem 1.1.

In the sequel, for a function f denote f⁽⁰⁾= f and f^(k)thek-th derivative of f and set P_k(s):= (−1)^kk!

i^k+1

1

(β+iα−i(s−1))^k+1− 1

(β−iα−i(s−1))^k+1

for any non-negative integerk.

2.1. The Riemann zeta function.

Theorem 2.1. Suppose (kn)_n≥1 is L^p-good universal for some p ∈[1,∞] and Hartman uniformly distributed. Then for any k≥0and s∈H₋1

2\L1we have

(2.1) lim

N→∞

1 N

N−1 n=0

∑

ζ^(k)(s+iT_α^kn_,β(x)) = α π

Z

R

ζ^(k)(s+iτ) α²+ (τ−β)²dτ for almost all x inR.

Denote the right hand side of this limit byl_α,β^(k) . Then

(2.2) l_α,β^(k) =







ζ^(k)(s+α+iβ) +P_k(s), if −¹₂<ℜ(s)<1, s6=1−α−iβ, (−1)^kγk−_(2α)^k!_k+1, if −¹₂<ℜ(s)<1, s=1−α−iβ,

ζ^(k)(s+α+iβ), ifℜ(s)>1, where

(2.3) γk= lim

N→∞

N n=1

∑

log^kn

n −log^k+1N k+1

! .

In the casek=0 we can extend the result to the lineL₁by setting (2.4) l_α⁽⁰⁾_,β =l_α,β⁽⁰⁾(1+it) =ζ⁽⁰⁾(1+α+i(t+β))− α

α²+ (t²+β²).

If we set k_n=n(n=1,2, . . .)Theorem 2.1 appears in [LS] and this is in the caseα = 1,β =0 andk=1 in [St2].

We also mention here that the casek_n=n(n=1,2, . . .)andT =T_1,0of Theorems 2.3 and 2.4 to follow, appeared in [St2], of Theorem 2.4 appeared first in [BSY] and of Theorems 2.6 and 2.7 appeared in [LS].

Proof. Note that fork≥0, we know that ζ^(k) is meromorphic and has an absolutely con- vergent Dirichlet series forℜ(s)>1. Thus the condition (1) of Theorem 1.1 is satisfied for c^′>1. This is because the Laurent expansion ofζ near the pole ats=1 is known [Br] and then that the Laurent expansion ofζ^(k)has the form

ζ^(k)(s) = (−1)^kk!

(s−1)^k+1+ (−1)^kγk+

∞ n=k

∑

(−1)ⁿ⁺¹γn

(n−k+1)!(s−1)^n−k+1.

Thus ifk≥0 the functionζ^(k) has a pole of orderk+1 ats=1. We can in addition show (Titchmarsh [T], pp. 95–96) that givenε>0 we have

ζ^(k)(σ+it)≪ |t|^µ(σ)+ε, where

µ(σ) =





0, ifσ >1,

(1−σ)

2 , if 0≤σ≤1,

1

2−σ, ifσ <0,

if k ≥0. Therefore Theorem 2.1 follows from Theorem 1.1 with c =−¹₂, s₀ = 1 and

m=k+1 toζ^(k)(s)as required.

We now state a formulation of the Lindel¨of Hypothesis in terms ofζ^(k)from [LS].

Lemma 2.2. The Lindel¨of Hypothesis(1.1)is equivalent, for everyε >0, to (2.5)

ζ^(k)(1

2+it) ≪ |t|^ε as|t|tends to∞, for any k≥0.

We now give a new characterization of the Lindel¨of Hypothesis for the Riemann zeta function, which generalizes the classical one. The classical case kn =n(n=0,1,2, . . .) appears in [LS].

Theorem 2.3. Suppose (kn)n≥1 is L^p-good universal for some p ∈[1,∞] and Hartman uniformly distributed. Suppose k is any non-negative integer. Then the statement, for any natural number l,

(2.6) lim

N→∞

1 N

N−1 n=0

∑

ζ^(k)

1

2+iT_α,β^kn (x)^2l= α π

Z

R

|ζ^(k)(¹₂+iτ)|^2l α²+ (τ−β)² dτ forµ_α,β -almost all x inR, is equivalent to the Lindel¨of Hypothesis.

