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de Bordeaux _{16(2004), 221–232}

Extremal values of Dirichlet

_L

-functions in the

half-plane of absolute convergence

parJ¨orn STEUDING

R´esum´e. On d´emontre que, pour toutθr´eel, il existe une infinit´e des=σ+itavecσ_→1+ ett_→+∞tel que

Re{exp(iθ) logL(s, χ)} ≥log log log logt

log log log logt +O(1).

La d´emonstration est bas´ee sur une version effective du th´eor`eme de Kronecker sur les approximations diophantiennes.

Abstract. We prove that for any realθthere are infinitely many values ofs=σ+itwithσ_→1+ andt_→+∞such that

Re{exp(iθ) logL(s, χ)} ≥log log log logt

log log log logt +O(1).

The proof relies on an effective version of Kronecker’s approxima-tion theorem.

1. Extremal values

Extremal values of the Riemann zeta-function in the half-plane of abso-lute convergence were first studied by H. Bohr and Landau [1]. Their results rely essentially on the diophantine approximation theorems of Dirichlet and Kronecker. Whereas everything easily extends to Dirichlet series with real coefficients of one sign (see [7], _§9.32) the question of general Dirichlet se-ries is more delicate. In this paper we shall establish quantitative results for DirichletL-functions.

Letq be a positive integer and letχ be a Dirichlet character modq. As usual, denote by s = σ+it with σ, t _∈ R, i2 = ₋1, a complex variable. Then the DirichletL-function associated to the characterχ is given by

L(s, χ) =

∞

X

n=1

χ(n) ns =

Y

p

1₋ χ(p) ps

−1

,

where the product is taken over all primes p; the Dirichlet series, and so the Euler product, converge absolutely in the half-planeσ >1. Denote by

χ0 the principal character modq, i.e., χ0(n) = 1 for all n coprime with q.

Then

(1) L(s, χ0) =ζ(s) Y

p|q

1₋ 1 ps

.

Thus we may interpret the well-known Riemann zeta-functionζ(s) as the Dirichlet L-function to the principal characterχ0mod 1. Furthermore, it

follows that L(s, χ0) has a simple pole at s = 1 with residue 1. On the

other side, anyL(s, χ) withχ₆=χ₀ is regular ats= 1 withL(1, χ)₆= 0 (by Dirichlet’s analytic class number-formula). Since L(s, χ) is non-vanishing inσ >1, we may define the logarithm (by choosing any one of the values of the logarithm). It is easily shown that forσ >1

(2) logL(s, χ) =X p

X

k≥1

χ(p)k kpks =

X

p χ(p)

ps +O(1).

Obviously,_|logL(s, χ)_{| ≤}L(σ, χ0) forσ >1. However

Theorem 1.1. For any ǫ > 0 and any real θ there exists a sequence of s=σ+it withσ >1 and t_→+_∞ such that

Re_{exp(iθ) logL(s, χ)_{} ≥}(1₋ǫ) logL(σ, χ0) +O(1).

In particular,

lim inf

σ>1,t≥1|L(s, χ)|= 0 and lim sup_σ>1,t≥1|L(s, χ)|=∞.

In spite of the non-vanishing ofL(s, χ) the absolute value takes arbitrarily small values in the half-planeσ >1!

The proof follows the ideas of H. Bohr and Landau [1] (resp. [8], _§8.6) with which they obtained similar results for the Riemann zeta-function (answering a question of Hilbert). However, they argued with Dirichlet’s homogeneous approximation theorem for growth estimates of _|ζ(s)_| and with Kronecker’sinhomogeneousapproximation theorem for its reciprocal. We will unify both approaches.

