ONTHEVALUE-DISTRIBUTIONOF L-FUNCTIONSsiauliaims.su.lt/pdfai/2003/ste-03.pdf · 2012-10-27 ·...

Fizikos ir matematikos fakultetoSeminaro darbai,�iauliu� universitetas,6, 2003, 87�119

ON THE VALUE-DISTRIBUTION OF L-FUNCTIONSJörn STEUDING

Johann Wolfgang Goethe-Universität Frankfurt,Robert-Mayer-Str. 10, 60054 Frankfurt, Germany;

e-mail: [email protected]

Abstract. We study the value distribution of L-functions from a subclassof the Selberg class. We prove Riemann-von Mangoldt type formulae forthe c-values and we compute their Nevanlinna functions.Key words and phrases: L-functions, Riemann-von Mangoldt formula, value-distribution.

Mathematics Subject Classi�cation: 11M06, 11M41.

"Une fonction entière, qui ne devient jamais ni à a ni à b estnécessairement une constante." Emile Picard

1. Introduction: the Riemann zeta-function

L-functions are fundamental and fascinating objects in number theory.They are generating functions formed out of local data associated with eitheran arithmetic object or with an automorphic form. They can be attached tosmooth projective varieties de�ned over number �elds, to irreducible (com-plex or p-adic) representations of the Galois group of a number �eld, or to acusp form or irreducible cuspidal automorphic representation. They all havein common that they are given by an Euler product, i.e., a product takenover prime numbers. In view of the unique prime factorization of the integersthey also have a Dirichlet series representation. The famous Riemann zeta-function ζ(s) may be regarded as the prototype. L-functions encode in theirvalue-distribution deep information about the underlying arithmetical or al-gebraical structure that are often not obtainable by elementary or algebraicalmethods. For instance, Dirichlet's class number formula gives informationon the deviation from unique prime factorization in the ring of integers ofquadratic number �elds by the values of certain Dirichlet L-functions L(s, χ)at s = 1. In particular, the distribution of zeros is of special interest with

88 On the value-distribution of L-functions

respect to many problems in multiplicative number theory, e.g. the Riemannhypothesis on the non-vanishing of the Riemann zeta-function in the righthalf of the critical strip and its impact on the distribution of prime numbers.We start with a brief introduction to the theory of the Riemann zeta-function.

The Riemann zeta-function is a function of a complex variable s = σ + itwhich is for σ > 1 given by

ζ(s) =∞∑

n=1

1ns

=∏p

(1− 1

ps

)−1

;

here and in the sequel p denotes always a prime number and the productis taken over all primes. The Dirichlet series, and so the Euler product,converges absolutely in the half-plane σ > 1 and uniformly in each compactsubset. The identity between the Dirichlet series and the Euler product wasdiscovered by Euler in 1737 and can be regarded as the analytic version ofthe unique prime factorization of the integers. The Euler product gives a �rstglance on the intimate connection between the zeta-function and the distri-bution of prime numbers. A �rst immediate consequence is Euler's proof ofthe in�nitude of the prime numbers. Assuming that there are only �nitelymany primes, the Euler product is �nite, and therefore convergent through-out the whole complex plane, contradicting the fact that the ζ(s)-de�ningDirichlet series reduces to the divergent harmonic series as s → 1+. Hence,there exist in�nitely many prime numbers. This is well known since Euclid'selementary proof, but the analytic access gives much deeper knowledge on theprime number-distribution. Riemann was the �rst to investigate the Riemannzeta-function as a function of a complex variable. In his only but outstandingpaper [28] on number theory from 1859 he outlined how the prime numberdistribution is linked to the analytic behaviour of ζ(s). However, the theoryof functions was not developped so far, but the open questions concerningthe zeta-function pushed the research in this �eld quickly forwards.

First of all one gets by partial summation

ζ(s) =∑

n6N

1ns

+N1−s

s− 1+ s

∞∫

N

[u]− u

us+1du.

This gives an analytic continuation for ζ(s) to the half-plane σ > 0 exceptfor a simple pole at s = 1 with residue 1. This process can be continuedto the left half-plane and shows that ζ(s) is analytic throughout the wholecomplex plane except for s = 1. Riemann found the functional equation

π−s2 Γ

(s

2

)ζ(s) = π−

1−s2 Γ

(1− s

2

)ζ(1− s),

J. Steuding 89

where Γ(s) denotes Euler's Gamma-function. In view of the Euler productit is easily seen that ζ(s) has no zeros in the half-plane σ > 1. It followsfrom the functional equation and basic properties of the Gamma-functionthat ζ(s) vanishes in σ < 0 exactly at the so-called trivial zeros s = −2nwith n ∈ N. All other zeros of ζ(s) are said to be nontrivial, and we denotethem by % = β + iγ. Obviously, they have to lie inside the critical strip0 6 σ 6 1, and it is easily seen that they are non-real. The functionalequation, in addition with the re�ection principle

ζ(s) = ζ(s),

shows some symmetries of ζ(s). In particular, the nontrivial zeros of ζ(s) haveto be distributed symmetrically with respect to the real axis and the verticalline σ = 1

2 . It was Riemann's ingenious contribution to number theory topoint out how the distribution these nontrivial zeros rule the distribution ofprime numbers. Riemann conjectured that the number N(T ) of nontrivialzeros % = β + iγ with 0 6 γ 6 T (counted according multiplicities) satis�esthe asymptotic formula

N(T ) ∼ T

2πlog

T

2πe.

This was proved in 1895 by von Mangoldt who found more precisely

N(T ) =T

2πlog

T

2πe+ O(log T ). (1)

Riemann worked with ζ(12 + it) and wrote that very likely all roots t are

real, i.e., all nontrivial zeros lie on the so-called critical line σ = 12 . This is

the famous, yet unproved Riemann hypothesis. Riemann's ideas led to the�rst proof of the celebrated prime number theorem by Hadamard and de laVallée-Poussin (independendly) in 1896. If π(x) counts the number of primesp 6 x, then

π(x) ∼ x

log x.

For more details and the current stage of knowledge towards Riemann's hy-pothesis we refer to Conrey's survey [6].

2. A class of L-functions

In the sequel we shall study the value distribution of general L-functions.It is our aim to prove Riemann-von Mangoldt-type formulae for the c-values


of these L-functions. Selberg [29] introduced in 1989 a class S of Dirich-let series having an Euler product, analytic continuation and a functionalequation of Riemann type in order to study the value-distribution of linearcombinations of L-functions. In the meantime this so-called Selberg classbecame an important �eld of research but yet its structure is not understoodvery well. We refer to the survey of Kaczorowski & Perelli [16] for the recentprogress. We restrict our observations to a subclass of Selberg's class.

The class S̃ consists of Dirichlet series

L(s) :=∞∑

n=1

a(n)ns

satisfying the hypotheses

• polynomial Euler product: for 1 6 j 6 m and each prime p, there existcomplex numbers αj(p) with |αj(p)| 6 1 such that

L(s) =∏p

m∏

j=1

(1− αj(p)

ps

)−1

;

• mean-square: there exist a positive constant κ such that

limx→∞

1π(x)

∑

p6x

|a(p)|2 = κ.