Proof. Via Lemma 2.2 the Lindel¨of Hypothesis implies that givenε>0 we have

ζ^(k) 1

2+.

2m

∈L^p(R,B,µ_α,β)

for each pair of natural numbers k,m. Here .represents the variablet ∈R. Theorem 1.1 and the ergodicity ofT_α,β implies that

N→∞lim 1 N

N−1 n=0

∑

ζ^(k)

1

2+iT_α^kn_,β(x)

2l

= α π

Z

R

|ζ^(k)(¹₂+iτ)|^2l α²+ (τ−β)²dτ. We now prove the converse. First

ζ^(k)(s) = (−1)^k−1 Z ∞

1

[x]−x+¹₂

x^s+1 (logx)^k−1(−slogx+k)dx+ (−1)^kk!

(s−1)^k+1. Thus there existsC_k>0, dependent only onk,α andβ, such that for|t| ≥1 we have

|ζ^(k)(1

2+it)|<C_k|t|.

Also evidently there existsc_α,β >0 such that if|τ| ≥1 1

α²+ (τ−β)² ≥c_α,β 1 1+τ^2.

Assuming that the Lindel¨of Hypothesis is false, implies there existsη>0 andτm→∞and C_α¹_,β such that

ζ^(m)(1

2+it)

>C_α¹_,βτ_m^η. Now,|ζ^(k)(¹₂+it)|<C_k|t|for any|t| ≥1 withC_k>0, and

ζ^(k)(1

2+iτ)−ζ(1

2+iτm) =

Z τ τm

|ζ^(k+1)(1

2+it)dt

<C_α²_,β|τ−τm|τ.

So |ζ(¹₂+iτ)| ≥ ¹₂C_α,β¹ τm^η for τ with|τ−τm| ≤τ_m⁻¹ withm large enough. LetL:= ²₃τm; then the intervalI:= (τm−τ_m⁻¹,τm+τ_m⁻¹)contains the interval(L,2L)for largem. Hence

Z 2L L

ζ^(k)

1

2+iτ^2l 1

2 dτ 1+τ²

≥ C₁

2 2lZ

Iτ_m^2lη−2dτ=2.

C₁ 2

2l

.τ_m^2lη−3,

which is≫T^2lη−3, and this is impossible asl→∞. So our theorem is proved.

We now specialize to the caseT =T_1,0and give a condition in terms of ergodic averages equivalent to the Riemann Hypothesis. We denote byρ=β+iγ, a representative complex zeros of the Riemann zeta function. See Titschmarsh [T], p. 30 for instance for details.

Theorem 2.4. Suppose(kn)n≥1 is both Hartman uniformly distributed and L^p-good uni- versal for some p>1. Then for almost all x in R with respect to Lebesgue measure we have

(2.7) lim

N→∞

1 N

N n=1

∑

log ζ

1 2+1

2iT^knx=

∑

ρ:ℜ(ρ)>¹₂

log ρ

1−ρ . Evidently, the Riemann Hypothesis follows if either side is zero.

To prove Theorem 2.4, we need the following lemma due to M. Balazard, E. Saias and M. Yor [BSY].

Lemma 2.5. We have

(2.8) 1

2π Z

ℜ(s)=¹₂

log|ζ(s)|

|s|² |ds|=

∑

ρ:ℜ(ρ)>¹₂

log

ρ 1−ρ

.

Proof. We wish to use Theorem 1.1 to deduce Theorem 2.4 using Lemma 2.5. To show Theorem 1.1 is relevant we need to show that

log ζ

1

2+i.∈L^p(R,µ1,0), (where.denotest∈R), i.e. that

Z

ℜ(s)=¹₂

|log|ζ(s))|^p|

|s|²

|ds|<∞.

We mentioned earlier that there existsC>0 such that if|t|>1,

ζ 1

2+it

≤C|t|.

Also notice that ^(log|ζ(s)|)_|s|2 ^p is continuous on an interval onℜ(s) =¹₂centred ons=¹₂. Away from that interval onℜ(s) = ¹₂ we use the observation thatζ(¹₂+it)≤C|t|. Hence (for s= ¹₂+it), givenδ >0 we have

(log|ζ(s)|)^p

|s|² ≪ 1

|s|^2−δ. This means

log ζ

1

2+i.∈L^p(R,µ1,0),

as required. Using the fact thatT preserves the measureµ1,0and is ergodic with respect to this measure, we have

N→∞lim 1 N

N n=1

∑

log ζ

1 2+1

2iT^knx= 1 2π

Z

ℜ(s)>¹₂

log|ζ(s)|

|s|² |ds|.

