Distribution of zeros of twisted motivic L-functions

Andrzej Da̧browski Jacek Pomykała

0. INTRODUCTION.

Let M be a pure motive over ℚ of weight w = w(M) and of the global conductor c(M). Also given Dirichlet character χ one can define the twisted motive M_{χ}; let L(M_{χ}, s) denote the corresponding L-function.

Let C be the set of real quadratic characters χ of conductors c(χ) which are relatively prime to c(M). Let

C(D) = {χ ∈ C : c(χ) ≤ D}.

and

m_{χ}(α) = ord_{s=α} L(M_{χ}, s).

The aim of this paper is to investigate the average (over the characters from C(D)) order of zeros ρ of L(M_{χ}, s) and the regularities in their distribution. Our results generalize those of [P-P] and [P] proved for the Hecke L-functions. We assume the good (conjectural) analytic properties of the motivic L-functions L(M_{χ}, s) (see section 1). On the other hand we require the suitable mean value estimate for L(M_{χ}, s) which we can prove only for the motives of rank d(M) ≤ 2 using the argument similar as in [H-B].

The starting point of the proof is the Riemann-Weil explicit formula in the form proved by Mestre [Me] (see also [Mor]). Using directly Mestre’s device we could obtain the nontrivial results only assuming the Generalized Riemann Hypothesis for the corresponding L-functions (cf. also [B] and [G]). We avoid this assumption by proving the corresponding density estimate for the zeros of L(M_{χ}, s) (see section 3). On the other hand we use the modified test function which transmits the investigation of zeros around the critical point \frac{w+1}{2} to those around any, fixed point s_{0} = \frac{w+1}{2} + it_{0} on the critical line (see [C-P]).

The required mean value estimate has the following form

∑_{χ∈C(D)} |L (M_{χ}, \frac{w + 1}{2} + it)|^{2} ≪_{ε} (D(|t| + 1)^{A})^{1+ε},  (0.1)

where A = A(M) ≥ 1 is some positive constant and ε > 0 is arbitrarily small. Let us remark that in the case of the classical Dirichlet L-functions (d(M) = 1) the estimate (0.1) is true with D^{ε} replaced by a power of log D (see [J]). In this case Theorem 1 below holds true with any δ = δ(D) ≥ \frac{log log D}{log D}. To formulate the main result we put

R_{χ}(s_{0}, δ) = ∑_{|ρ−s_{0}|≤δ} m_{χ}(ρ)

Under the above assumptions we have Theorem 1. There exist some absolute, positive constants c_{1} and c_{2} such that for any, sufficiently small δ > 0, we have

∑_{χ∈C(D)} Rχ(s_{0}, δ) ≤ c_{1}δD log D,  (0.2)

∑_{χ∈C(D)} Rχ(s_{0}, δ) ≥ c_{2}δD log D,  (0.3)

∑_{χ∈C(D)} Rχ(0, δ) = o(D log D).  (0.4)

as D tends to infinity.

All the results of this paper could be stated (and proved) without using any motivic language. Almost all we need to know about motives concerns the conjectural description of their L- functions (see section 1).

Acknowledgements. Part of the final version of the paper was prepared when the second named author has been visiting the Max-Planck Institut für Mathematik in Bonn (Germany) in October 1997. He wishes to thank the Institute for the hospitality and support.

1. PRELIMINARIES ON MOTIVIC L-FUNCTIONS.

We will define a pure motive M over ℚ of weight w = w(M) and rank d = d(M), by specifying Betti, de Rham and l-adic realisations (for each prime l)

H_{B}(M), H_{DR}(M), H_{l}(M)

which are, respectively, vector spaces over ℚ, ℚ and ℚ_{l} of dimension d, and which are endowed with the additional structures and comparison isomorphisms [CP-R,D]. In particular H_{B}(M) admits an involution ρ_{B}, H_{l}(M) is Gal(\overline{ℚ}/ℚ)-module, and there is a Hodge decomposition into ℂ-vector spaces

H_{B}(M) ⊗ ℂ = ⊕_{i+j=w}H^{i,j}(M),

where, letting ρ_{B} act on the vector space on the left via the first factor in the tensor product, we have

ρ_{B}(H^{i,j}(M)) = H^{j,i}(M).