• analytic continuation: there exists a non-negative integer k such that(s− 1)kL(s) is an entire function of �nite order;

• functional equation: there are numbers Q > 0, λj > 0, µj with Re µj >0, and some complex number ω with |ω| = 1 such that

ΛL(s) = ωΛL(1− s),

whereΛL(s) := L(s)Qs

r∏

j=1

Γ(λjs + µj).

The hypothesis on the polynomial Euler product implies that the coef�cientsa(n) are multiplicative, and that each Euler factor has a Dirichlet seriesrepresentation.

Similar subclasses were introduced by several mathematicians, e.g. Bom-bieri & Hejhal [1], Bombieri & Perelli [2], Conrey & Ghosh [7], [8], and Perelli

J. Steuding 91

[26]. All known examples of functions in the Selberg class are automorphicor at least conjecturally automorphic L-functions, and for all of them it turnsout that the related Euler factors Lp are the inverse of a polynomial in p−s.This pattern �ts to the Langlands reciprocity conjecture which relates L-functions attached to group representations and automorphic L-functions.The axiom on the mean square is closely related to Selberg's conjectures[29].

The degree of L ∈ S̃ is de�ned by

dL = 2r∑

j=1

λj .

The quantity dL is well-de�ned. If NL(T ) counts the number of zeros ofL ∈ S in the region 0 6 σ 6 1, |t| 6 T (according to multiplicities), then onecan show by standard contour integration

NL(T ) ∼ dLπ

T log T (2)

in analogy to the Riemann-von Mangoldt formula (1) for Riemann's zeta-function. We will give a more precise asymptotic formula in Theorem 14below. It is conjectured that all L have integral degree. The functions ofdegree one are the Riemann zeta-function and shifts of Dirichlet L-functionsL(s + iθ, χ) attached to primitive characters χ with θ ∈ R. Examples ofdegree 2 are normalized L-functions associated with holomorphic newforms.Normalized L-functions attached to non-holomorphic newforms are expectedto lie in S (but the so-called Ramanujan-Petersson conjecture is not veri�edyet). Higher examples are Dedekind zeta-functions to number �elds K; theirdegree is equal to the degree of the underlying �eld extension K/Q. Furtherexamples are Rankin-Selberg L-functions and Artin L-functions (subject totheir holomorphy predicted by Artin's conjecture). This can be shown byanalogues of the prime number theorem with error terms which rely on sui-table zero-free regions for the functions in question. Anyway, it can be shownthat if one is willing to accept some widely believed conjectures, then a largeclass of functions belongs to S̃.

In view of the Euler product it is clear that any element L of S̃ does notvanish in the half-plane of absolute convergence σ > 1. This gives rise tothe notions of critical strip and critical line. The zeros of L(s) located atthe poles of gamma-factors appearing in the functional equation are calledtrivial. Clearly, they all lie in σ 6 0, and it is easily seen that they are locatedat

s = −k + µj

λjwith k ∈ N ∪ {0} and 1 6 j 6 m.


All other zeros are said to be nontrivial. We cannot in general exclude thepossibility that L(s) has a trivial zero and a nontrivial at the same point. Itis expected that for every function in the Selberg class the analogue of theRiemann hypothesis holds. The grand Riemann hypothesis states that, forany L ∈ S,

L(s) 6= 0 for σ >12.

3. The Phragmén-Lindelöf principle

The order of growth of a meromorphic function is of special interest. Wede�ne

µL(σ) = lim supt→±∞

log |L(σ + it)|log |t| .

Taking into account the absolute convergence of the de�ning Dirichlet seriesin the half-plane σ > 1 we get immediately µL(σ) = 0 for σ > 1. The orderof growth to the left of the critical strip is ruled by the functional equationwhich we may rewrite as

L(s) = ∆L(s)L(1− s),

where∆L(s) := ωQ1−2s

r∏

j=1

Γ(λj(1− s) + µj)Γ(λjs + µj)

.

In view of Stirling's formula,

log Γ(s) =(

s− 12

)log s− s +

12

log 2π + O(

1|s|

)

for |arg | 6 π − ε and |s| > 1, we get after a short computation

Lemma 1. For t > 1,

∆L(σ + it) =(λQ2tdL

) 12−σ−it

exp(

itdL +iπ(µ− dL)

4

) (ω + O

(1t

)),

whereµ := 2

r∑

j=1

(1− 2µj) and λ :=r∏

j=1

λ2λj

j .

Now we are in the position to obtain upper bounds for the order of growthinside the critical strip.

J. Steuding 93

Theorem 2. As |t| → ∞,

L(σ + it) ³ |t|( 12−σ)dL |L(1− σ + it)|.

In particular,

µL(σ) 6

0 if σ > 1,12dL(1− σ) if 0 6 σ 6 1,

(12 − σ)dL if σ < 0.

Our argument is the so-called Phragmén-Lindelöf principle.Proof. The �rst assertion follows immediately from the functional equa-

tion and Lemma 1. This estimate implies for σ < 0

µL(σ) =(

12− σ

)dL.

The value of µL(σ) for 0 6 σ 6 1 is a more di�cult task. Here we applythe theorem of Phragmén-Lindelöf which is a kind of maximum principle forunbounded regions:

Theorem 3. Let f(s) be analytic in the strip σ1 6 σ 6 σ2 with f(s) ¿exp(ε|t|). If

f(σ1 + it) ¿ |t|c1 and f(σ2 + it) ¿ |t|c2 ,

then f(s) ¿ |t|c(σ) uniformly in σ1 6 σ 6 σ2, where c(σ) is linear withc(σ1) = c1 and c(σ2) = c2.

A proof can be found in Titchmarsh's book [31]. Note that there arecounterexamples if the growth condition is not ful�lled. However, by theaxiom on the analytic continuation any L ∈ S̃ satis�es this condition.

We continue with the proof. Theorem 3 shows that µL(σ) is non-increa-sing and convex downwards. By the estimates of µL(σ) for σ < 0 and σ > 1the second assertion of the theorem follows.

In view of the functional equation, resp. the convexity of µL, the valuefor σ = 1

2 is essential. In particular, we obtain µ(12) 6 dL

4 , or equivalently,

L(

12

+ it

)¿ |t| dL4 +ε (3)

for any ε > 0 as |t| → ∞; this bound is also known as convexity bound. Thebest known upper bound for the Riemann zeta-function is µζ(1

2) 6 89570 due

to Huxley [11].


4. The mean-square of the coe�cients

For our studies on the value-distribution of functions in S̃ we have to proveseveral mean-square formulae. We start with an estimate for the coe�cientsa(n) in the Dirichlet series representation of L. Recall that m is the degreeof the polynomial in p−s in the local Euler factor of L.

Lemma 4. As x →∞,∑

n6x

|a(n)|2 ¿ x(log x)m2−1.