µ1,0 almost everywhere. Theorem 2.4 now follows from Lemma 2.5.

2.2. DirichletL-functions.

Theorem 2.6. Let L(s,χ) denote the L-functions associated to the characterχ. Suppose (kn)n≥1is Hartman uniformly distributed and L^p-good universal for some p>1. Then, for k≥1,

(i) if χis non-principal, for s∈H₋1

2 we have

N→∞lim 1 N

N−1

∑

n=0

L^(k)(s+iT_α,β^kn (x),χ) = α π

Z

R

L^(k)(s+iτ,χ) α²+ (τ−β)²dτ

(2.9) = L^(k)(s+α+iβ,χ) for almost all x inR, (ii) if χ(=χ0)is principal, for s∈H₋1

2\L1we have

(2.10) lim

N→∞

1 N

N−1 n=0

∑

L^(k)(s+iT_α^kn_,β(x),χ) =α π

Z

R

L^(k)(s+iτ,χ) α²+ (τ−β)²dτ, for almost all x inR.

Denote this limit, i.e. the right hand side of (2.10), byl_α^(k)_,β(s,χ0). Then (2.11)

l_α^(k)_,β(s,χ0) =







L^(k)(s+α+iβ,χ0) +γ−1(χ0)Pk(s), if −¹₂<ℜ(s)<1, s6=1−α−iβ, γ_k(χ0)−^k!γ_(2α)^−1(χ_k+1⁰⁾, if −¹₂<ℜ(s)<1, s=1−α−iβ, L^(k)(s+α+iβ,χ0), ifℜ(s)>1,

where γ₋₁(χ0),γ_k(γ0), are constants that depend on χ0. These are the coefficients of the Laurent expansion ofL^(k)(s,χ0)nears=1. Ifk=0 we can extend the result to the lineL₁ by defining

l_α,β⁽⁰⁾ =l_α,β⁽⁰⁾(1+it,χ0) =L(1+α+i(t+β),χ0)− αγ−1(χ0) α²+ (t²+β²).

Proof. We know that L^(k)(s,χ) has a Dirichlet series expansion for ℜ(s)>1, for each non-negative integerk. From this we can show that

L^(k)(s,χ)≪_k,ε|t|^µ(σ)+ε where

µ(σ) =





0, ifσ >1,

(1−σ)

2 , if 0≤σ≤1,

1

2−σ, ifσ <0.

Ifχ is non-principal thenL^(k)(s,χ)is entire for allk≥0, soL^(k)(s,χ)satisfies Theorem 1.1 for alls∈H₋1

2. Ifχ=χ0is principal,L^(k)(s,χ0) (k≥1)has a pole of orderk+1 ats=1.

We can therefore apply Theorem 1.1 withc=−¹₂,s₀=1 andm=k+1, toL^(k)(s,χ0)with

Laurent coefficients coming from [IK].

2.3. The Dedekind zeta function of a number field. We now consider the Dedekind ζK(s)function of a number field Kof degreedK over the rationalsQwhich is defined by as follows. Fors∈Csuch thatℜ(s)>1, let

ζ_K(s) =

∑

I⊂O_K

1 (N_K/Q(I))^s.

HereI runs over the ideals contained in the ring of integers ofK denotedO_K andN_K/Q(I) denotes the absolute norm ofI inK, which is the cardinality of the quotientO_K/I. In the caseK=Q, the functionζK(s)reduces to the Riemann zeta functionζ(s). The complex functionζK(s)can be extended meromorphically to the entire complex plane with a simple pole ats=1. See [Coh] p. 216, for more details.

Theorem 2.7. Suppose (kn)n≥1 is L^p-good universal for some p ∈[1,∞] and Hartman uniformly distributed. For any k≥0and any s∈H1

2−_d¹

K

\L1we have

(2.12) lim

N→∞

1 N

N−1

∑

n=0

ζ_K^(k)(s+iT_α^kn_,β(x)) = α π

Z

R

ζ_K^(k)(s+iτ) α²+ (τ−β)²dτ for almost all x inR.