The archetype of a motive arises from a smooth projective variety X defined over ℚ; here we consider

H_{B}^{m}(X(ℂ)), H_{DR}^{m}(X), H_{l}^{m}(X),

(the Betti cohomology with coefficients in ℚ of the complex manifold X(ℂ), the algebraic de Rham cohomology of the algebraic variety X/ℚ, and for each prime number l, the l-adic cohomology of X viewed over \overline{ℚ}). Another example is the motive M(f) attached to a newform f by Deligne, Scholl and Jannsen.

Also, we have linear algebra operations on motives. In particular, one can consider M^{∨}, the dual motive: its l-adic representation is contragredient to that of M. Given two motives M_{1} and M_{2} one can consider tensor product M_{1} ⊗ M_{2}: its l-adic representation is the tensor product of those for M_{1} and M_{2}. We let M(m) = M ⊗ ℚ(m), where ℚ(m) denotes the Tate motive.

The L-function L(M, s) of M is defined for s with large Re s as the following Euler product

L(M, s) = ∏_{p} L_{p}(M, p^{−s}),

extended over all primes p and where

\begin{aligned}L_{p}(M, X)^{−1} & = det (1 − X · r_{l}(Fr^{−1})|_{H_{l}(M)^{Ip}} ) \\ & = (1 − α_{p}^{(1)}X)...(1 − α_{p}^{(d)}X); \end{aligned}

here r_{l} is the representation giving Gal(\overline{ℚ}/ℚ)-module structure on H_{l}(M), and Fr_{p} ∈ Gal(\overline{ℚ}/ℚ) is the Frobenius element at p, defined modulo conjugation and modulo the inertia subgroup I_{p} ⊂ G_{p} ⊂ Gal(\overline{ℚ}/ℚ) of the decomposition group G_{p}. We make the standard hypothesis that the coefficients of L_{p}(M, X)^{−1} do not depend on l ≠ p.

We next state the conjectural functional equation for L(M, s). First let’s recall the definition of the Euler factor L_{∞}(M, s) at infinity. We suppose that if w(M) is even and H^{\frac{w}{2} , \frac{w}{2}} (M) ≠ 0, the involution ρ_{B} acts on it by a scalar ±1. Put

Γ_{ℝ}(s) = π^{− \frac{s}{2}} Γ(\frac{s}{2}), Γ_{ℂ}(s) = 2(2π)^{−s}Γ(s), h(i, j) = dim_{ℂ}H^{i,j}(M),

and, if w(M) is even and H^{\frac{w}{2} , \frac{w}{2}} (M) ≠ 0, we put

r^{α}(M) = dim_{ℂ}\{x ∈ H^{\frac{w}{2} , \frac{w}{2}} (M) : ρ_{B}(x) = α(−1)^{\frac{w}{2}} x\}, α = ±1.

We define

L_{∞}(M, s) = L_{∞,ℝ}(M, s)L_{∞,ℂ}(M, s),

where

L_{∞,ℝ}(M, s) = Γ_{ℝ}(s − \frac{w}{2} )^{r^{+}(M)}Γ_{ℝ}(s + 1 − \frac{w}{2} )^{r^{−}(M)},

and

L_{∞,ℂ}(M, s) = ∏_{i<j} Γ_{ℂ}(s − i)^{h(i,j)}.

We put

Λ(M, s) = L_{∞}(M, s)L(M, s).  (1.1)

We also assume the existence of a finite set of bad primes S such that, for each prime l and each r ∉ S ∪ {l}, the inertia group I_{r} acts trivially on H_{l}(M). For each r ∈ S, and each l ≠ r, we additionally assume that the conductor of H_{l}(M) as a representation of the local Galois group D_{r} does not depend on l. The global conductor c(M) of M is then defined as the product of local conductors at primes r ∈ S.