Proof. From the identity

∞∑

n=1

a(n)ns

=∏p

m∏

j=1

(1− αj(p)

ps

)−1

=∏p

m∏

j=1

(1 +

∞∑

k=1

αj(p)k

pks

),

valid for σ > 1, we deduce

a(n) =∏

p|n

∑

k1,...,km>0k1+...+km=ν(n;p)

m∏

j=1

αj(p)kj ,

where ν(n; p) is the exponent of the prime p in the prime factorization of theinteger n. Taking into account |αj(p)| 6 1 we �nd

|a(n)| 6∏

p|n

∑

k1,...,km>0k1+...+km=ν(n;p)

1 =: dm(n), (4)

say. Thus it is su�cient to �nd a mean-square estimate for dm(n); note thatthe divisor function dm(n) is multiplicative (and appears in the Dirichletseries expansion of ζ(s)m). Consequently,

dm(n)2 =∑

d|ng(d)

with some multiplicative function g. Since

dm(pν) = #{(k1, . . . , km) ∈ Nm0 : k1 + . . . + km = ν} =

(m + ν − 1)!ν!(m− 1)!

,

J. Steuding 95

we �nd

g(1) = dm(1)2 = 1, g(p) = dm(p)2 − dm(1)2 = m2 − 1,

and by induction

g(pν) = dm(pν)2 − dm(pν−1)2 ∼ m2ν

ν!,

as ν →∞. Hence we obtain

∑

n6x

dm(n)2 6 x∑

d6x

g(d)d

6 x∏

p6x

(1 +

∞∑

ν=1

g(pν)pν

).

In view of the above estimates the right hand side above equals

x∏

p6x

(1 +

m2 − 1p

+∞∑

ν=2

m2ν

ν!pν

)= x

∏

p6x

(1 +

m2 − 1p

)+ O(x).

Now a classic estimate due to Mertens,

∏

p6x

(1 +

1p

)¿

∏

p6x

(1− 1

p

)−1

¿ log x,

gives the estimate of the lemma.

5. The quadratic moment

Now we shall use this result to obtain a mean-square estimate for theintegral over L(s) on vertical lines to the right of the critical line. In thehalf-plane of convergence it is easily shown that

1T

∫ T

1|L(σ + it)|2dt ∼

∞∑

n=1

|a(n)|2n2σ

.

Carlson [4] obtained for Dirichlet series an analogue of Parseval's theoremfor Fourier series:

Theorem 5. LetA(s) =

∞∑

n=1

an

ns


be regular and of �nite order for σ > α, and

limT→∞

12T

∫ T

−T|A(σ + it)|2dt ¿ 1

Then,

limT→∞

12T

∫ T

−T|A(σ + it)|2dt =

∞∑

n=1

|an|2n2σ

,

for σ > α, and uniformly in any strip α < σ1 6 σ 6 σ2.

A proof can also be found in Titchmarsh`s monography [31].However, for the �rst we will apply the following re�nement of Carlson's

classic theorem on the mean-square of Dirichlet series due to Potter [27]:

Theorem 6. Suppose that the functions

A(s) =∞∑

n=1

an

nsand B(s) =

∞∑

n=1

bn

ns

have a half-plane of convergence, are of �nite order, and that all singulari-ties lie in a subset of the complex plane of �nite area. Further, assume theestimates ∑

n6x

|an|2 ¿ xb+ε and∑

n6x

|bn|2 ¿ xb+ε,

as x →∞, and that A(s) and B(s) satisfy

A(s) = h(s)B(1− s),

where h(s) ³ |t|c(a2−σ) as |t| → ∞, and c is some positive constant. Then

limT→∞

12T

∫ T

−T|A(σ + it)|2dt =

∞∑

n=1

|an|2n2σ

for σ > max{a2 , 1

2(b + 1)− 1c}.

Taking into account Theorem 2 and Lemma 4 we deduce

Corollary 7. For σ > max{

12 , 1− 1

dL

},

limT→∞

1T

T∫

0

|L(σ + it)|2dt =∞∑

n=1

|a(n)|2n2σ

.

J. Steuding 97

The series on the right hand side converges with respect to the estimate

a(n) ¿ nε, (5)

which is a simple consequence of (4); note that also this estimate wouldhave been su�cient to deduce the latter corollary from Potter's theorem.Corollary 7 should be compared with the mean-square estimate of Perelli[26] for his class of L-functions; his argument di�ers from our argument.

Thus, the quadratic-mean of L on vertical lines with real part > max{12 ,

1− 1dL} exists. For our later purpose it would be desireable to extend this to

the half-plane σ > 12 . However, this is a very di�cult task. The di�culties

come up with large degrees dL. For instance, Chandrasekharan and Nara-simhan [5] obtained for Dedekind zeta-functions the estimate

T∫

0

|ζK(σ + it)|2dt ¿ T d(1−σ)(log T )d

for 12 6 σ 6 1− 1

d , where d is the degree of ζK(s). Potter's theorem yields onlyan asymptotic formula throughout σ > 1

2 if the degree d is less or equal two.The di�culties become more obvious by noting that any result on the mean-square of an L-function from the Selberg class of degree d is comparable tothe corresponding result for the 2d-th moment of the Riemann zeta-function.The fourth moment of ζ(s) is quite well understood (by the work of Hardy& Littlewood, Ingham and others) but higher moments are still unsettled;there are only estimates known. For more details on mean-value results werefer to Ivi¢' monography [12], Matsumoto's survey [24] and Conrey [6].

6. Lindelöf's hypothesis

Lindelöf [22] conjectured that the order of growth of the zeta-functionis much smaller than the one the Phragmén-Lindelöf principle gives. Moreprecisely, he expressed his belief that ζ(s) is bounded if σ > 1

2 + ε with any�xed positive ε. In terms of the µ-function from Section 3 this would implyµζ(1

2) = 0 or equivalently

ζ

(12

+ it

)¿ tε

as t → ∞. The last statement is now known as Lindelöf's hypothesis andit is yet unproved though the boundedness conjecture is false. As it was


proved by Littlewood, the Lindelöf hypothesis follows from the truth of theRiemann-hypothesis. The converse implication, however, does not hold. LetN(σ, T ) count the number of zeros % = β + iγ with β > σ and 0 < γ 6 T .Backlund proved that the Lindelöf hypothesis is equivalent to the much lessdrastic but yet unproved hypothesis that for every σ > 1

2

N(σ, T + 1)−N(σ, T ) = o(log T ).

Furthermore, the Lindelöf hypothesis implies the so-called density hypothesiswhich states that

N(σ, T ) ¿ T 2(1−σ)+ε.

Garunk²tis & Steuding [9] proved that the Lerch zeta-function has for certainparameters in�nitely many zeros to the right of the critical line; more pre-cisely, there are À T many zeros up to level T . This indicates that in caseof the Lerch zeta-function has the density hypothesis does not follow fromthe Lindelöf hypothesis (if it does not obey an Euler product). On the otherside Garunk²tis & Steuding [10] showed that the analogue of the Lindelöfhypothesis for Lerch zeta-functions seems reasonable.

In [8] Conrey & Ghosh generalized the Lindelöf hypothesis to the Selbergclass. Suppose that L ∈ S is an entire function which satis�es the Riemannhypothesis, the Euler product is of the form of S̃, and that all λj appearingin the functional equation are equal to 1

2 . Then they proved that for anyε > 0 there exists a constant c, depending only on ε and dL, such that

∣∣∣∣L(

12

+ it

)∣∣∣∣ 6 c

Q(1 + |t|)dL

2

m∏

j=1

(1 + |µj |)

ε

.

