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On the value-distributions of logarithmic derivatives of Dedekind zeta functions

We study the distributions of values of the logarithmic derivatives of the Dedekind zeta functions on a fixed vertical line. The main object is determining and investigating the density functions of such value-distributions for any algebraic number field. We construct the density functions as the Fourier inverse transformations of certain functions represented by infinite products that come from the Euler products of the Dedekind zeta functions.


Introduction
Let s = σ + it be a complex variable, and let ζ(s) denote the Riemann zeta function. The study of the value-distribution of ζ(s) is a classical topic in number theory. After several results on the values of log ζ(s) and (ζ ′ /ζ)(s), Bohr and Jessen [1] arrived at the following result. Fix a rectangle R in the complex plane whose edges are parallel to the axis. For any T > 0, let V σ (T, R) be the Lebesgue measure of the set of all t ∈ [−T, T ] for which log ζ(σ + it) belongs to R. Then they proved the existence of the limit for any σ > 1/2. The method of Bohr and Jessen was improved by Jessen and Wintner [13]. Kershner and Wintner [14] proved an analogous result for (ζ ′ /ζ)(s). Bohr and Jessen also proved that the limit W σ (R) can be written as an integral containing a certain continuous and non-negative function F σ (z) as follows: where z = x + iy. The existence of the limit (1.1) was generalized to L-functions of cusp forms and Dedekind zeta functions by Matsumoto [18,19,20]. On the other hand, the generalization of the integral formula (1.2) has been restricted within the case of Dedekind zeta functions associated with Galois extensions of Q [20]. Then, in the last decade Ihara studied the various averages of the values of log L(s, χ) and (L ′ /L)(s, χ) of L-functions L(s, χ) associated with characters χ. Let K be any global field and χ a character on K, and fix s = σ + iτ . In [8], he considered the existence of the function M σ (z) for which the formula (1.3) Ave χ Φ L ′ (s, χ) L(s, χ) = C Φ(z)M σ (z)|dz| holds for a sufficiently wide class of test functions Φ, where |dz| denotes the measure (2π) −1 dxdy. The meanings of the averages Ave χ are given in [8] for the three cases: (A) The field K is either Q, an imaginary quadratic field, or a function field over F q . The character χ runs over Dirichlet characters on K; (B) The field K is a number field having at least two archimedean primes, and χ runs over normalized unramified Grössencharacters, which are defined in [8], of K; (C) K = Q and χ = χ t , where t ∈ R and χ t (p) = p −it for each prime number p.
Here we refer only to the case (C), and there If we take Φ as the characteristic function of a rectangle R, we obtain a result analogous to the integral formula (1.2) for (ζ ′ /ζ)(s). In each cases, Ihara constructed M σ (z) and succeeded to prove the formula (1.3) for σ > 1 or σ > 3/4 if K is a number field or a functional field, respectively. Ihara also studied the Fourier transformM σ (w) defined bỹ is the additive character of C defined by using the inner product z, w = RzRw + IzIw = R(zw).
In the above three cases, Ihara noted that the functionM σ (w) has an infinite product representation that comes from the Euler product of L(s, χ).
The analogue of Ihara's results for log L(s, χ) was proved by Ihara and Matsumoto [9,10] in the case (A) and (C). Moreover, the condition for σ was weakened to σ > 1/2 unconditionally under some revisions of the meanings of averages and restrictions for a class of test functions [9,10,12]. Today there are several variations of the results of Ihara-Matsumoto studied by Ihara-Matsumoto [11], Lebacque-Zykin [16], Matsumoto-Umegaki [21], and Mourtada-Murty [23], however, a generalization of the case (C) to an arbitrary number field is not known by a technical difficulty as well as the formula (1.2). We solve such difficulties by the method of Guo [4] below.
The functions F σ from Bohr-Jessen and M σ from Ihara-Matsumoto in the case (C) are regarded as the density functions corresponding to the value-distribution of log ζ(σ + it) or (ζ ′ /ζ)(σ + it). Such a density function had been also studied by Guo [4], who proved the following. Write Φ(x + iy) = φ(x, y), where φ : R 2 → R is an infinitely differentiable and compactly supported function. Let σ 1 be a large fixed positive real number, and let θ, δ > 0 with δ + 3θ < 1/2. Then the following formula holds: andχ denotes the Fourier transform of χ. The error term E is estimated as Guo [5] also proved several results on the density of zeros of ζ ′ (s) as applications of his studies on the density functionχ(u, v). Guo's method is different from the ones of Bohr-Jessen and Ihara-Matsumoto in some aspects (see § §2.3 below). By the differences, we may apply Guo's method to cases where the generalization has technical difficulty through the other two methods. In the present paper, we prove an analogue of Guo's result in the case of Dedekind zeta function associated with an arbitrary number field, mixed some notations from the papers of Ihara and Matsumoto. It seems difficult to prove such a result by the method of [18,19,20].
Although our proof is largely same as Guo's original proof, sometimes we arrange the argument for a cost of the treatment of number fields.
The paper consists of six sections. In Section 1 as above, we discussed the previous works to the theory of the value-distributions that we consider. In Section 2, we states the results of the paper after setting up some notations. The main result is Theorem 2.3 bellow, which is an analogue of the result in [4] for the Dedekind zeta functions. This theorem is deduced from Propositions 2.1 and 2.2. We also discuss differences between the methods of the present paper and of Bohr-Jessen or Ihara-Matsumoto. In Section 3, we prove Proposition 2.1. The method of the proof is almost the same as [4], except for using some results on the density of the zeros of the Dedekind zeta functions. In Section 4, we prove Proposition 2.2. We prove this proposition by following the method of [4] again. At the final step in the proof, it is necessary to give a lower bound on a certain Dirichlet series supported on prime powers, which is more complicated in the case we consider than in [4]. To give an available lower bound, we use the Chebotarev density theorem, and this idea is one of the most important parts at the present paper. Section 5 is the reminder work to the proof of Theorem 2.3. This is just a simple Fourier analysis. In Section 6, we discuss further results obtained by the same method as in the proof of Theorem 2.3. We have the analogue of Theorem 2.3 for L-functions of a primitive Dirichlet character or a primitive cusp form.