Proof. Using (2) we have for x_≥2

(3) Re_{exp(iθ) logL(s, χ)_}

≥X

p≤x χ0(p)

pσ Re{exp(iθ)χ(p)p

−it

} −X p>x

χ0(p)

Denote byϕ(q) the number of prime residue classes modq. Since the val-uesχ(p) are ϕ(q)-th roots of unity ifpdoes not divide q, and equal to zero otherwise, there exist integersλp (uniquely determined modϕ(q)) with

χ(p) =

In view of the unique prime factorization of the integers the logarithms of the prime numbers are linearly independent. Thus, Kronecker’s approxi-mation theorem (see [8], _§8.3, resp. Theorem 3.2 below) implies that for any given integerω and anyx there exist a real numberτ >0 and integers hp such that

Obviously, with ω _{→ ∞} we get infinitely many τ with this property. It follows that

provided that ω_≥4. Therefore, we deduce from (3)

Re_{exp(iθ) logL(σ+iτ, χ)_{} ≥}cos

in view of (2). Obviously, the appearing series converges. Thus, sendingω and xto infinity gives the inequality of Theorem 1.1. By (1) we have

(7) logL(σ, χ0) = log

assertions of the theorem follow.

of absolute convergence:

if α > 0 is transcendental; note that in the case of transcendental α the Lerch zeta-function has zeros inσ >1 (see [3] and [4]).

In view of Theorem 1.1 we have to ask for quantitative estimates. Let π(x) count the prime numbers p_≤x. By partial summation,

X

The prime number theorem implies for x_≥2

X By the second mean-value theorem,

Z y Substituting (7) in formula (8) yields

Re_{exp(iθ) logL(σ+iτ, χ)_}

thenx tends to infinity asσ_→1+. We obtain for x sufficiently large

(9) Re_{exp(iθ) logL(σ+iτ, χ)_{} ≥}(1 +O(ω−2)) log logx

2. Effective approximation

H. Bohr and Landau [2] (resp. [8],_§8.8) proved the existence of aτ with 0_≤τ _≤exp(N6) such that

cos(τlogpν)<−1 + 1

N for ν= 1, . . . , N,

wherepνdenotes theν-th prime number. This can be seen as a first effective version of Kronecker’s approximation theorem, with a bound forτ (similar to the one in Dirichlet’s approximation theorem). In view of (5) this yields, in addition with the easier case of bounding_|ζ(s)_|from below, the existence of infinite sequencess_±=σ_±+it_± withσ_±→1+ andt_±→+_∞for which

(10) _|ζ(s+)| ≥Alog logt+ and

1 |ζ(s−)| ≥

Alog logt−,

whereA >0 is an absolute constant. However, for DirichletL-functions we need a more general effective version of Kronecker’s approximation theorem. Using the idea of Bohr and Landau in addition with Baker’s estimate for linear forms, Rieger [6] proved the remarkable

Theorem 2.1. Let v, N _∈ N, b _∈Z,1 _≤ω, U _∈R. Let p1 < . . . < pN be prime numbers (not necessarily consecutive) and

uν ∈Z, 0<|uν| ≤U, βν ∈R for ν = 1, . . . , N.

Then there existhν ∈Z,0≤ν ≤N, and an effectively computable number C=C(N, pN)>0, depending onN and pN only, with

(11)

h0

uν

v logpν−βν−hν

<

1

ω for ν = 1, . . . , N and b_≤h0≤b+ (2U vω)C.

We need C explicitly. Therefore we shall give a sketch of Rieger’s proof and add in the crucial step a result on an explicit lower bound for linear forms in logarithms due to Waldschmidt [9].

LetK be a number field of degree D overQ and denote byL_K the set of logarithms of the elements ofK_{\ {}0_}, i.e.,

L_K=_{ℓ_∈C : exp(ℓ)_∈K_}.

If a is an algebraic number with minimal polynomial P(X) over Z, then define the absolute logarithmic height ofa by

h(a) = 1 D

Z 1

0

log_|P(exp(2πiφ))_|dφ;

Theorem 2.2. Let ℓν ∈ LK and βν ∈ Q for ν = 1, . . . , N, not all equal zero. Define aν = exp(ℓν) for ν = 1, . . . , N and

Λ =β₀+β₁loga₁+. . .+βNlogaN.

Let E, W and Vν,1≤ν ≤N, be positive real numbers, satisfying W _≥ max

1≤ν≤N{h(βν)}, 1

D ≤V1 ≤. . .≤VN,

Vν ≥max

h(aν),| logaν|

D

for ν = 1, . . . , N

and

1< E _≤min

exp(V1), min 1≤ν≤N

4DVν |logaν|

.

