Uroš Kač

Franc Novak

Florence Azaïs

Pascal Nouet

Michel Renovell

Implementation of an experimental IEEE 1149.4 mixed-signal test chip

The paper reports current results in the design, implementation and applications of an IEEE 1149.4 test chip with extended ABM functionality. As no previous experience or information on Dot 4 compatible integrated circuits design was available, the project has been approached in two stages. A simplified preliminary test chip was designed "from scratch" and has recently become available. Debug and evaluation of the preliminary design will provide useful information for the design and implementation of a second test chip version. Besides correcting identified design deficiencies the second version will also include minor ABM modifications which will allow the development of enhanced measurement procedures as described in the paper.

Introduction

Since the early 1980s electronics manufacturers have been looking for a solution to the problem of limited access circuit testability. As increasingly dense and complex electronic designs made the established incircuit test techniques more costly and difficult to implement, the innovative boundary scan method was proposed to improve design controllability and observability. On the initiative of Joint Test Action Group (JTAG) the proposed approach was adopted in 1990 by IEEE as the 1149.1 standard [START_REF]IEEE Standard Test Access Port and Boundary-Scan Architecture, IEEE Standard[END_REF]. Since IEEE 1149.1 addressed solely digital circuits, the development of equivalent test structures for incorporation into mixedsignal circuits and corresponding measurement methodologies [START_REF] Parker | Structure and metrology for an analog testability bus[END_REF] started soon after. Standardization efforts on a IEEE 1149.1 compatible test bus that would improve mixed-signal design testability at both device and assembly levels finally resulted in the adoption of the IEEE 1149.4 (Dot 4) standard in 1999 [START_REF]IEEE Standard for a Mixed-Signal Test Bus[END_REF].

The IEEE 1149.1 boundary scan quickly became a widely accepted design for testability technique in digital circuits, gaining support from component manufacturers as well as EDA tools and test equipment providers. This resulted in extensive use of 1149.1 infrastructure in various designs throughout the past decade. On the other hand, its mixed-signal twin seems to have a much harder time finding the way into real life applications. The apparent lack of interest for 1149.4 from major electronics manufacturers, with few exceptions [START_REF]Mixed-Signal Test and the IEEE 1149.4 Standard[END_REF], results in the absence of Dot 4 compatible catalogue devices and consequently the inability of designers to include standardized mixed-signal test infrastructure into their systems.

A probable reason for this situation is the complexity of on-chip Dot 4 infrastructure and the impact it would have on proven analog and mixed-signal designs. The efforts necessary to include overhead logic into the circuit must be justified by the benefits it provides. One can learn from the history of 1149.1 boundary scan technology that its acceptance was greatly helped by applications, which demonstrated its value not only in the design testability domain but also in areas such as design validation and debugging or in-system configuration of programmable devices.

Similarly, we believe that wider popularization of the Dot 4 mixed-signal test bus will only come with the demonstration of its benefits in actual designs and also with the presentation of other interesting applications based on the Dot 4 infrastructure [START_REF] Lofstrom | Early Capture For Boundary Scan Timing Measurments[END_REF]. To facilitate our efforts in developing innovative 1149.4 based test and measurement procedures and in the absence of ready to use solutions we decided to design and implement an experimental Dot 4 chip including certain extended functionality.

Implementation of the Dot 4 test chip

Since no information on the implementation of onchip Dot 4 infrastructure other than IEEE 1149.4 standard guidelines was available, we approached the design of the test chip in two stages. First, a preliminary chip adherent to the standard 1149.4 architecture would be implemented, which will allow us to assess actual characteristics of complex mixed-signal cells (Analog Boundary Module / ABM, Test Bus Interface Circuit / TBIC) and identify possible design inefficiencies and necessary modifications. To simplify design debugging, the digital part (Test Access Port / TAP controller, instruction and bypass registers and decoder) is implemented off-chip using a programmable logic device.

In next stage the final version of the Dot 4 test chip will be implemented, which will add the digital control circuitry logic to the design and provide some extended ABM functionality with respect to the standard architecture. As mentioned above, only ABM and TBIC cells, which represent the key architectural features of 1149.4 standard, were implemented on-chip (Figure 1). The 0.8µm AMS CYE technology was chosen for the implementation of the test chip.