The proof follows the lines of Littlewood's proof that the Riemann hypothesisimplies the Lindelöf hypothesis (it relies on a combination of the Phragmén-Lindelöf principle, the Borel-Carathéodory theorem and Hadamard's threecircle theorem). Note that if L ∈ S̃ has a pole of order k at s = 1, thenit is easily seen that ζ(s)k appears in the factorization of L into primitivefunctions. Therefore, the result of Conrey & Ghosh applies also to thosefunctions. In view of the Phragmén-Lindelöf principle one can prove uncon-ditionally ∣∣∣∣L

(12

+ it

)∣∣∣∣ ¿ (c(λ, µ)Q2(1 + |t|)dL)14+ε

for some positive constant c depending on the data of the functional equation.We already obtained with the estimate (3) a similar bound in the t-aspect.

J. Steuding 99

The so-called subconvexity problem is to �nd a δ > 0 such that the exponenton the right hand can be replaced by 1

4 − δ. For certain L-functions subcon-vexity bounds are known but in general this seems to be a di�cult problem.However, solutions of the subconvexity problem (in the Q-aspect) lead toseveral important applications in number theory; for details and exampleswe refer to the survey of Iwaniec & Sarnak [14]. Anyway, for our later appli-cation we are only interested in the t-aspect. Thus, we state the generalizedLindelöf hypothesis by

L(

12

+ it

)¿ tε as t →∞.

This coincides with Perelli's Lindelöf hypothesis for his class of L-functionswhich is quite similar to S̃. Among other interesting results Perelli [26]showed that the analogue of Riemann's hypothesis implies the Lindelöf hy-pothesis, and that Backlund's reformulation of the Lindelöf hypothesis interms of their zero-distribution o� the critical line holds also for his L-functions.

There are several further interesting reformulations of the Lindelöf hy-pothesis in case of the Riemann zeta-function. A classic one in terms ofmoments on the critical line was found by Hardy and Littlewood. Theyproved that the Lindelöf hypothesis is true if and only if for any k ∈ N

1T

T∫

1

∣∣∣∣ζ(

12

+ it

)∣∣∣∣2k

dt ¿ T ε. (6)

Another one is due to Laurin£ikas [17]. He proved that the Lindelöf hypoth-esis is equivalent to the asymptotic formula

1T

meas{

t ∈ [0, T ] :∣∣∣∣ζ

(12

+ it

)∣∣∣∣ < xT ε

}= 1−O

(δ(T )

1 + xA

),

where ε,A and x are positive numbers, x su�ciently large, and δ(T ) is anarbitrary positive function which tends to zero as T → ∞; the appearingmeasure is the Lebesgue measure.

7. The quadratic moment revisited

Now we will prove an asymptotic formula for the mean-square on verticallines to the right of the critical line subject to the Lindelöf hypothesis. Thiswill also provide an unconditional proof of Corollary 7.


Theorem 8. The Lindelöf hypothesis implies that, for any σ > 12 ,

12T

T∫

−T

|L(σ + it)|2dt ∼∞∑

n=1

|a(n)|2n2σ

.

The proof is very similar to Carlson's approach to mean-values of theRiemann zeta-function [4]. Therefore we only give a sketch of proof followingTitchmarsh's presentation in [32], �9.7.

Proof. With regard to Carlson's theorem 5 it su�ces to show thatT∫

−T

|L(σ + it)|2dt ¿ T. (7)

For this aim we consider for δ > 0 and σ > 1 the Dirichlet series∞∑

n=1

a(n)ns

exp(−δn).

Let c be a constant satisfying 1 < c < σ. By Mellin's inversion formula,

1ns

=1

Γ(s)

∞∫

0

xs−1 exp(−nx)dx, (8)

valid for σ > 0, we get

∞∑

n=1

a(n)ns

exp(−δn) =∞∑

n=1

a(n)ns

12πi

c−σ+i∞∫

c−σ−i∞Γ(z)(δn)−zdz.

Interchanging summation and integration, which is allowed in view of theabsolute convergence, shows that the latter expression equals

12πi

c+i∞∫

c−i∞Γ(z − s)

∞∑

n=1

a(n)nz

δs−zdz =1

2πi

c+i∞∫

c−i∞Γ(z − s)L(z)δs−zdz.

Now we move the path of integeration to the left. For σ1 with σ−1 < σ1 < σwe pass the simple pole of Γ(z−s) at z = s with residue L(z), and the possiblepole of L(z) at z = 1 of order k, where the residue is a �nite sum of termsof the form

cu,vΓ(u)(1− s)δs−1(log δ)v,

J. Steuding 101

where u, v are non-negative integers and the cu,v's are certain constants withu + v = k. By Stirling's formula the latter residue is

¿ δσ−1+ε exp(−BT ),

where B is a positive absolute constant. Thus, for δ > |t|−B we obtain bythe calculus of residues

L(s) =∞∑

n=1

a(n)ns

exp(−δn)− 12πi

σ1+i∞∫

σ1−i∞Γ(z − s)L(z)δs−zdz

+O(exp(−B|t|)). (9)

Now we integrate this expression with respect to t. For σ > 12 we get by

squaring out the modulus

T∫

12T

∣∣∣∣∣∞∑

n=1

a(n)ns

exp(−δn)

∣∣∣∣∣2

dt

¿ T∞∑

n=1

|a(n)|2n2σ

exp(−2nδ) +∞∑

m=1

∞∑

n=1n 6=m

a(m)a(n)(mn)σ

exp(−(m + n)δ).

The �rst series is convergent independent of δ > 0. The double series turnsout to be bounded by δ2σ−2−ε. Thus,

∫ T

12T

∣∣∣∣∣∞∑

n=1

a(n)ns

exp(−δn)

∣∣∣∣∣2

dt ¿ T + δ2σ−2−ε. (10)

Let z = σ1 + iτ , then

12πi

σ1+i∞∫

σ1−i∞Γ(z − s)L(z)δs−zdz ¿ δσ−σ1

∞∫

−∞|Γ(z − s)L(z)|dτ.

By the Cauchy-Schwarz inequality this is bounded by

¿ δσ−σ1

∞∫

−∞|Γ(z − s)|dτ ·

∞∫

−∞|Γ(z − s)L(z)2|dτ

12

.

J. Steuding 103

Using this with 21−nT instead of T and summing up over all n ∈ N, we �ndthat the same bound holds for the integral with limits 0 and T . Putting

δ = T− 1+2µL(σ1)

2(1−σ1) ,

the estimate (7) is valid for

σ > 1− 1− σ1

1 + 2µL(σ1).

The right hand side coincides for σ1 = max{12 , 1− 1

dL} with the unconditional

bound from Corollary 7, and if Lindelöf's hypothesis is true we have µL(12) =

0 and thus may take σ > 12 . This proves the theorem.

There are no big di�culties to do the same for the 2k-th moments, andalso to prove that this leads to an equivalent statement for Lindelöf's hy-pothesis analogously to (6) for ζ(s).

8. Sums over c-values

Many beautiful results on the value-distribution of L-functions followfrom the general theory of Dirichlet series like the Big Picard-theorem (see[23]), but more advanced statements can only be proved by exploiting thecharacterizing properties (the functional equation and the Euler product).In the sequel we shall study the distribution of the values taken by L ∈ S̃and their frequencies. Our argument follows Levinson [20] and Levinson &Montgomery [21], respectively, who proved similar results for the Riemannzeta-function.