Statements of results
2.1. Dedekind zeta functions. Let K be an algebraic number field of degree d. The Dedekind zeta function of K is defined by where a and p run through all non-zero ideals and prime ideals of K, respectively, and N (a) denotes the norm of a. The series and product converge absolutely for σ > 1, and the function has a holomorphic continuation to the whole complex plane except for a simple pole at s = 1. Moreover, it satisfies the functional equatioñ Here D K denotes the discriminant of K, r 1 is the number of real places of K, and r 2 is the number of complex places of K.
For a rational prime number p, we write the prime ideal factorization of p in K as with N (p j ) = p f (pj ,p) for j = 1, . . . , g(p). Then the product in (2.1) is written as Hence we have in the half-plane σ > 1, where the function Λ K (n) is supported on prime powers and Due to the fundamental equation e(p 1 , p)f (p 1 , p) + · · · + e(p g(p) , p)f (p g(p) , p) = d, the function Λ K (n) is bounded by dΛ(n), where Λ(n) = Λ Q (n) denotes the usual von Mangoldt function. This upper bound for Λ K (n) is an essential tool in the following study.

Notations and statements of results. Let
for any ǫ > 0, and let ψ w (z) denote the additive character as in (1.4). We define for w = u + iv. At first, we prove the following result using the function m K (u, v, σ).
The main result of the present paper is stated by using the Fourier inverse transformation of m K (u, v, σ). Thus we next consider the estimates on m K (u, v, σ). Proposition 2.2. Let σ 1 be a large fixed positive real number, and suppose σ ∈ (1/2, σ 1 ]. Then there exists J K (σ) > 0 such that for all u, v ∈ R with |u| + |v| ≥ J K (σ) and for all non-negative integers n and m, we have Here the constant J K (σ) depends only on K and σ, and the positive constant c K (σ 1 ) depends only on K and σ 1 . The implied constant depends only on n and m.
To state the main result, we prepare the Fourier transformations and the class of test functions. The measures |dz| and |dw| denote (2π) −1 dxdy and (2π) −1 dudv, where z = x + iy and w = u + iv, respectively. For an integrable function f (z) on C, we define the Fourier transformation f ∧ (w) and the Fourier inverse transformation f ∨ (w) as We usually write f ∧ asf . Then, according to the definition in [10], let Λ be the set of all functions f such that f and f ∧ belong to L 1 ∩ L ∞ , and the formula (f ∧ ) ∨ = f holds. We see that the Schwartz class on C is included in the class Λ, hence any infinitely differentiable and compactly supported function belongs to Λ.
By Proposition 2.2, m K (u, v, σ) belongs to Λ as a function of w = u + iv for u, v ∈ R. Therefore, we define the function M K,σ (z) as the Fourier inverse transformation of m K (u, v, σ), that is We finally state the main result of the paper.
Theorem 2.3. Let σ 1 be a large fixed positive real number. Let θ and δ be two positive real numbers with δ + 3θ < 1/2. Then for each ǫ, there exists a positive real number T 0 = T 0 (θ, δ, σ 1 , ǫ, K) such that for all T ≥ T 0 and for all σ in the interval The implied constant depends only on K and σ 1 . 2)] has introduced the functionM for any global field K. Then the formula (1.3) holds in the three cases described in Section 1 for M is a common density function applied in both cases (A) and (C). Otherwise, Ihara noted that M however, if K = Q, these functions do not coincide. As a result, some arguments being valid in the method of Ihara-Matsumoto are impossible, for instance, we know fewer structure of the function M K,σ (z) thanM K,σ (w).
We next consider the relation to generalizations of Bohr-Jessen type research [20]. For any fixed positive integer N , we define where P N denotes the set of rational primes until the N -th. Let V N (T, R) be the Lebesgue measure of the set of all t ∈ [0, T ] for which log ζ K (σ + it; N ) belongs to R. In [20], the proof of the existence of the limit (1.1) start with showing the limit exists. We extend W N to a probabilistic measure on C, and let Λ(w; N ) be its Fourier transform. This is calculated as If the following estimate Λ(w; N ) = O(|w| −(2+η) ) holds uniformly in N for some η > 0, we can write W (R) as in (1.2) due to Lévy's inversion formula. In order to prove this estimate, the result of Jessen and Wintner [13, Theorem 10] has been used, which we only apply to the case the Euler product is convex in the meaning in [19]. This is why the generalization of the integral formula (1.2) is restricted. Now, except for the difference between log ζ K (s) and (ζ ′ K /ζ K )(s), these functions Λ(w) and Λ(w; N ) coincide with the functionM K,σ (w) andM K,σ (w; N ), respectively, whereM K,σ (w; N ) is defined by a suitable partial product of (2.1). In this paper, we consider only the estimate ofM K,σ (w) without using any results of Jessen-Wintner. This estimate does not tell ofM K,σ (w; N ) at all.
Recently, Matsumoto and Umegaki [22] considered an alternative method for the estimate of Λ(w; N ), and they succeeded to prove the limit formula (1.2) for automorphic L-functions. This new method may be applied to prove the limit formula (1.2) for various zeta or L-functions with non-convex Euler product. On the other hand, the generalization of Ihara-Matsumoto theorem remains to be considered.

Proof of Proposition 2.1
In this section, we prove Proposition 2.1. At first, we approximate the logarithmic derivative (ζ ′ K /ζ K )(s) by some Dirichlet polynomials and replace the left hand side of (2.5) with under an acceptable error, where R is any positive real number greater than T , and x is a certain function of T that tends to infinity as T tends to infinity. Next, we consider the limit of the integral (3.1) as R tends to infinity. This limit will be written as where m K,p (u, v, σ) are functions in (2.4) and w = u + iv. Finally, we achieve the proof of Proposition 2.1 by estimating the difference between the function (3.2) and m K (u, v, σ).