The realization of conceptual analog switches and comparators in both ABM and TBIC cells requires particular attention. While the standard allows conceptual switches to be realized in different ways, we have chosen to use exclusively transmission gates since the chip is implemented in CMOS technology and does not include a functional analog core. This would also produce a more generic ABMs structure usable both as an input or output module in experimental designs. The analog switching architecture of an ABM cell is illustrated in Figure 2. The relatively high internal resistance exhibited by standard AMS library switch cells required design modifications (Table 1) in order to obtain transmission gates suitable for inclusion into the ABM/TBIC analog switching structure. Similarly, available standard cell comparators occupied an unacceptably large silicon area, therefore a full-custom comparator was designed, which satisfies our requirements both in terms of silicon area and electrical characteristics (Table 2). Table 1: Comparison of area and resistance between library and modified switch cell The second version of the Dot 4 test chip will include some modifications, which address certain deficiencies in the preliminary design as well as facilitate the simulation of optional functions included in experimental 1149.4 compatible designs. The later will also allow the implementation of alternative test and measurement procedures based on existing Dot 4 infrastructure.

C e l l O f f s e t ( µ V ) A v ( d B ) V o u t m i n V o u t m a x V t h m i n V t h m a x S i z e ( µ m 2 ) C o m p 0 1 B ( l i b r a r y ) 2 7 8 7 3 n V 5 V 0 1 V 4 4 V 2 5 0 0 0 C o m p a - r a t o r ( f u l l c u s t o m ) 9 2 6 1 1 7 m V 5 V 0 1 V 4 6 V 2 8 0 0
Considering both implementation guidelines and measurement methodology proposed by the standard, we decided to apply the following modifications to the ABM switching architecture:

The core disconnect switch was excluded from the original cell implementation and will be implemented as a separate cell consisting of a low resistance (R ON < 50 Ω) CMOS transmission gate. This fature will allow the user to decide whether to use the on-chip switch, to include the core disconnect facility into the functional output of his analog design or not to implement it at all. Three modified ABM cells were also designed with additional switching resources, which allow to connect the compare voltage input of the comparator to either V th or one of the V H , V L or V G reference voltages (Figure 4, Table 3).

Finally, the test control circuitry will be implemented on-chip, releasing valuable pad resources required by input/output control signals and simplifying the use of the Dot 4 test chip. A schematic representation of the modified test chip is illustrated in Figure 5.

Applications of extended ABM functionality

Besides mandatory instructions (PROBE, BYPASS, SAMPLE/PRELOAD and EXTEST) various optional test instructions (INTEST, RUNBIST, HIGHZ,...) are also proposed by the IEEE 1149.4 standard, which can be implemented in Dot 4 compliant components [START_REF]IEEE Standard for a Mixed-Signal Test Bus[END_REF].

These instructions basically follow the rules set by the IEEE 1149.1 standard. There are however certain differences in how they apply to the digital and to the analog part of a mixed-signal device, which is a consequence of the functional differences between ABMs and digital boundary modules (DBM). This may in turn affect the efficiency of mixed-signal test procedures based on Dot 4 infrastructure.

The 1149.4 digital boundary modules provide the same functionality as the 1149.1 boundary scan cells. These allow to apply test stimulus and monitor the response of the digital part of the mixed-signal core during an internal test (INTEST) with the core being isolated from input/output pins. On the other hand, the implementation of the ABM, as proposed by the standard, does not allow to disconnect the analog core from input/output pins while maintaining the ability to access the core functional inputs/outputs through the on-chip analog test bus. Consequently, analog INTEST can only be performed with external circuitry connected to the analog core. This requires some means of control over external on-board circuitry in order to provide appropriate operating conditions to the core or to ensure that the inputs are quiescent.

To overcome this deficiency we propose an additional test access node, which is placed on the "core" side of the core disconnect switch (SD). This node would provide access to the analog core functional input/output through the on-chip analog test bus (AB1, AB2) and two additional small-size CMOS switches (SB1_INT, SB2_INT), which bypass the core disconnect switch.