Let c be a complex number and denote the roots of the equation

L(s) = c (11)

by %c = βc + iγc. Our �rst aim is to proveTheorem 9. Let c 6= 1. Then, for b > max{1

2 , 1− 1dL},

∑

βc>bT6γc62T

(βc − b) ¿ T,

and under assumption of the truth of Lindelöf 's hypothesis∑

βc> 12

T6γc62T

(βc − 1

2

)= o(T log T ).


The case c = 1 is somehow exceptional since

L(s) = 1 + O(2−σ), (12)

as σ →∞. We will brie�y discuss this case at the end of Section 9.Proof. In view of (12) there exists a positive real number A depending

on c such that all βc < A. Put

`(s) =L(s)− c

1− c.

Obviously, the zeros of `(s) correspond exactly to the c-values of L(s). Nextwe will apply Littlewood's lemma:

Lemma 10. Let f(s) be regular in and upon the boundary of the rectangle Rwith vertices a, a+ iT, b+ iT, b, and not zero on σ = b. Denote by ν(σ, T ) thenumber of zeros % = β + iγ of f(s) inside the rectangle with β > σ includingthose with γ = T but not γ = 0. Then

∫

Rlog f(s)ds = −2πi

a∫

b

ν(σ, T )dσ.

This is an integrated version of the principle of the argument; for a proofsee [32], �9.9.

Let ν(σ, T ) denote the number of zeros %c of `(s) with βc > σ and T 6γc 6 2T (counting multiplicities). Now let a be a parameter with a >max{A + 1, b}. Then Littlewood's lemma 10, applied to the rectangle Rwith vertices a + iT, a + 2iT, b + iT, b + 2iT , gives

∫

Rlog `(s)ds = −2πi

a∫

b

ν(σ, T )dσ.

Sincea∫

b

ν(σ, T )dσ =∑

βc>bT6γ62T

βc∫

b

dσ =∑

βc>bT6γc62T

(βc − b),

we get

2π∑

βc>bT6γc62T

(βc − b)

J. Steuding 105

=

2T∫

T

log |`(b + it)|dt−2T∫

T

log |`(a + it)|dt+

−a∫

b

arg `(σ + iT )dσ +

a∫

b

arg `(σ + 2iT )dσ

=4∑

j=1

Ij , (13)

say. To de�ne log `(s) we choose the principal branch of the logarithm on thereal axis, as σ →∞; for other points s the value of the logarithm is obtainedby analytic continuation.

We start with the vertical integrals. Obviously,

I1 = I1(T, b) =

2T∫

T

log |L(b + it)− c|dt− T log |1− c|. (14)

By Jensen's inequality the appearing integral is

6T

2log

1

T

2T∫

T

|L(b + it)|2dt

+ O(T ).

By Corollary 7 this is ¿ T for b > max{12 , 1 − 1

dL}. Hence, I1(T, b) ¿ T .

An immediate consequence of Lindelöf's hypothesis is

2T∫

T

∣∣∣∣L(

12

+ it

)∣∣∣∣2

dt ¿ T 1+ε

for any positive ε. Thus, assuming the truth of Lindelöf's hypothesis we get

I1

(T,

12

)¿ εT log T.

Next we consider I2. Since a > 1 we have

`(a + it) = 1 +1

1− c

∞∑

n=2

a(n)na+it

. (15)


The absolute value of the series is less than 1 for su�ciently large a. Thereforewe �nd by the Taylor expansion of the logarithm

log |`(a + it)| = Re∞∑

k=1

(−1)k

k(1− c)k

∞∑

n1=2

. . .∞∑

nk=2

a(n1) · . . . · a(nk)(n1 · . . . · nk)a+it

.

In view of (5) this leads to the estimate

I2 = Re∞∑

k=1

(−1)k

k(1− c)k

∞∑

n1=2

. . .∞∑

nk=2

a(n1) · . . . · a(nk)(n1 · . . . · nk)a

2T∫

T

dt

(n1 · · ·nk)it

¿∞∑

k=1

1k

( ∞∑

n=2

1na−ε

)k

¿ 1, (16)

for su�ciently large a. It remains to estimate the horizontal integrals I3, I4.Suppose that Re `(σ + iT ) has N zeros for b 6 σ 6 a. Then divide [b, a]

into at most N + 1 subintervals in each of which Re `(σ + iT ) is of constantsign. Then

|arg `(σ + iT )| 6 (N + 1)π. (17)To estimate N let

g(z) =12

(`(z + iT ) + `(z + iT )

).

Then we have g(σ) = Re `(σ + iT ). Let R = a− b and choose T so large thatT > 2R. Now, Im (z + iT ) > 0 for |z − a| < T . Thus `(z + iT ), and henceg(z), is analytic for |z− a| < T . Let n(r) denote the number of zeros of g(z)in |z − a| 6 r. Obviously, we have

2R∫

0

n(r)r

dr > n(R)

2R∫

R

dr

r= n(R) log 2.

With Jensen's formula (see [32], �3.61),2R∫

0

n(r)r

dr =12π

2π∫

0

log∣∣∣g

(a + 2Re iθ

)∣∣∣dθ − log |g(a)|,

we deduce

n(R) 61

2π log 2

2π∫

0

log∣∣∣g

(a + 2Re iθ

)∣∣∣dθ − log |g(a)|log 2

.

J. Steuding 107

By (15) it follows that log |g(a)| is bounded. By Theorem 2 we have in anyvertical strip of bounded width

L(s) ¿ |t|B

with a certain positive constant B. Obviously, the same estimate holds forg(z). Thus, the integral above is ¿ log T , and n(R) ¿ log T . Since theinterval (b, a) is contained in the disc |z − a| 6 R, we have N 6 n(R).Therefore, with (17), we get

|I4| 6a∫

b

|arg `(σ + iT )|dσ ¿ log T.

Obviously, I3 can be bounded in the same way.Collecting together all estimates the assertions of the theorem follow.Now we will include most of the c-values into our observations. In view

of Lemma 1 and Theorem 2 there exist positive constants C ′, T ′ such thatthere are no c-values in the region σ < −C ′, T > T ′. Therefore, assume thatb < −C ′ − 1 and T > T ′ + 1. By the functional equation (in the form ofSection 3)

log |L(s)− c| = log |∆L(s)|+ log |L(1− s)|+ O

(1

|∆L(s)L(1− s)|

).

By Lemma 1

log |∆L(s)| =(

12− σ

)(dL log t + log(λQ2)) + O

(1t

).

Thus∫ 2T

Tlog |L(b + it)− c|dt

=(

12− b

) 2T∫

T

(dL log t + log(λQ2))dt

+

2T∫

T

log |L(1− b− it)− c|dt + O(log T ).


Now suppose that c 6= 1. The �rst integral on the right hand side is easilycalculated by elementary means. The second integral equals by (14)

−2T∫

−T

log |L(1− b + it)− c|dt =

−2T∫

−T

log |`(1− b + it)|dt− T log |1− c|.