For each p ∉ S, we assume that the reciprocal complex roots of L_{p}(M, X)^{−1} have absolute value equal to p^{\frac{w}{2}} ; according to the monodromy conjecture, a more refined version of the purity should hold also at bad primes. It implies, that L(M, s) converges in the complex half-plane Re s > \frac{w}{2} + 1.

Conjecture 1. [D, CP-R] The function Λ(M, s) has a meromorphic continuation over the whole complex plane, and satisfies the functional equation

Λ(M, s) = ϵ(M)c(M)^{\frac{w+1}{2} −s}Λ(M^{∨}(−w), w + 1 − s),

where ϵ(M) is a complex number of modulus 1 which is independent of s. Moreover, L(M, s) is holomorphic unless both w(M) is even and ℚ (−\frac{w}{2} ) is a direct summand of M, in which case L(M, s) may possibly have a pole at the point s = \frac{w}{2} + 1.

We also conjecture that Λ(M, s) minus the singular part is bounded in every vertical strip:

−∞ < σ_{0} ≤ Re(s) ≤ σ_{1} < ∞.

By Stirling’s formula there exist an absolute constant c > 0 such that for σ_{0} ≤ Re s ≤ σ_{1} we have

L_{∞}(M^{∨}(−w), w + 1 − s)(L_{∞}(M, s))^{−1} = O((Im s)^{c}),

Using the above functional equation and Phragmen-Lindelöf principle we obtain for s in the above strip

L(M, s) = O((Im s)^{c_{0}})),  (1.2)

where c_{0} is some positive constant.

Let χ be a quadratic character of conductor c(χ). To such χ one can associate a motive [χ] over ℚ of weight 0 and rank 1 so that L([χ], s) = L(s, χ). Also given any (pure) motive M, we can define M_{χ} = M ⊗ [χ]; it is again a motive over ℚ of the same weight and rank so that we have

L(M_{χ}, s) = ∏_{p} L_{p}(M, χ(p)p^{−s}).

Moreover we have Lemma 1. ([CP-R]; lemma 3.1 (3) ) Assume that (c(χ), c(M)) = 1. Then

c(M_{χ}) = c(M)c(χ)^{d(M)}.

The above conjectures will be assumed in the following sections.

2. THE MEAN VALUE ESTIMATE.

We slightly modify Heath-Brown’s arguments ([H-B], section 10). The main tool is the following large sieve type estimate ([H-B], Corollary 3)

∑_{χ∈C(D)}| ∑_{n≤N} a_{n}χ(n)|^{2} ≪ (DN)^{ε}N(D + N) max_{n≤N} |a_{n}|^{2}.  (2.1)

Define A to be the number of Γ_{ℝ}′s plus twice number of Γ_{ℂ}′s factors in L_{∞}(M, s). We will prove (cf. [P-P]) Theorem 2. Let ε > 0 and assume that d = d(M) ≤ 2. Then for any s = σ + it with σ ∈ [\frac{w+1}{2}, \frac{w+2}{2}] we have

∑_{χ∈C(D)} |L(M_{χ}, σ + it)|^{2} ≪_{ε} (D + (D(|t| + 1)^{A})^{w+2−2σ})(D(|t| + 1))^{ε}.

Proof. Let

S^{+}(D, s) = ∑_{χ∈C(D)} |L(M_{χ}, s)|^{2} ,

S^{−}(D, s) = ∑_{χ∈C(D)} |L(M_{χ}^{∨}(−w), s)|^{2}

and ν^{±}(σ) be the infimum of those exponents ν^{±} respectively, for which

S^{±}(D, σ + it) ≪ (D + (DT^{A})^{w+2−2σ})(DT^{A})^{ν^{±}}

uniformly in D and t, where T = |t| + 1. We wil show that ν^{±} ≤ 0. Similarly as in [H-B] p.268 we have for any X > 0 and \frac{w+1}{2} ≤ α ≤ σ ≤ \frac{w+2}{2}