Finally, defineV_ν+= max_{Vν,1} for ν =N and ν =N−1, with V₁+ = 1 in the caseN = 1. If Λ₆= 0, then

|Λ_|>exp₋c(N)DN+2(W + log(EDV_N+)) log(EDV_N+₋₁)_×

×(logE)−N−1 N

Y

ν=1

Vν

withc(N)_≤28N+51_N2N_.

This leads to

Theorem 2.3. With the notation of Theorem 2.1 and under its assump-tions there exists an integerh0 such that (11) holds and

b_≤h0 ≤b+ 2 + ((3ωU(N + 2) logpN)4+ 2)N+2×

×exp 28N+51N2N(1 + 2 logpN)(1 + logpN−1)

N

Y

ν=2

logpν

!

; (12)

if pN is the N-th prime number, then, for any ǫ > 0 and N sufficiently large,

(13) b_≤h0 ≤b+ (ωU)(4+ǫ)Nexp

N(2+ǫ)N.

Proof. For t_∈Rdefine

f(t) = 1 + exp(t) + N

X

ν=1

exp2πituν

v logpν −βν

With γ−1 := 0, β−1 := 0, γ0 := 1, β0 := 0 andγν := u_vν logpν,1 ≤ν ≤ N,

By the multinomial theorem,

f(t)k= X

Setting β0 = j0 −j0′, βν = u_vν(jν −jν′) and aν = pν for ν = 1, . . . , N, we have, with the notation of Theorem 2.2,

Λ =

Thus we may take

(15) A= exp 28N+51_N2N_{(1 + 2 logp}

application of the Cauchy Schwarz-inequality to the first multiple sum and of the multinomial theorem to the second multiple sum on the right hand side of (16) yields

This gives

(17) _|f(τ)_|> N + 2₋2µ, where µ:= (N + 2)

2_logk

3k ;

note that µ <1 for k_≥11. By definition

f(t) = 1 + exp(2πi(tγν −βν)) + N

X

m=0

m6=ν

exp(2πi(tγm−βm)).

Therefore, using the triangle inequality,

|f(t)_{| ≤}N +_|1 + exp(2πi(τ γν−βν))| for ν= 0, . . . , N, and arbitraryt_∈R. Thus, in view of (17)

|1 + exp(2πi(τ γν−βν))|>2−2µ for ν= 0, . . . , N. Ifhν denotes the nearest integer toτ γν −βν, then

|τ γν−βν−hν|<

r

µ

2 for ν = 0, . . . , N. Forν = 0 this implies_|τ₋h0|<√µ. Replacing τ by h0 yields

|h₀γν−βνhν|<√µ

1 + max ν=1,...,N|γν|

for ν= 1, . . . , N.

Puttingk= [(3wU(N + 2) logpN)4] + 1 we get

b₋1_≤h0≤b+ 1 +B =b+ 1 +A([(3ωU(N+ 2) logpN)4] + 2)N+2. Substituting (15) and replacingb₋1 byb, the assertion of Theorem 2.1 fol-lows with the estimate (12) of Theorem 2.3; (13) can be proved by standard

estimates.

3. Quantitative results

We continue with inequality (9). LetpN be theN-th prime. Then, using Theorem 2.3 with N =π(x), v =uν = 1, and

βν = λpν

ϕ(q) + θ

2π for ν= 1, . . . , N, yields the existence ofτ = 2πh0 with

(18) b_≤ τ

2π ≤b+ω

(4+ǫ)N_exp(N(2+ǫ)N₎

such that (4) holds, asN and x tend to infinity. We choose ω = log logx, then the prime number theorem and (18) imply

Theorem 3.1. For any realθ there are infinitely many values ofs=σ+it withσ _→1+and t_→+_∞ such that

Re_{exp(iθ) logL(s, χ)_{} ≥}log log log logt

log log log logt +O(1).

Using the Phragm´en-Lindel¨of principle, it is even possible to get quantita-tive estimates on the abscissa of absolute convergence. We write f(x) = Ω(g(x)) with a positive function g(x) if

lim inf x→∞

|f(x)_| g(x) >0;

hence, f(x) = Ω(g(x)) is the negation of f(x) = o(g(x)). Then, by the same reasoning as in [8],_§8.4, we deduce

L(1 +it, χ) = Ω

log log logt log log log logt

,

and

1

L(1 +it, χ) = Ω

log log logt log log log logt

.