The integral on the right hand side turns out to be bounded for su�cientlylarge negative b by the same reasoning as in (16). Thus we get

I1 =(

12− b

)(dLT log

4T

e+ T log(λQ2)

)− T log |1− c|+ O(log T ).

By (13) in addition with the estimates for the Ij 's from the proof of theprevious theorem we obtain

Theorem 11. Let c 6= 1. Then, for su�ciently large negative b,

2π∑

T6γc62T

(βc − b) =(

12− b

)(dLT log

4T

e+ T log(λQ2)

)

−T log |1− c|+ O(log T ).

9. Riemann-von Mangoldt-type formulae

We can rewrite the sums over c-values from the previous section as follows∑

βc

(βc − b) =(

12− b

)∑

βc

1 +∑

βc

(βc − 1

2

).

The �rst sum on the right counts the number of c-values and the second onemeasures the distances of the c-values from the critical line. Let N c(T ) countthe number of c-values of L(s) with T 6 γc 6 2T . Then

Corollary 12. Let c 6= 1. As T →∞,

N c(T ) =dL2π

T log4T

e+

T

2πlog(λQ2) + O(log T ),

and ∑

T6γc6T

(βc − 1

2

)= − T

2πlog |1− c|+ O(log T ).

J. Steuding 109

Thus, the c-values, weighted with respect to their distance to the criticalline, lie asymmetrically distributed (which is not too surprising in view ofthe increasing of µL(σ) as σ → −∞). Nevertheless, our next aim is to showthat most of the c-values lie close to the critical line. Unfortunately, for thispurpose we have to assume the Lindelöf hypothesis. De�ne the countingfunctions (according multiplicities)

N c+(T, σ) = #{%c : T 6 γc 6 2T, βc > σ}

andN c−(T, σ) = #{%c : T 6 γc 6 2T, βc < σ}.

Then

Theorem 13. Let c 6= 1. Then, for any σ > max{12 , 1− 1

dL},

N c+(T, σ) ¿ T, (18)

and under assumption of the Lindelöf hypothesis, for any δ > 0,

N c−

(T,

12− δ

)+N c

+

(T,

12

+ δ

)¿ δT log T.

Proof. First of all, let σ > max{12 , 1 − 1

dL} and �x σ1 ∈ (max{1

2 , 1 −1

dL}, 1). Then

N c+(T, σ) 6

1σ − σ1

∑

βc>σT6γc62T

(βc − σ1).

The sum on the right side is less than

∑

βc>σ1

T6γc62T

(βc − σ1) ¿2T∫

T

log |`(σ1 + it)|dt + O(log T ),

where we used Littlewood's lemma and the techniques from the previoussection for the latter inequality. With view to the unconditional estimateof (14) in the proof of Theorem 9 we obtain (18). Under assumption of theLindelöf hypothesis we get analogously

N c+

(T,

12

+ δ

)¿ ε

δT log T

for any positive ε.


Next we consider N c−. Let b be a su�ciently large constant. We have

∑

βc> 12−δ

T6γc62T

(βc − b) 6(

12− b

) ∑

βc> 12−δ

T6γc62T

1 +∑

βc> 12

T6γc62T

(βc − 1

2

).

Hence∑

T6γc62T

(βc − b) =∑

βc< 12−δ

T6γc62T

(12− b + βc − 1

2

)+

∑

βc> 12−δ

T6γc62T

(βc − b)

6(

12− b

)N c(T ) +

∑

βc< 12−δ

T6γc62T

(βc − 1

2

)

+∑

βc> 12

T6γc62T

(βc − 1

2

).

The second sum on the right is bounded by εT log T by Theorem 9 for anypositive ε. Since any term in the �rst sum on the right hand side is 6 −δ,we obtain

−δN c−

(T,

12− δ

)>

∑

T6γc62T

(βc − b)−(

12− b

)N c(T ) + O(εT log T ).

In view of Theorem 11 and Corollary 12 we get

N c−

(T,

12− δ

)¿ ε

δT log T.

Putting ε = δ2 we obtain the assertion of the theorem.Thus, subject to the truth of the Lindelöf hypothesis we get by comparing

Corollary 12 and Theorem 13 for any positive ε

N c−

(T,

12− ε

)+N c

+

(T,

12

+ ε

)¿ εN c(T ),

so the c-values are clustered around the critical line for any c. In particular,we see that density theorems do not indicate the truth of the Riemann hy-pothesis. This extraordinary value distribution shows that the critical line isa so-called Julia line from the classical theory of functions. Julia improved

J. Steuding 111

the Big Picard-theorem by showing that if the analytic function f has anessential singularity at a, then there exist a real θ0 and a complex z suchthat for every su�ciently small ε > 0

C− {z} ⊂ f({a + r exp(iθ) : |θ − θ0| < ε, 0 < r < ε}).

The ray {a + r exp(iθ0) : r > 0} is called Julia line.The distribution of the c-values close to the real axis is quite regularly. It

can be shown that there is always a c-value in a neighborhood of any trivialzero of L(s), and with �nitely many exceptions there are no other in theleft half-plane (the main ingredients for the proof are Rouché's theorem andStirling's formula). Consequently, the number of these c-values having realpart in between −R and −2 is asymptotically dL

2 R. On the other side, by(12) the behaviour nearby the positive real axis is very regularly. Further,note that all results from above hold as well with respect to c-values fromthe lower half-plane.

Now let N cL(T ) count the number of c-values with respect to the region

b 6 σ 6 1, |t| 6 T . Using Corollary 12 with 2−nT for n ∈ N instead of T andadding up we get

N cL(T ) = 2

∞∑

n=1

N c(2−nT ) =(

dLπ

T logT

e+

T

πlog(λQ2)

) ∞∑

n=1

12n

+dLπ

T∞∑

n=1

log 4− n log 22n

+ O(log T ).

The appearing series are easily evaluated by 1 and 0, respectively. Hence,this summation removes the factor 4 in the logarithmic term, and we haveproved

Theorem 14. For any complex c 6= 1,

N cL(T ) =

dLπ

T logT

e+

T

πlog(λQ2) + O(log T ).

The case of the zeros of L(s) is a precise Riemann-von Mangoldt formula(1). An immediate consequence of Theorem 14 is that the multiplicity ofthe nontrivial zeros % is bounded by 1 + log(1 + |γ|). More advanced resultson the multiplicities of the zeros were obtained by Ivi¢ [13] in case of theRiemann zeta-function.

Theorem 14 should be compared with the results of Perelli [26] andLekkerkerker [19]. In the general case of c-values we have to mention Tsang.


Let NL(σ, T ) count the c-values %c = βc + iγc of L(s) with βc > σ and|γc| 6 T . Tsang [33] proved under assumption of the Riemann hypothesisthat for su�ciently small ε, �xed σ < 1

2 , T12+ε 6 H 6 T , and any complex

number c with ε 6 |1− c| 6 1ε ,

N cζ (σ, T + H)−N c

ζ (σ, T ) ∼ H

πlog

T

2π

with an error term depending on ε,H and T ; his result holds unconditionallyprovided σ 6 0. We conclude with some related results due to Selberg [29].Under assumption of the generalized Riemann hypothesis he obtained forc 6= 1 the asymptotic formula

∑

βc> 12

0<γc<T

(βc − 1

2

)=

√nL

4π32

T√

log log T

+T

4πlog

|c|1− |c|2 + O

(T

(log log log T )3√log log T

).