S^{+}(D, s) ≪ ∑_{χ∈C(D)} |∑_{n} a_{n}χ(n)n^{−s}e^{− \frac{n}{X}} |^{2} + X^{2(α−σ)} ∫_{ℝ} S^{+}(D, α + iu)e^{−|u−t|}du. (2.2)

Using the functional equation we obtain

S^{+}(D, α + iu) = ∑_{χ∈C(D)} |U(D, α + iu)|^{2}|L(M_{χ}^{∨}(−w), w + 1 − α − iu)|^{2}

where

U = U(D, α+iu) = ϵ(M_{χ})c(M_{χ})^{\frac{w+1}{2} −(α+iu)}L_{∞}(M_{χ}^{∨}(−w), w+1−s)(L_{∞}(M_{χ}, s))^{−1} (2.3)

The last quotient contains the Γ-factors of type

Γ(\frac{1}{2}(w + 1 − s + a))(Γ(\frac{1}{2}(s + a))^{−1}

or the similar ones with double arguments. In the second case we use the duplication formula

Γ(s) = Γ(\frac{s}{2})Γ(\frac{s + 1}{2})2^{s−1}π^{− \frac{1}{2}}

to have \frac{s}{2} in all arguments of Γ-factors. Since A is the total number of them, then using the equality

Γ(σ + it) = \sqrt{2π}|t|^{σ− \frac{1}{2}} e^{− \frac{π}{2} |t|}(1 + O(|t|^{−1})

valid for σ_{0} ≤ σ ≤ σ_{1} and t > t_{0}(σ_{1}, σ_{2}), we deduce that the ratio of the L_{∞}-terms in (2.3) is ≪_{σ} |t|^{A( \frac{w+1}{2} −σ)}. Hence by Lemma 1 we have

\begin{aligned}|U|^{2} & ≪ (c(χ)^{d})^{w+1−2α}|t|^{A(w+1−2α)} \\ & ≪ (D^{d}|t|^{A})^{w+1−2α}. \end{aligned}

The summation over n ≥ X(log DT)^{2} in (2.2) is negligible due to the decay of exp(−\frac{n}{X}). We split the remaining part of the sum into the intervals of type (N, 2N] and apply (2.1) to obtain

∑_{χ∈C(D)} | ∑_{N<n≤2N} a_{n}χ(n)n^{−s}e^{− \frac{n}{X}} |^{2} ≪ (D + X^{w+2−2σ})(DX)^{ε}

for \frac{w+1}{2} ≤ σ ≤ \frac{w+2}{2}. Let Δ = σ − \frac{w+1}{2} and η > 0 be sufficiently small and X = (DT^{A})^{1+η}. By the definition of ν^{±} and the above estimate we obtain

\begin{aligned}S^{+}(D, s) & ≪ ((DT^{A})^{w+2−2σ} + D)(DT^{A})^{η+3ε} \\ & + (DT^{A})^{2(α−σ)(1+η)}(D^{d}T^{A})^{w+1−2α}(D + (DT^{A})^{2α−w})(DT^{A})^{ν^{−}(w+1−α)+ε}. \end{aligned}

If \frac{w+1}{2} + η ≤ σ ≤ \frac{w+2}{2} then we choose α = w + 1 − σ getting

\begin{aligned}S^{+}(D, s) & ≪ ((DT^{A})^{w+2−2σ} + D)(DT^{A})^{η+3ε} \\ & + (DT^{A})^{−4Δ(1+η)}(D^{d}T^{A})^{2Δ}D(DT^{A})^{ν^{−}(w+1−α)+ε} \end{aligned}  (2.4)

and by the inequality d ≤ 2 we obtain, by comparing the exponents that ν^{+}(σ) ≤ max(η + 3ε, −4η^{2} +ε+ν^{−}(σ)). Writting the same for the dual motive we conclude by the symmetry that ν^{−}(σ) ≤ max(η+3ε, −4η^{2}+ε+ν^{+}(σ)). Choosing 2η > \sqrt{ε} we conclude that ν^{+}(σ) ≤ 0 in this case.