However, the method of Ramachandra [5] yields better results. As for the Riemann zeta-function (10) it can be shown that

L(1 +it, χ) = Ω(log logt), and 1

L(1 +it, χ) = Ω(log logt), and further that, assuming Riemann’s hypothesis, this is the right order (similar to [8],_§14.8). Hence, it is natural to expect that also in the half-plane of absolute convergence for Dirichlet L-functions similar growth es-timates as for the Riemann zeta-function (10) should hold. We give a heuristical argument. Weyl improved Kronecker’s approximation theorem by

Theorem 3.2. Let a1, . . . , aN ∈Rbe linearly independent over the field of rational numbers, and let γ be a subregion of the N-dimensional unit cube with Jordan volume Γ. Then

lim T→∞

1

Tmeas{τ ∈(0, T) : (a1t, . . . , aNt)∈γ mod 1}= Γ.

We conclude with some observations on the density of extremal values of logL(s, χ). First of all note that if

|L(1 +iτ, χ)_|±1 _≥f(T)

holds for a subset of values τ _∈[T,2T] of measure µT, where f(T) is any function which tends withT to infinity, then

Z 2T

T |

L(1 +it, χ)_|±2dt_≥µT f(T)2.

In view of well-known mean-value formulae we have µ= 0, which implies

lim T→∞

1

Tmeas{τ ∈[0, T] : |L(σ+iτ)|

±1

≥f(T)_}= 0.

This shows that the set on which extremal values are taken is rather thin. The situation is different for fixedσ >1. LetQ be the smallest primep for which χ0(p)6= 0. Then

|logL(s, χ)_{| ≤}logL(σ, χ0) =Q−σ

1 +O

Q Q+ 1

σ

;

note that the right hand side tends to 0+ asσ_→+_∞, and thatQ_≤q+ 1.

Theorem 3.3. Let 0< δ < 1₂. Then, for arbitrary θ and fixedσ >1, lim inf

M→∞

1

M♯{m≤M : (1−δ) logL(σ, χ0)−Re{exp(iθ) logL(σ+2πim, χ)}} ≥Q−2σ

1 +24 σ

≥δ2Q2+8(2Q)−8Q2−32exp₋23Q2+51Q4Q2+2.

Proof. We omit the details. First, we may replace (2) by

logL(s, χ)₋X p

χ(p) ps

≤ X

p,k≥2

χ0(p)

kpkσ .

This gives with regard to (8)

Re_{exp(iθ) logL(σ+2πim, χ)_{} ≥}(1₋δ) logL(σ, χ0)−2

x1−σ σ₋1−8

Q2−2σ 2σ_(σ−1) for some integerh0 =m, satisfying (12), whereN =π(x) and cos2_ωπ = 1−δ.

Puttingx=Q2, proves (after some simple computation) the theorem.

For example, if χ is a character with odd modulusq, then the quantity of Theorem 3.3 is bounded below by

≥ δ

16

Acknowledgement. The author would like to thank Prof. A. Laurinˇcikas, Prof. G.J. Rieger and Prof. W. Schwarz for their constant interest and encouragement.

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[5] K. Ramachandra,On the frequency of Titchmarsh’s phenomenon for ζ(s) - VII. Ann. Acad. Sci. Fennicae14_{(1989), 27–40.}

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[7] E.C. Titchmarsh,The theory of functions. Oxford University Press, 1939 2nd ed. [8] E.C. Titchmarsh,The theory of the Riemann zeta-function. Oxford University Press, 1986

2nd ed.

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[10] H. Weyl,_{Uber ein Problem aus dem Gebiete der diophantischen Approximation}¨ _{. G¨ottinger}

Nachrichten (1914), 234-244.

J¨ornSteuding

Institut f¨ur Algebra und Geometrie Fachbereich Mathematik

Johann Wolfgang Goethe-Universit¨at Frankfurt Robert-Mayer-Str. 10

60 054 Frankfurt, Germany