Furthermore, for

σ(T ) :=12− µ

√log log T

log Tand λ :=

dLµ

2√

πnL

with positive µ he proved∑

βc>σ(T )0<γc<T

(βc − σ(T ))

=12

√nLπ

exp(−πλ2)

2π+ λ− λ

∞∫

λ

exp(−πx2)dx2

T

√log log T

+

log |c|

∞∫

λ

exp(−πx2)dx− log |1− c| T

2π

+O(

T(log log log T )3√

log log T

).

From these results it can be deduced that about half of the c-values lie tothe left of the critical line, statistically well distributed at distances of order

√log log T

log T

J. Steuding 113

o� σ = 12 , and

N cL(σ(T ), T ) ∼ N c

L(T )

∞∫

−λ

exp(−πx2)dx.

Of the rest most lie quite close to the critical line at distances of order notexceeding

(log log log T )3

log T√

log log T.

This improves some results due to Selberg (unpublished) and Joyner [15]and gives a much more detailed description of the clustering of the c-valuesaround the critical line.

In the exceptional case c = 1 one has to consider

`(s) =qs

a(q)(L(s)− 1),

where q is the smallest integer greater than one such that a(q) 6= 0. Then,by a similar reasoning as for Theorem 14, one gets analogous results.

10. Nevanlinna theory

Nevanlinna theory was introduced by Rolf Nevanlinna [25] in the 1920'sto tackle the value-distribution of meromorphic functions in general. Werecall some basic facts which, for example, can be found in Nevanlinna'smonography [25], sections VI and IX.

Let f be a meromorphic function and denote the number of poles of f(s)in |s| < r by n(f,∞, r) (counting multiplicities). The number of c-values isgiven by

n(f, c, r) = n(

1f − c

,∞, r

).

Then the integrated counting function is de�ned by

N(f, c, r) =

r∫

0

(n(f, c, %)− n(f, c, 0))d%

%+ n(f, c, 0) log r.

The proximity function is de�ned as

m(f, r) =12π

2π∫

0

log+ |f(r exp(iθ))|dθ,


andm(f, c, r) = m

(1

f − c, r

),

where log+ x = max(0, log x). This function measures how close f is to thec-value on the circle |s| = r. The characteristic function is de�ned by

T(f, r) = N(f,∞, r) + m(f, r).

Furthermore, let

T(f, c, r) = N(f, c, r) + m(f, c, r).

The �rst main theorem in Nevanlinna theory states that for any c ∈ C

N(f, c, r) + m(f, c, r) = T(f, r) + O(1).

The right hand side is up to a bounded quantity independent of c. Thisremarkable result shows that the characteristic function is indeed a charac-teristic of f . The quantity

δ(f, c) := 1− lim supr→∞

N(f, c, r)T(f, r)

is called the de�ciency of the value c of f . This de�ciency is positive onlyif there are relatively few c-values. The second main theorem in Nevanlinnatheory implies the so-called de�ciency relation which states that

∑

c∈C∪{∞}δ(f, c) 6 2;

note that only for countably many values of c the de�ciency can di�er fromzero.

Only recently Ye [35] computed the Nevanlinna functions for the Riemannzeta-function. Without big e�ort we can extend his results to functionsL ∈ S̃. Actually, the Nevanlinna functions for those L(s) are determined bythe Gamma-factors in the functional equation.

Firstly, let σ0 > 1 be �xed. We write s = r exp(iθ), so σ = r cos θ. It iseasily seen that, for σ > σ0,

12π

∫

{θ : r cos θ>σ0}

log+ |L(r exp(iθ))|dθ ¿ 1.

J. Steuding 115

Further, in view of Theorem 212π

∫

{θ : 1−σ06r cos θ6σ0}

log+ |L(r exp(iθ))|dθ ¿ log r

for 1− σ0 6 σ 6 σ0; note that the Lebesgue measure of the set {θ ∈ [0, 2π] :σ = r cos θ ∈ [1 − σ0, σ]} is bounded by 1

r . Finally, we have for σ 6 1 − σ0

by the functional equation in the form of Section 3

log+ |L(r exp(iθ))| 6r∑

j=1

(log+ |Γ(λj(1− r exp(iθ)) + µj)|

+ log+ |Γ(λjr exp(iθ)) + µj)|) + O(r).

With view to Stirling's formula and Ye's estimates we get in this case12π

∫

{θ : r cos θ<1−σ0}

log+ |L(r exp(iθ))|dθ 6dLr

πlog r + O(r).

Adding the estimates for the other cases we obtain for the proximity function

m(L, r) 6dLr

πlog r + O(r).

SinceN(L,∞, r) ¿ log r, (19)

we getT(L, r) 6

dLr

πlog r + O(r). (20)

It follows from Theorem 14 that

N(L, 0, r) =dLr

πlog r + O(r). (21)

The �rst main theorem implies

N(L, 0, r) 6 T(L, 0, r) = N(L,∞, r) + m(L, r) + O(r).

In view of (20) and (21) we get an asymptotic formula for the characteristicfunction.

Theorem 15. As r →∞,

T(L, r) =dLr

πlog r + O(r).


A positive function t(r) is said to be of �nite order λ if

lim supr→∞

log t(r)log r

= λ;

t(r) is of maximum, mean or minimum type of order λ according to the upperlimit

lim supr→∞

t(r)rλ

is in�nite, �nite and positive, or zero. A meromorphic function is de�ned tobe of the same order and the same type as its characteristic function T(r, f).Consequently, L ∈ S̃ is of order one and maximum type.

Further we deduce from this and (19) for the de�ciency value of in�nity

δ(L,∞) = 1− lim supr→∞

N(L,∞, r)T(L, r)

= 1.

This is not too surprising. In view of Theorem 14 the de�ciency values forc 6= 1,∞ are equal zero.

Theorem 15 has in combination with the �rst main theorem of Nevanlinnatheory for the exceptional case c = 1 the consequence

N1L(T ) 6

dLT

πlog T + O(T ).

A more sophisticated analysis would show that this is actually an equality.However, we do not go into the details.

11. Universality

In 1975 Voronin [34] proved a remarkable universality theorem for ζ(s)which roughly states that any non-vanishing analytic function can be approxi-mated uniformly by certain purely imaginary shifts of the zeta-function in thecritical strip. In the meantime it was shown by Bagchi, Gonek, Laurin£ikas,Reich and others that there exist a plenty of universal Dirichlet series, forexample Dirichlet L-functions, Dedekind zeta-functions, Lerch zeta-functionsand L-functions associated with cusp forms. We refer to Laurin£ikas' survey[18] for a detailed overview. The author [30] proved that any function L ∈ S̃is universal.

J. Steuding 117

Theorem 16. Let K be a compact subset of the strip max{12 , 1− 1

dL} < σ <

1 with connected complement, and let g(s) be a non-vanishing continuousfunction on K which is analytic in the interior of K. Then, for any ε > 0,

lim infT→∞

1T

meas{

τ ∈ [0, T ] : maxs∈K

|L(s + iτ)− g(s)| < ε

}> 0.