If \frac{w+1}{2} ≤ σ ≤ \frac{w+1}{2} + η then choosing α = \frac{w+1}{2} − η we have by the above that ν^{+}(w + 1 − α) ≤ 0 and by (2.4) we obtain

\begin{aligned}S^{+}(D, s) & ≪ ((DT^{A})^{w+2−2σ} + D)(DT^{A})^{η+3ε} + (D^{d}T^{A})^{w+1−2α}(D + (DT^{A})^{2α−w}) \\ & ≪ ((DT^{A})^{w+2−2σ} + D)(DT^{A})^{η+3ε} + D(DT^{A})^{4η+ε}, \end{aligned}

hence in this case also ν^{+}(σ) ≤ 0 . Analogously we prove that ν^{−}(σ) ≤ 0. This completes the proof of Theorem 2.

3. APPLICATION OF THE ZERO DETECTION METHOD.

In this section we show how to deduce from (0.1) and Theorem 2, the zero density estimate (see Theorem 3 below). We apply the technique familiar from the Montgomery’s book [Mo] (see Chapter 12). Since we are not interested in obtaining the best exponent here, the corresponding steps are essentially simpler. Let

N(σ, T, χ) = ♯{ρ : L(M_{χ}, ρ) = 0, Re ρ ≥ σ, |γ| ≤ T }.

The result is the following Theorem 3. Let 0 ≤ δ ≤ 1/2 be fixed. Then for any ε > 0 and D > D_{0}(M, ε) we have

∑_{χ∈C(D)} N (\frac{w + 1}{2} + δ, T, χ) ≪_{ε} (DT^{A+3})^{1+ε−δ}.

Proof. We start from the Euler product representation

L(M_{χ}, s) = ∏_{p}(1 − α_{p}^{(1)}χ(p)p^{−s})^{−1} ...(1 − α_{p}^{(d)}χ(p)p^{−s})^{−1},

in the corresponding half-plane, where |α_{p}^{(i)}| ≤ p^{\frac{w}{2}} (1 ≤ i ≤ d). In this half-plane the inverse

\frac{1}{L(M_{χ}, s)} = ∑_{n≥1} b_{n}χ(n)n^{−s},

is determined by the coefficients b_{n} satisfying the bound

|b_{n}| ≤ τ_{d}(n)n^{\frac{w}{2}} ,

where τ_{d}(n) is the number of representations of n as a product of d-factors.

Let B_{X}(M_{χ}, s) = ∑_{n≤X} b_{n}χ(n)n^{−s}. Then

L(M_{χ}, s)B_{X}(M_{χ}, s) = ∑_{n≥1} e_{n}χ(n)n^{−s},

where

e_{n} = e_{n}(X) = \{ \begin{matrix}1 & n = 1 \\ 0 & 1 < n ≤ X \\ ≪ τ_{2d}(n)n^{\frac{w}{2}} , & n > X. \end{matrix}

Therefore applying the Mellin transform we obtain ( for sufficiently large Y ≤ \frac{X}{log^{2} X})

\begin{aligned}\frac{1}{2πi} ∫_{(α)} L(M_{χ}, s + v)B_{X}(M_{χ}, s + v)Y^{v}Γ(v)dv & = ∑_{n≥1} e_{n}χ(n)n^{−s}e^{− \frac{n}{Y}} \\ & = e^{\frac{−1}{Y}} + O(X^{−1}). \end{aligned}

where the integration above is taken along the vertical line Re v = α with α > \frac{w}{2} + 1. Moving the line of integration Re v = α to Re v = \frac{w+1}{2} − Re s, we pass a simple pole of gamma function at v = 0 with residuum L(M_{χ}, s)B_{X}(M_{χ}, s), and possibly a pole of L(M_{χ}, s + v) at v = \frac{w}{2} + 1 − s. Since the last condition may happen at most for one character χ ∈ C, the corresponding contribution does not depend on D. Therefore