Denote the right hand side of this limit byl_K^(k)

α,β. We have (2.13)

l_K^(k)

α,β(s) =







ζ_K^(k)(s+α+iβ) +γ−1(K)Pk(s), if ¹₂−_d¹

K <ℜ(s)<1, s6=1−α−iβ, k!γ_k(K)−^k!γ_(2α)⁻¹k+1^(K), if ¹₂−_d¹_K <ℜ(s)<1, s=1−α−iβ,

ζ^(k)(s+α+iβ), ifℜ(s)>1.

Here γ₋₁(K)and γ_k(K)are constants dependent only on K, which are coefficients of the Laurent expansion ofζ_K^(k)nears=1. Also in the casek=0 we can extend the result to the lineL₁by setting

(2.14) l_K⁽⁰⁾

α,β(1+it) =ζ⁽⁰⁾(1+α+i(t+β))− αγ−1(K) α²+ (t²+β²).

Proof. We refer to [St1] for a bound forLon the half line and to [HIKW] for the coeffi- cients of the Laurent expansionζK(s)near the poles=1. We then proceed as earlier with Theorem 2.1,c= ¹₂−_d¹

K,s₀=1 andm=k+1.

Remark 2.8. Results analogous to the ergodic characterisation of the Lindel¨of Hypothesis just given, can be proved forL-functions associated to primitive characters and Dedekind ζ function. In both cases, the statements of ergodic characterisation are of the type: there exist constantsk,ε>0 such that

f 1

2+it

≪f,k,ε |t|^ε,

as|t| →∞, for suitableL-functions f. This inequality when f is anL-function arising from Dirichlet series is called the Generalized Lindel¨of Hypothesis.

2.4. The Hurwitz zeta function. Recall that the Hurwitz zeta function is defined fora>0 ands∈Cwithℜ(s)>1 by

ζ(s,a) =

∞ n=1

∑

1 (n+a)^s.

It is continued meromorphically to the whole ofCwith a single pole ats=1.

Theorem 2.9. Suppose (kn)n≥1 is L^p-good universal for some p ∈[1,∞] and Hartman uniformly distributed. Then for any s such thatℜ(s)>−¹₂,s6=1, with0≤a<1and k a non-negative integer, we have

(2.15) lim

N→∞

1 N

N−1 n=0

∑

ζ^(k)(s+iT_α^kn_,β(x),a) =α π

Z

R

ζ^(k)(s+iτ,a) α²+ (τ−β)²dτ, for almost all x inR.

Denote the right hand side of this limit byl_α,β^(k) (s,a). Then

(2.16) l_α^(k)_,β(s,a) =







ζ^(k)(s+α+iβ,a) +P_k(s), if −¹₂<ℜ(s)<1, s6=1−α−iβ, (−1)^kγk(a)−_(2α^k!₎k+1, if −¹₂<ℜ(s)<1, s=1−α−iβ,

ζ^(k)(s+α+iβ,a), ifℜ(s)>1, where

(2.17) γk(a) = lim

N→∞

N n=1

∑

log^k(n+a)

n+a −log^k+1(N+a) k+1

! .

In the casek=0 we can extend the result to the lineL₁by setting (2.18) l_α⁽⁰⁾_,β(1+it,a) =ζ⁽⁰⁾(1+α+i(t+β),a)− α

α²+ (t²+β²).

If we setk_n=n(n=1,2, . . .)Theorem 2.1 appears in [LS] and this is the caseα=1,β=0, andk=1 in [St2].

Proof. Following the proof of Theorem 2.1 choosec=−¹₂,s₀=1 andm=k+1. For the bound on|ζ^(k)(s,a)|on the half line and the coefficients of the Laurent series ofζ^(k)(s,a)

nears=1 we refer to [St1].

3. PROOF OF THEOREM 1.1

The proof is in the continuation, and a generalization, of a Theorem of [LS] in which conditions (1), (2) and (3) are used. We first recall a special case of a Theorem of S. Sawyer [Sa]:

Suppose for a dynamical system(R,B,µ_α,β,T_α,β)thatg∈L^p(R,B,µ_α,β)and letkgk= (^R_X|f|^pdµ_α,β)^1p. Set

Mg(x) =sup

N≥1

1

N

N n=1

∑

g(s+iT_α,β^kn (x)). (N=1,2, . . .)