By a standard use of Rouché's theorem we get as an immediate conse-quence of Theorem 16 with g(s) constant c 6= 0 that the number of c-valuesup to level T is À T . Let N c

L(α, β, T ) count the number of c-values of L(s)in the region α < σ < β and 0 < t < T . In conjunction with the estimate(18) of Theorem 13 we obtain

Corollary 17. Let c be a complex number 6= 0, 1. Then, for any α, βsatisfying max{1

2 , 1− 1dL} < α < β < 1,

N cL(α, β, T ) ³ T.

This should be compared with a theorem of Bohr & Jessen [3]. Theyproved for the Riemann zeta-function that for any α and β satisfying 1

2 <α < β < 1, the limit

limT→∞

1T

N cζ (α, β, T )

exists and is positive. In conjunction with well known classic density theo-rems it follows that zeros are indeed exceptional values.

References

[1] E. Bombieri, D.A. Hejhal, On the distribution of zeros of linear combinationsof Euler products, Duke Math. J. 80, 821�862 (1995).

[2] E. Bombieri, A. Perelli, Distinct zeros of L-functions, Acta Arith. 83, 271�281(1998).

[3] H. Bohr, B. Jessen, Über die Werteverteilung der Riemannschen Zetafunk-tion, zweite Mitteilung, Acta Math. 58, 1�55 (1932).

[4] F. Carlson, Contributions a la theorie des series de Dirichlet, Arkiv för Mat.Astr. och Fysik 16, No. 18 (1922), and 19, No. 25 (1926).

[5] K. Chandrasekharan, R. Narasimhan, The approximate functional equationfor a class of zeta-functions, Math. Ann. 152, 30�64 (1963).


[6] J. B. Conrey, The Riemann hypothesis, Notices Amer. Math. Soc. 50, 341�353 (2003).

[7] J. B. Conrey, A. Ghosh, On the Selberg class of Dirichlet series: small degrees,Duke Math. J. 72, 673�693 (1993).

[8] J. B. Conrey, A. Ghosh, Remarks on the generalized Lindelöf hypothesis,Preprint, available at www.math.okstate.edu/∼conrey/papers.shtml

[9] R. Garunk²tis, J. Steuding, On the zero distribution of Lerch zeta-functions,Analysis 22, 1�12 (2002).

[10] R. Garunk²tis, J. Steuding, Does the Lerch zeta-function satisfy the Lindelöfhypothesis?, in: Analytic and probabilistic methods in Number Theory. Proc.3rd Palanga Conf., A.Dubickas et al. (Eds.), Vilnius, TEV, 61�74 (2002).

[11] M. N. Huxley, Exponential sums and the Riemann zeta-function. IV, Proc.London Math. Soc. 66, 1�40 (1993).

[12] A. Ivi¢, The theory of the Riemann zeta-function with applications, New York,John Wiley (1985).

[13] A. Ivi¢, On the multiplicity of zeros of the zeta-function, Bull. Acad. SerbeSci. Arts, Classe Scie. Math. 24, 119�132 (1999).

[14] H. Iwaniec, P. Sarnak, Perspectives on the analytic theory of L-functions,Geom. funct. anal., special volume - GAFA, 705�741 (2000).

[15] D. Joyner, Distribution theorems of L-functions, Harlow, Essex, Pitman Re-search Notes in Mathematics (1986).

[16] J. Kaczorowski, A. Perelli, The Selberg class: a survey, in: Number theoryin progress. Proc. Zakopane Conf., Vol. 2: Elementary and analytic numbertheory, Berlin, De Gruyter, 953�992 (1999).

[17] A. Laurin£ikas, A probabilistic equivalent of the Lindelöf hypothesis, in: Ana-lytic and probabilistic methods in Number Theory. Proc. 3rd Palanga Conf.,A.Dubickas et al. (Eds.), Vilnius, TEV, 157�161 (2002).

[18] A. Laurin£ikas, The universality of zeta-functions, Acta App. Math. 78, 251�271 (2003).

[19] C. G. Lekkerkerker, On the zeros of a class of Dirichlet series, Proefschrift,van Gorcum & Comp. N.V. (1955).

[20] N. Levinson, Almost all roots of ζ(s) = a are arbitrarily close to σ = 1/2,Proc. Nat. Acad. Sci. U.S.A. 72, 1322�1324 (1975).

[21] N. Levinson, H. L. Montgomery, Zeros of the derivative of the Riemann zeta-function, Acta Math. 133 (1974), 49�65.

J. Steuding 119

[22] E. Lindelöf, Quelques remarques sur la croissance de la fonction ζ(s), Bull.Sci. Math. 32, 341�356 (1908).

[23] S. Mandelbrojt, Dirichlet series, Dordrecht, Reidel Publishing Company(1972).

[24] K. Matsumoto, Recent developments in the mean square theory, in: NumberTheory. Trends Math., Basel, Birkhäuser, 241�286 (2000).

[25] R. Nevanlinna, Analytic functions, Berlin, Springer (1970).

[26] A. Perelli, General L-functions, Ann. Mat. Pura Appl. 130, 287�306 (1982).

[27] H. S. A. Potter, The mean values of certain Dirichlet series I, Proc. LondonMath. Soc. 46, 467�468 (1940).

[28] B. Riemann, Über die Anzahl der Primzahlen unterhalb einer gegebenenGrösse, Montasber. Preuss. Akad. Wiss. Berlin, 671�680 (1859).

[29] A. Selberg, Old and new conjectures and results about a class of Dirichletseries, in: Proc. Amal� Conf. on Analytic Number Theory, E. Bombieri et al.(Eds.), Salerno, Università di Salerno, 367�385 (1992).

[30] J. Steuding, Universality in the Selberg class, in: Proc. "Analytic Num-ber Theory and Diophantine equations". Max Planck-Institut Bonn, Bonnermath. Schriften, (to appear) = preprint www.math.uni-frankfurt.de/∼steu-ding/steuding.shtml

[31] E. C. Titchmarsh, The theory of functions, Oxford University Press, 2nd ed.(1939).

[32] E. C. Titchmarsh, The theory of the Riemann zeta-function, Oxford Univer-sity Press, 2nd ed., revised by D.R. Heath-Brown (1986).

[33] K. M. Tsang, Distribution of values of the Riemann zeta-function, PhD thesis,Princeton (1984).

[34] S. M. Voronin, Theorem on the 'universality' of the Riemann zeta-function,Izv. Akad. Nauk SSSR, Ser. Matem. 39, 475�486 (1975) (in Russian) = Math.USSR Izv. 9, 443�445 (1975).

[35] Z. Ye, The Nevanlinna functions of the Riemann zeta-function, J. Math. Ana-lysis Appl. 233, 425�435 (1999).

Apie L-funkciju� reik²miu� pasiskirstym¡J. Steuding


Straipsnyje nagrinejamas L-funkciju� i² Selbergo klases poklasio reik²miu� pasis-kirstymas. I�rodoma Rymano-von Mangoldto tipo formule c-reik²mems ir apskai-£iuojamos ju� Nevanlinos funkcijos.

Rankra²tis gautas2003 07 04

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