\begin{matrix}1 & ≪ L(M_{χ}, s)B_{X}(M_{χ}, s)|+ \\ \frac{1}{2π} & ∫ & {∞ \atop −∞} & |(LB_{X})(M_{χ}, \frac{w + 1}{2} + iu)Y ^{\frac{w+1}{2} −σ+i(u−t)} Γ (\frac{w + 1}{2} − σ + i(u − t))du)| \end{matrix} (3.1)

where s = σ + it. We apply the above inequality with s = ρ such that Re ρ ≥ \frac{w+1}{2} + δ. Moreover we assume that δ ≥ (log D)^{−1} since otherwise Theorem 3 is trivial. Let |t| + 1 ≤ T. Then for the gamma factor we have

Γ(\frac{w + 1}{2} − σ + i(u − t)) ≪ log D exp (−c|u|/T)

with some c > 0. Summing both sides of (3.1) over the zeros ρ and then over χ ∈ C we see that the RHS of (3.1) contributes at most to

∑_{χ∈C} Y^{−δ}N(\frac{w + 1}{2} + δ, T, χ) ∫_{−∞}^{∞} |(LB_{X})(M_{χ}, \frac{w + 1}{2} + iu)e^{−c|u|/T} | du.

Hence by the the trivial bound

N(\frac{w + 1}{2} + δ, T, χ) ≪ T log DT

and the Cauchy’s inequality we obtain that

\begin{aligned}∑_{χ∈C(D)} N (\frac{w + 1}{2} + δ, T, χ) & ≪ Y^{−δ}T log(DT) ∫_{−∞}^{+∞} ( ∑_{χ∈C(D)} |L(M_{χ}, \frac{w + 1}{2} + iu)|^{2})^{\frac{1}{2}} \\ & × (∑_{χ} |B_{X}(M_{χ}, \frac{w + 1}{2} + iu)|^{2})^{\frac{1}{2}} e^{−c|u|/T} du \end{aligned}

(3.2)

Now taking X = DT^{A} and Y = \frac{X}{log^{2} X} , Theorem 3 follows directly from (0.1) and Theorem 2.

4. THE EXPLICIT FORMULA.

Here we follow the approach by Mestre [Me]. Given an integrable function f : ℝ → ℂ with compact support, define by the capital letter

F (s) = ∫_{−∞}^{+∞} f(x)e^{sx}dx

the suitable Laplace transform. For a function f of real or complex variable and for λ > 1 we define f_{λ}(z) = f(\frac{z}{λ}). Let

\begin{aligned}Λ_{1}(s, χ) & = Λ(M_{χ}, s)(c(M)c(χ)^{d(M)}))^{\frac{s}{2}} , \\ Λ_{2}(s, χ) & = Λ(M^{∨}(−w)_{χ}, s)(c(M)c(χ)^{d(M)})^{\frac{s}{2}} , \end{aligned}  (4.1)

Applying Cauchy’s integration formula we have (for a suitable weight function Φ) the equality

∑_{ρ} m_{χ}(ρ)Φ(ρ) = \frac{1}{2πi} ∫_{( \frac{w+1}{2} )} (\frac{Λ′_{1}}{Λ_{1}} (s, χ)Φ(s) + \frac{Λ′_{2}}{Λ_{2}} (s, χ)Φ(w + 1 − s)) ds

where the above sum is understood as a symmetric limit of | Im ρ| ≤ T as T tends to infinity. Mestre’s explicit formula is derived when spliting the logarithmic derivative of Λ_{1} and Λ_{2} into the conductor term, term containing the gamma-factors (archimedean) and term containing the L-function (non-archimedean). For Φ(s) = F (s − \frac{w+1}{2}) it has the following form (see [Me] p.213)