If(kn)n≥1isL^p-good universal for p>1, then there existsC>0 such that

(3.1) Mg

p≤Ckgkp. Because

N¹∑^N_n=1g(s+iT_α^kn_,β(x))≤Mg(x) (N=1,2, . . .)andMg∈L^p, the dominated con- vergence theorem implies

h(x) = lim

N→∞

1 N

N

∑

n=1

h(s+iT_α^kn_,β(x))

exists in L^p-norm. Our next order of business is to show that h(s+iT_α,β(x)) =h(s+ix).

LetU_α,βg(s+ix) =g(s+iT_α,β(x)). This is a norm preserving operator onL^p asT_α,β is µ_α,β measure preserving. Also letU_α⁻¹_,β denote theL² adjoint of U_α,β. Recall that we say any sequence (cn)_n∈Z is positive definite if given a bi-sequence of complex numbers (zn)_n∈Z, only finitely many of whose terms are non-zero, we have ∑n,m∈Zcn−mznzm≥0.

Herezis the conjugate of the complex numberz. Letha,bi=^R_Rabdµ_α,β (i.e. the standard inner product on L²). Notice that D

U_αⁿ_,βg,gE

n∈Z is positive definite. Recall that the Bochner-Herglotz theorem [Kt] says that there is a measureω_gonT, such that

D

U_αⁿ_,βg,gE

= Z

T

zⁿdωg(z). (n∈Z)

This tells us that 1

N

N n=1

∑

g(s+iT_α^kn_,β⁺¹(x))− 1 N

N n=1

∑

g(s+iT_α^kn_,β(x))₂

= Z

T(2−z−z⁻¹)1 N

N

∑

n=1

z^kn

²dω_g(z) using the parametrizationz=e^2πiθ forθ ∈[0,1), this is

=4 Z

T

sin² θ

2 1

N

N n=1

∑

z^kn

²dωg(z).

Using the fact that sin^θ₂ =0 ifθ =0 and the fact that(kn)_n≥1 is Hartman uniformly dis- tributed, by (1.9) and (1.10), we see that g(s+iT_α,β(x)) =g(x). A standard fact from ergodic theory is that if T_α,β is ergodic and g(s+iT_α,β(x)) =g(x)for measurable gthen g(x)is constant, which must be^R_Rgdµ_α,β. This extends to theL^p-norm for allp>1.

All we have to do now is to show that the pointwise limit is the same as the norm limit, i.e. that g(x) =g(x) =^R_Rgdµ_α,β. We consider the sequence of natural numbers(Nt)t≥1

such that

1

N_t

Nt

n=1

∑

g(s+iT_α^kn_,β(x))− Z

Rg(x)dµ_α,β

p≤ 1 t. Thus

∞ t=1

∑

Z

X

1

N_t

Nt

n=1

∑

g(s+iT_α^kn_,β(x))− Z

Rg(x)dµ_α,β^pdµ<∞.

Fatou’s lemma gives Z

R

^∞

t=1

∑

1

N_t

Nt

n=1

∑

g(s+iT_α^kn_,β(x))− Z

Rg(x)dµ_α,β^pdµ <∞.

This implies

∞ t=1

∑

1

N_t

Nt

n=1

∑

g(s+iT_α,β^kn (x))− Z

Rg(x)dµ_α,β^p<∞.

almost everywhere. This means

1 N_t

Nt

n=1

∑

g(s+iT_α,β^kn (x))− Z

X

g(x)dµ_α,β=o(1).

µ_α,β almost everywhere. As(kn)n≥1isL^p-good universal we must have

N→∞lim 1 N

N−1 n=0

∑

g(s+iT_α^kn_,βx) = Z

Rg(s+ix)dµ_α,β µ_α,β almost everywhere as required to prove Theorem 1.1.

4. PROOF OFTHEOREMS 1.2 TO1.5

We begin by recalling the spectral regularization method of Lifshits and Weber [LW1]

[LW2], and some technical preliminaries. Assume (X,β,ν)is a measure space with fi- nite measure ν and that S is a ν measure preserving transformation of (X,β,ν). For

f ∈L¹(X,β,ν)let

(4.1) B_N(f) = 1

N

N−1 k=0

∑

f◦T^k. (N=1,2, . . .)