Approximation by Dirichlet polynomials.
In this subsection, we sometimes assume σ > D ǫ . We define for any s ∈ C and x > 0. The goal of this subsection is the following lemma.
The proof of this lemma consists of four steps. At first, we replace (ζ ′ K /ζ K )(σ + it) in the left hand side of (3.3) with the Dirichlet polynomials Next, we consider the differences between Λ K (n) and Λ K,x (n), and replace the Dirichlet polynomials (3.4) with The third step looks different from the other ones. We replace the interval of the integrals Then the integral which we consider is the one in the right hand side of (3.3).
Step1 We prove the following lemma.
and h and x are any functions of T that tend to infinity with T . The constants c and C are positive and depend only on K and ǫ, and the implied constant depends only on K and σ 1 .
If σ > 1, we can directly approximate (ζ ′ K /ζ K )(s) from the Dirichlet series (2.2). If σ ≤ 1, we connect (ζ ′ K /ζ K )(s) to Dirichlet polynomials by supplementing terms that come from zeros and poles of zeta function. The following lemma is an analogue of the result of Selberg [26,Lemma 2]. Lemma 3.3. Let x > 1. Then for any s ∈ C excluding zeros or the pole of ζ K (s), we have Proof. Let α = max{2, 1 + σ}. We consider the integral This integral converges to as T tends to infinity, due to (2.2) and the well known formula for any c > 0. We replace the vertical line of the integral in (3.7) by the other three sides of the rectangle with vertices at Here, as usual, the function ⌊x⌋ denote the greatest integer less than or equal to x. We first consider the integrals On these contours, we have [15,Satz 194]. Then we see the integrals (3.9) and (3.10) tend to zero as T tends to infinity. The reminder work is estimating the integral We also have ζ ′ those at z = −q and z = ρ are respectively. Then the result follows.
In order to examine the term of non-trivial zeros in (3.6), we prepare the following lemma.
Lemma 3.4. Let N K (α, T ) denote the number of zeros ρ = β + iγ of ζ K (s) with β ≥ α and 0 ≤ γ ≤ T . Then for any ǫ > 0, we have where c and C are positive constants that depend only on K and ǫ, and the implied constant depends only on K.
Proof. If K = Q, the Dedekind zeta function is just the Riemann zeta function. In this case, there is a classical result of Selberg [27], which is we have Heath-Brown's result [6]: for any ǫ > 0, there exists a positive constant C = C(ǫ, K) such that The implied constant depends only on K. We again obtain the estimate (3.12), since The case of quadratic fields remains. The Dedekind zeta function of any quadratic field is the product of the Riemann zeta function and a Dirichlet L-function attached to a quadratic character. Fujii [2] showed the estimate Here N χ (α, T ) denotes the number of zeros ρ = β + iγ of the Dirichlet L-function L(s, χ) with β ≥ α and 0 ≤ γ ≤ T . Therefore, N K (α, T ) satisfies the same estimate as the case K = Q.
Proof of Lemma 3.2. We approximate (ζ ′ K /ζ K )(s) with the first term of (3.6) by estimating the reminder terms. In order to avoid the pole at s = 1, we assume t ≥ 2. Furthermore, we also assume t / ∈ B h for the purpose of the investigation of non-trivial zeros, where B h = B h (σ, T ) is the set of all t ∈ [0, T ] for which there exists a zero ρ = β + iγ of ζ K (s) such that β ≥ 1 2 (σ + D ǫ ) and |γ − t| ≤ h. Here, h is a function of T that tends to infinity as T tends to infinity.