∑_{ρ} m(ρ)F (ρ − \frac{w + 1}{2}) = log(c(M)c(χ)^{d(M)/2}) − ∑_{i} I(a_{i}, b_{i}) − ∑_{i} I(a′_{i}, b′_{i})

− ∑_{l≥1,p−prime,i=1,2,...d(M)}(α_{p}^{(i)l} + β_{p}^{(i)l})f(l log p)\frac{log p}{p^{(w+1)l/2}}

(4.2)

Here both, the contribution of the gamma factors I(a′_{i}, b′_{i}) and β_{p}^{(i)} (the inverse p-roots of the dual motive) correspond to the dual part Λ_{2}(s, χ) (see (1.1) and (4.1)). We apply (4.2) with f(x) multiplied by an additive character e^{it_{0}x}, where f(x) ∈ C_{0}^{k}(ℝ) (k ≥ 2) is a real, even function such that f(0) = 1. Regarding f(x)e^{it_{0}x} as a new test function (which is not real as in Mestre’s formulation) the formula (4.2) still holds if we replace F (s − \frac{w+1}{2}) by F (s − s_{0}) and multiply the coefficients α_{p}^{(i)}, β_{p}^{(i)} by the corresponding additive characters (cf.[P]). To see this let us consider only the archimedean part. The integral to compute is the following

\begin{aligned}I_{s_{0}}(a, b) & = ∫_{(𝔠)} ψ(as + b)F (s − s_{0})ds \\ & = ∫_{−∞}^{+∞} ψ((a𝔠 + b) + ia(t + t_{0}))f̂(t)dt \end{aligned}

where ψ(s) = \frac{Γ′}{Γ} (s), a = 1/2 or 1, b is a half-integer and 𝔠 = \frac{w+1}{2}. To evaluate it we apply the equality due to Gauss

ψ(σ + iat) − ψ(σ) = ∫_{−∞}^{+∞} k(x/a)\frac{1 − e^{itx}}{x} dx,

where σ = a\frac{w+1}{2} + b, k(x) = xe^{−σx}/(1 − e^{−x}) for x > 0 and 0 otherwise. Therefore changing the order of integration we obtain

\frac{1}{2π} ∫_{−∞}^{+∞} f̂(t)(ψ(σ + ia(t + t_{0}) − ψ(σ))dt

\begin{matrix}= ∫_{−∞}^{+∞} \frac{k(x/a)}{x} ∫_{−∞}^{+∞} f̂(t)(1 − e^{i(t+t_{0})x})dtdx \\ = ∫_{−∞}^{+∞} (\frac{f(ax)e^{−σx}}{1 − e^{−x}} e^{iat_{0}x} − \frac{f(0)e^{−x}}{x} ) dx \end{matrix}

Hence we get the equality

\frac{1}{2πi} ∫_{(𝔠)} ψ(as + b)G(s − \frac{w + 1}{2})ds = − ∫_{−∞}^{+∞} (\frac{g(ax)e^{−σx}}{1 − e^{−x}} − g(0)\frac{e^{−x}}{x} ) dx,

where g(x) = f(x)e^{it_{0}x}. Replacing (f, F) by (f_{λ}, λF_{\frac{1}{λ}}) and summing both sides of (4.2) over χ ∈ C(D) we obtain by Lemma 2 of [P-P] the equality

λ ∑_{χ∈C(D)} ∑_{ρ} m_{χ}(ρ)F_{\frac{1}{λ}} (ρ − s_{0}) = d(M)f_{λ}(0) ∑_{χ∈C(D)} log c(χ) + O_{s_{0}}(Dλ + \sqrt{D}e^{2λ}). (4.3)