We need the following lemma, due to R. Jones, R. Kaufman, J. Rosenblatt and M. Wierdl [JKRW].

Lemma 4.1. Suppose(Np)p≥ is a strictly increasing sequence of positive integers. Then there exists an absolute constant C>0such that

(4.2)

∞ p=1

∑

sup

Np≤N<N_p+1

|BN(f)−BNp(f)|²

2 ≤Ckfk2.

Write log₊(u) =max(1,log(u))foru≥1. Letω denote the spectral measure associated to the element f and define its regularised measure ˆω via its Radon-Nykodim derivative by

dωˆ dx(x) =

Z π

−πQ(θ,x)ω(dθ), where

Q(θ,x) =

|θ|⁻¹log²₊(^θ_x) if|x|<|θ|, θ²|x|⁻³ if|θ| ≤ |x| ≤π. The following is a theorem of Lifshits and Weber [LW1].

Lemma 4.2. Suppose N0 and N1with N0<N1 are positive integers. Then there exists an absolute constant C>0such that

(4.3) sup

N₀≤N<N₁

|BN(f)−BN₀(f)|²

2 ≤ Cωˆ [ 1 N₁

1 N₀)

.

Suppose 0<p<∞. For a sequence of real numbersx={xn:n≥1}the quantitykxkp,∞

is defined in (1.16). Also ifr<pwe have kxkp,∞≤ kxkp≤

p p−r

¹_p

kxkp,∞.

We also consider

N_x^∗=sup

m≥1

#{n:^x_nⁿ > _m¹}

m .

In our considerations,x_n= f(τⁿ(x)) (n=1,2, . . .)and x=O_f(x) = (f(τⁿ(x))n≥1which is the value of f along the orbit of a point x∈X. Also let N_f(x) = N_O^∗

f(x). We have the following Lemma due to I. Assani [As] (see also [W] - we refer to Jamison, Orey and Pruitt [JOP] for more on the random variable case, and [RR] for Birnbaum-Orlicz spaces and the

“Llog₊L” notation).

Lemma 4.3. (i) For any non-negative f ∈L¹(µ)the function N_f(x)is weak-(p,p) for all p∈(1,∞). Further

(4.4) lim

m→∞

#{n: ^f(τn_n^(x)) > _m¹}

m =

Z

X

f dµ, µ almost everywhere and also in L¹(µ)-norm.

(ii) For f ∈Llog₊L

(4.5)

f(τⁿ(x)) n

_1,∞

1

<∞.

Note that, for two positive constantsC₁,C2, C₁N_f(x)≤

f(τⁿ(x))

n ,n≥1_1,∞≤C₂N_f(x).

We now turn to the proof of Theorem 1.2.

By Lemma 4.1,

∞

∑

k=1

sup

N_k≤N<N_k+1

|YN(σ)−YN_k(σ)|²

2≤C π

Z

R

|ζ(σ+iτ)|² 1+τ^{2 d}τ, where as beforeCis universal. We note that, forσ ∈(¹₂,1),

Z

R

|ζ(σ+it)|² 1+t² dt=

Z ∞ 0

{x}

x^2σ+1dx=−1 σ

ζ(2σ)

2 +ζ(2σ−1) 2σ−1

.

Here again{x}=x−[x]denotes the fractional part ofx. The quantity in the bracket on the right hand side is negative. Further,

1 2π

Z

R

|ζ(¹₂+it)|²

1

4+t² dt=log(2π)−γ.

whereγ is Euler’s constant. See Prop. 7, Cor. 8 and (1.26) in Coffey [C]. Thus

(4.6) C

π Z

R

|ζ(σ+iτ)|²

1+τ^{2 d}τ≤ C πσ

ζ(2σ)

2 +ζ(2σ−1) 2σ−1

. completing the proof of Theorem 1.2.

We now turn to the proof of Theorem 1.3. The first assertion is a consequence of Lemma 4.3 (i). The limit is

1 π

Z

R

|ζ(σ+it)|² 1+τ^{2 dt.}

This integral is a special case of integrals calculated in [St1]. As a consequence of Theorem 1.1 in [St2],

1 π

Z

R

|ζ(σ+it)|²

1+τ^{2 dt=}ζ(σ+1)− 2 σ(2−σ).