Before proceeding to the approximation of (ζ ′ K /ζ K )(s), we estimate the measure of We come back to the approximation of (ζ ′ K /ζ K )(s). Let t ∈ B c h ∩ [2, T ]. Then the second term of (3.6) is estimated as , since any trivial zero is located at non-positive integer, whose multiplicity is at most d. For the last term, we divide the set of non-trivial zeros into two sets.
One of them consists of zeros ρ = β + iγ with |γ − t| ≥ h. If ρ belongs to such a set, we have [15,Satz 180]. If ρ belongs to the set of remaining non-trivial zeros, we have β < (σ + D ǫ )/2. We divide the reminder sum further into those zeros with |γ − t| ≥ 1 and those without. For the first sum, and for the second sum, The last inequalities in (3.13) and (3.14) are deduced from [15,Satz 181].
To complete the proof, we see that the difference By the fact |ψ w (z 1 )−ψ w (z 2 )| ≪ |w| |z 1 −z 2 | and the above estimates, we have the desired result.
Step2 The goal of this step is proving the following lemma.
and x is any function of T that tends to infinity with T . The implied constant is absolute.
Proof. By using the estimate |ψ w (z 1 ) − ψ w (z 2 )| ≪ |w| |z 1 − z 2 |, the difference is estimated as The last inequality is due to Cauchy's inequality. Then we apply the following result of [25, Theorem 6.1]: for any real T 0 and T , we have here we use the bound Λ K (n) ≤ dΛ(n) ≤ d log n. This proves Lemma 3.5.
Step3 At the third step, we prove the following lemma.
Lemma 3.6. Let σ 1 be a large fixed positive real number and A > 0. Then there exists an absolute constant T 0 > 0 such that for all R ≥ T ≥ T 0 , for all σ ∈ (1/2, σ 1 ], and for all |w| ≤ A, we have , and x and N are functions that grows with T , N being an even integer. The implied constant is absolute.
We use the following result in the proof.
Proof of Lemma 3.6. To simplify notation, let F K (t) = F K (t, σ; x) denotes the function We have by the Taylor series expansion, where N is any function of T that takes even values and increases with T . Then the integral By using the expansion the first term of (3.17) is equal to Therefore, the difference we should consider is equal to Then we investigate H(n, m). Let l = n−m. At first, suppose m and l are both non-zero. Then we have ..,n1m≤x 2 n21,...,n 2l ≤x 2 Λ K (n 11 ) · · · Λ K (n 1m )Λ K (n 21 ) · · · Λ K (n 2l ) (n 11 · · · n 1m n 21 · · · n 2l ) σ n 11 · · · n 1m n 21 · · · n 2l −it .
By using Lemma 3.7, the Dirichlet polynomial (3.20) is estimated as Consequently, we obtain that Hence the second term of (3.18) is Then Lemma 3.6 follows.
Step4 At the final step, we prove the following lemma.
Lemma 3.8. Let σ 1 be a large fixed positive real number and A > 0. Then there exists an absolute constant T 0 > 0 such that for all R ≥ T ≥ T 0 , for all 1/2 < σ ≤ σ 1 , and for all |w| ≤ A, we have , and x is any function of T that grows with T . The implied constant is absolute.
is estimated as For the first term of the right hand side of (3.21), the inner sum is calculated as Thus the first term is ≪ d(log x)x 1−2σ . The inner sum of the second term of (3.21) is ≪ p −2σ , hence the second term is Therefore, we complete the proof of Lemma 3.8.