5. COMPLETION OF THE PROOF.

In order to use the density estimate we first split the RHS of (4.3) into sums over | Re(ρ − s_{0})| ≤ 1/λ (main term) and | Re(ρ − s_{0})| > 1/λ (error term). From the classical arguments (see e.g. [Pe]) we know that m(ρ) < log(D| Im ρ|), hence the error term restricted additionally by the condition | Im ρ − t_{0}| ≤ 1 contributes at most to

log(DT_{0}) ∑_{χ} ∑_{| Re ρ−𝔠|> \frac{1}{λ} ,| Im ρ−t_{0}|≤1} e^{λ(Re ρ−𝔠)} ≪ log(DT_{0}) ∑_{χ} ∫𝔠+λ^{−1}^{\frac{w+2}{2}} e^{λ(σ−𝔠)}dN(σ, T_{0}, χ)

≪ log(DT_{0}) ∑_{χ}(N(𝔠 + λ^{−1}, T_{0}, χ) + λ ∫𝔠+λ^{−1}^{\frac{w+2}{2}} e^{λ(σ−𝔠)}N(σ, T_{0}, χ))dσ

where T_{0} = |t_{0}| + 1 and χ varies in the set C(D). Now using the density estimate we conclude that the corresponding contribution is

O_{ε}(λT_{0}(DT_{0}^{A+3})^{1+ε− \frac{1}{2λ}} ).

Choosing f ∈ C_{0}^{k}(ℝ) with sufficiently large k = k(A), the decay of F (s) over the imaginary lines shows that the above estimate holds without the additional restriction | Im ρ−t_{0}| ≤ 1. To complete the proof of Theorem 1 we choose F_{1} and F_{2} such that both Re F_{1}(s) and Re F_{2}(s) are ≫ 1 for |s| ≤ 1, both are positive for | Re s| ≤ 1 and | Im s| ≤ 1, and in the case when | Re s| ≤ 1 and | Im s| > 1 we have Re F_{1}(s) > 0 and Re F_{2}(s) < 0. In the first case we obtain by (4.3)

λ ∑_{χ} ∑_{| Re ρ−s_{0}|≤1/λ} m_{χ}(ρ) ≤ c_{1}D log D + O_{ε,s_{0}}(Dλ + \sqrt{D}e^{2λ} + λD^{1+ε− \frac{1}{2λ}} )

while in the second

λ ∑_{χ} ∑_{| Re(ρ−s_{0}|≤1/λ,| Im(ρ−s_{0})|≤1/λ} m_{χ}(ρ) ≥ c_{2}D log D + O_{ε,s_{0}}(Dλ + \sqrt{D}e^{2λ} + λD^{1+ε− \frac{1}{2λ}} )

where c_{1} and c_{2} are some absolute, positive constants. Then the conclusions follow if we take λ = \frac{1}{δ} in the first case, λ = \frac{2}{δ} in the second one, ε small enough and D tending to infinity. It remains to prove the existence of the functions f_{1} and f_{2} with Laplace transforms F_{1} and F_{2} satisfying the required conditions. The construction of f_{1} was given in [P-P]. To construct f_{2} assume that h ∈ C_{0}^{∞}(ℝ) satisfies the condition ĥ ≥ 0 and define h_{∗} = h+ch′′ with a suitable, positive constant c. Then we have ĥ_{∗}(ξ) = ĥ(ξ)−cξ^{2}ĥ(ξ) < 0 for |ξ| ≫ 1 and taking f_{2}(x) = h_{∗}(x)(ch(2x))^{−1} we obtain, by Proposition 3, p.59 [S-Z] that

F_{2}(σ + it) = π^{2} ∫_{−∞}^{+∞} \frac{ĥ_{∗}(t − x)}{ch(π(x − iσ)/4)}dx

= π^{2} cos(πσ/4) ∫_{−∞}^{+∞} ĥ_{∗}(t − x)\frac{ch(πx/4)}{| ch(π(x − iσ)/4)|^{2}} dx

hence the required properties follow.

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Andrzej Da̧browski University of Szczecin Department of Mathematics ul. Wielkopolska 15 70-451 Szczecin Poland E-mail: dabrowsk@sus.univ.szczecin.pl

Jacek Pomykała University of Warsaw Institute of Mathematics ul. Banacha 2 02-097 Warsaw Poland E-mail: pomykala@mimuw.edu.pl