For the second assertion, it follows from Lemma 4.3 (ii). The function ζ(s)∈Llog₊L is integrable, with respect to the Cauchy measure, because of the estimate

ζ(σ+it)≪εt^{1−σ2 +ε}.

See Titchmarsh, [T] section 5.1, for the details of this. Therefore Theorem 1.3 is proved as required.

To prove Theorem 1.4 we need the following two lemmas.

Suppose(X,β,µ)is a measure space and thatT :X →X is a measure preserving map.

Let(ak)^∞_k=0be a sequence of natural numbers and for any measurable f onX set

(4.7) C_Nf(x) := 1

N

N−1 k=0

∑

f(T^akx), (N=1,2,· · ·) that is the ergodic averages corresponding to the sequence and let

M f(x) := sup

N≥1

|CNf(x)|.

Further from the data(X,β,µ,T)and f we construct the ergodicq-variation function Vqf(x) = (

∑

N≥1

|C_N+1f(x)−CNf(x)|^q)^1q. (q ≥ 1) Our first lemma is Theorem 1 from [NW1].

Lemma 4.4. Suppose for a sequence of natural numbers(a_k)^∞_k=0that for some p > 1and f

C_p > 0depending only on p and(X,β,µ,T)we have

(4.8) M f_p ≤ Cf_pkfkp.

Then there exists another constant Dp > 0depending only on p and(X,β,µ,T)such that if q > 1then

(4.9) V_qf

p ≤ D_pkfkp.

Further suppose there is a constantCb > 0depending only on (X,β,µ,T) such that we have

(4.10) µ({x:M f(x) > λ}) ≤ Cb λ Z

X|f|dµ.

Then there is a constant D > 0depending only on(X,β,µ,T)such that if q > 1then (4.11) µ({x:V_qf(x) > λ}) ≤ D

λ Z

X|f|dµ.

Proof. Suppose(X,L,η)denote a finite measure space i.e. η(X)<∞. Now, let(Tn)n≥1

denote a sequence of linear transformations ofL^p(X,L,η)into measurable functions on X, such that each T_n is continuous in measure, that is, such that if ||f − f_m||p tends to 0 as m tends to ∞, then ||Tnf −T_nf_m||p also tends to 0 as m tends to ∞. Set T^∗f(x) = sup_n≥1|Tnf(x)| for f ∈L^p(X,L,η). We will say the family (Tn)n≥1 commutes with w: X →X ifT^∗f(w(x))≤T^∗g(x)where g(x) = f(w(x)) onX. Now supposeF is a family of ergodic η preserving transformations on X, closed under composition. We will call (Tn)n≥1distributive if it commutes with all the elements of some familyF onX. We have the following result of S. Sawyer [Sa].

Lemma 4.5. Let (Tn)n≥1 be a distributive sequence of linear operators on L^p(X,L,η), where each T_nis continuous in measure and maps L^p(X,L,η)to measurable functions on X . If p∈[1,2]and if T^∗f(x)<∞ηalmost everywhere, then there exists a uniform constant C>0such that

(4.12) η {x:T^∗f(x)≥λ}

≤ C λ^p

Z

X|f(x)|^pdη, for allλ >0and f ∈L^p(X,L,η).

The conclusion of Lemma 4.5 is that if T^∗f(x) <∞ η almost everywhere, then the operator T^∗ satisfies a weak^∗(p,p). If there is a C>0 such that ||T^∗f||p≤C||f||p we sayT^∗satisfies astrong(p,p)inequality. It is easy to check that strong(p,p)inequalities imply the corresponding weak(p,p)inequalities for the operatorT^∗. On the other hand it follows from the Marcinkiewicz interpolation theorem [SW] that a weak(p,p)inequality implies a strong(q,q)inequality ifq>p. We now specialize (4.1) to the situation where f is defined onX=R,L is the Lebesgue algebra onX =Randη=µ_α,β:

T_Nf(x):= 1 N

N n=1

∑

f(T_α,β^kn (x)) (n=1,2, . . .) with f(x) =ζ(σ+ix). One checks readily that(TN)_N≥1 commutes with(T_αⁿ_,β)_n≥1 and is therefore distributive. In light of Lemma 4.5, we see that Theorem 1.4 follows by recalling