3.2.
Limit formula of a special integral. In the previous subsection, we finally reached the integral (3.1). Next, we prove that the limit of the integral can be written as a certain finite product.
Lemma 3.9. For any σ, x > 0, and for any w = u + iv ∈ C, where the right hand side is defined by This is a simple consequence of the following result of Heath-Brown [7, Lemma 2].
Lemma 3.10. Let b i (t) (1 ≤ i ≤ k) be continuous functions of period 1 from R to C. Then there exists the limit for any real constants γ 1 , . . . , γ k . Moreover, if the numbers γ i are linearly independent over Q, then the limit is equal to Proof of Lemma 3.9. We take b p (t) as the function and γ p = − log p/2π, then Heath-Brown's lemma yields Lemma 3.9.

3.3.
Behavior of m K,p (u, v, σ) for large p. To prove Proposition 2.1, the remainder work is the replacement of m K (u, v, σ; x) with m K (u, v, σ). Keeping in mind that |m K,p (u, v, σ)| ≤ 1, we show the following estimate. Proof. We calculate m K,p (u, v, σ) in (2.4) as (2πmt) and We expand the exponential in (3.24). Due to the identities We assume p ≥ x 2 , where x = exp((log T ) 5 3 θ ). Then we find that µ and R are small when T is sufficiently large. In fact, we have for p ≥ x 2 . By the conditions for x, A, and σ, and this tends to zero as T tends to infinity. The argument for R is similar. Hence we can define log m K,p (u, v, σ), where log means the principal blanch of the logarithm. By using (3.27) log 2 p p 2σ , where the implied constant depends only on d. This deduces the following estimate and we have p≥x 2 log 2 p p 2σ ≪ σ1 x 2(1−2σ) log x (σ − 1 2 ) 2 for σ > 1/2 by the Prime Number Theorem. After adopting the choices of the functions x and A, we obtain the desired estimate.

Proof of Proposition 2.2
In this section, we regard m K,p (u, v, σ) in (3.24) as a function of complex variables u and v. At first, we examine the differentiability of m K (u, v, σ). Proof. We only consider the case of the first derivative with respect to u, since other cases are proved in a similar way. Fix a complex number v arbitrarily. We show that converges uniformly in {u ; |u| ≤ R} as y → ∞ for any fixed R > 0. As before we express m K,p (u, v, σ) as 1 − µ + R with We see that µ and R are small for large p. In fact, |µ| is less than 1/4 for p > p 0 , since |µ| ≪ v (dR log p p −σ ) 2 , here p 0 does not depend on u. The argument for R is similar. Then we can define log m K,p (u, v, σ) for p > p 0 . Hence we have here A is a positive constant that depends only on v. The summation of the right hand side taken over p converges, therefore the product (4.1) uniformly converges for u as y tends to infinity. This proves that m K (u, v, σ) is an entire function of u.
By Lemma 4.1, applying Cauchy's integral formula to m K (u, v, σ), then Proposition 2.2 comes down to the following lemma.
For given constants α, β, and c 0 = c 0 (K), let and define m and m (2) At first, we consider the estimate of m K (σ 1 ) such that for all α, β ∈ R with |α| + |β| ≥ J (1) K (σ 1 ) depends only on K and σ 1 , and the constant A K > 0 depends only on K.
Proof. We first see that By the conditions for u and v, the absolute values of their imaginary parts are less than 1/2. Hence we have d log p p σ with an absolute implied constant. Summing up over p ≤ P 0 , we obtain |m (1) Here A and A 1 > 0 are absolute constants. With the choice of P 0 , the desired estimate follows if |α| + |β| ≥ J (1) The estimate for m for all (u, v) ∈ D(α, 1/2) × D(β, 1/2). The constant J K (σ) depends only on K and σ, and the constant c K (σ 1 ) > 0 depends only on K and σ 1 .
Proof. As before we express m K,p (u, v, σ) as 1−µ+R. We want to define log m K,p (u, v, σ) again. We let ǫ > 0 sufficiently small, and we assume |α|+|β| is greater than some positive absolute constants. For p ≥ P 0 , we have (4.2) (|α| + |β|)d log p p σ ≤ (|α| + |β|)d log P 0 P σ 0 , hence (4.2) is less than ǫ if we choose c 0 = c 0 (K) appropriately. Then |µ| and |R| are less than 1/4, and log m K,p (u, v, σ) is defined and calculated as We first replace µ with the real number The error of this replacement is estimated as with an absolute implied constant, since (u, v) ∈ D(α, 1/2) × D(β, 1/2). For the estimate on R, we have for p ≥ P 0 , where all the implied constants are absolute. We next estimate |µ| 2 . Since (|α| + |β|)d log P 0 /P σ 0 is sufficiently small, we also have with an absolute implied constant. Furthermore, we clearly see that By the above estimates, we obtain where A > 0 is an absolute constant. We assume that |α| + |β| is greater than some absolute constants. Then the first term of (4.3) is less than (|α| + |β|) 2 16 As a result, we obtain Hence we have Then, we consider the lower bound on the following Dirichlet series For a rational prime number p, we write the prime ideal factorization of p as , with N (p j ) = p f (pj,p) for j = 1, . . . , g(p). Suppose that p is any rational prime that completely splits in K. Then the coefficient Λ K (p m ) is equal to d log p for any m by the formula (2.3). Thus we have where ′ means the summation restricted to primes that completely split in K. The Chebotarev density theorem gives #{p ≤ x | p is a prime number that completely splits in K} #{p ≤ x | p is a prime number}  in (4.4). At first, we calculate this term as ≤ Ad 2 (|α| + |β|)d log P 0 P σ 0 (|α| + |β|) 2 p≥P0 log 2 p p 2σ .
We replace the constant c 0 = c 0 (K) smaller so that

Proof of the Theorem 2.3
By Proposition 2.2, the function m K (u, v, σ) is integrable over C as a function of w = u + iv, then we define M K,σ (z) = C m K (u, v, σ)ψ −z (w)|dw|.
Then we prove Theorem 2.3. By the assumption on the test function Φ, we have the inversion formula Φ(z) = CΦ (w)ψ −w (z)|dw|.
Hence the integral 1 where A = (log T ) δ . By Proposition 2.1, the first term of (5.1) is equal to By the fact |M K,σ (−w)| ≤ 1 coming from the definition, the error when we replace the integral over |w| ≤ (log T ) δ with the integral over C is ≪ |w|>A |Φ(w)| |dw|.
Therefore, we complicate the proof of Theorem 2.3.

Further results on M -functions by Guo's method
According to Theorem 2.3, the function M K,σ (z) is regarded as the density function corresponding to the value-distribution of (ζ ′ K /ζ K )(σ + it). Such a density function is sometimes called "M -function" since the paper of Ihara [8]. As we have seen, Theorem 2.3 is deduced from Proposition 2.1 and 2.2. Hence, if we have the analogue of these propositions for a given zeta or L-function, then it is possible to prove the existence of the M -function on such a zeta or L-function.
For L-functions with some nice properties, the result corresponding to Proposition 2.1 holds if we suppose a result analogous to Lemma 3.4. Hence the main difficulty should exist in the poof of Proposition 2.2. See the proof of Lemma 4.4. There, we investigated the lower bound of (6.1) p≥M ∞ m=1 Λ K (p m ) 2 p 2mσ .
In general, we should consider The implied constant depends only on * and σ 1 .
The proof is completely the same as Theorem 2.3.