The proposed ABM enhancement would allow to perform analog core internal test (INTEST) without the need for control over external circuitry, therefore simplifying the overall test procedures. Furthermore, functional test of chip cores and external circuitry could be also performed by means of on-board IEEE 1149.4 infrastructure. As demonstrated in [START_REF] Kač | Alternative Test Methods Using IEEE 1149[END_REF], selected analog functional blocks can be reconfigured into a self-testing structure by establishing connections via the Dot 4 analog test bus and efficient GO/NO-GO functional tests can then be performed. The enhanced ABM structure would also allow to perform reconfiguration based functional test procedures internally to the chip and could be exploited for the implementation of efficient analog built-in self-tests (ABIST), [START_REF] Arabi | Testing analog and mixedsignal integrated circuits using oscillation test method[END_REF].

The necessary ABM modifications would minimally impact its original structure: with core disconnect switches (SD) being controlled by a global chip signal, an additional global signal provided by the 1149.4 instruction decoder during INTEST would be sufficient to activate the core analog test bus switches (SB1_INT, SB2_INT) and disable the standard switches (SB1, SB2). Only minor modifications in ABM control logic are therefore required in order to generate adequate control signals. The modified Dot 4 test chip, which implements the core disconnect switch as a separate block, can be exploited to simulate the enhanced ABM. Two ABMs and one SD are connected as illustrated in Figure 6, providing a structure similar to the proposed enhanced ABM. Such a configuration can be primarily used to experimentally evaluate alternative test procedures, based on analog blocks reconfiguration.

The second modification in the ABM cell consist of additional switching resources, which allow to connect the compare voltage input of the comparator to either V th or one of the V H , V L or V G reference voltages. This feature could prove useful during interconnect test as it provides augmented diagnostic capabilities, allowing one to distinguish between multiple input voltage levels. An illustrative example is given in the following.

Consider the situation of a bridging fault encountered during interconnect test (EXTEST) as illustrated in Figure 7. In the preliminary chip implementation, SH and SL are implemented with identical CMOS transmission gates which exhibit substantial ON-resistance connected in series with reference voltage sources. As noted by the IEEE 1149.4 standard, in this case the voltage appearing at the combined (shorted) net is likely to be at some value between the V H and V L levels. The compare voltage V th should be chosen such that it is clearly different from possible input voltage levels while on the other hand the standard limits V th to a value in the range (V H + V L )/2 ± (V H -V L )/4, which might cause intermediate input values to overlap with the uncertainty region of the comparator, resulting in possible fault masking. This is confirmed by SPICE simulation (Figure 7) which places the resulting voltage uncomfortably close to (V H + V L )/2. A possible solution to this problem is to use the proposed modified ABM switching scheme to shift the comparator treshold level away from the hazardous intermediate area by switching the compare voltage to either V H ' or V L ', according to the expected fault-free input value (V H or V L ), Figure 8.

According to IEEE 1149.4 standard, V H and V L should always be available on both input and output pins and they can be pin specific i.e. they are not necessarily the same on all pins. Therefore adequate V H ' and V L ' values can be chosen on the input pins In 1 and In 2 , such that a sufficient noise margin (NM) is maintained for both fault-free input voltage levels.

Although other approaches are possible, such as providing dominant/recessive V L /V H drivers in the ABM design, the presented solution might prove advantageous both in terms of parasitics impact on the functional pin and the required additional implementation area.

Conclusion

The paper reports current results in the design, implementation and applications of an IEEE 1149.4 test chip with extended ABM functionality. The work is performed by LIRMM and JSI in the frame of bilateral french-slovenian PROTEUS project. The first lot of the preliminary test chip version has recently become available and is currently in debug and evaluation stage. We are currently evaluating the functionality and identifying deficiencies of both analog and the digital parts of the design before integrating them in the second Dot 4 chip version. Besides, the prototype will be used in the early development stage of enhanced measurement procedures briefly described in Section 3. Corresponding software support for our home-brewed boundary scan tester is currently under development.

Figure 1 :

 1 Figure 1: Preliminary test chip block diagram

Figure 2 :

 2 Figure 2: Schematic representation of the ABM

  C e l l D i m e n s i o n s ( µ m ) S u r f a c e ( µ m 2 ) R O N ( Ω ) a t V D D T G 2 B ( l i b r a r y ) 1 n v ( m o d i f i e d , i n t e g r a t e d i n v e r t e r ) 2

Table 2 :

 2 Comparison between library and full-custom comparator implementation Synthesis of the control logic and registers was relatively straight forward, requiring only logic equations optimization in order to allow the use of standard library cells (inv, nand2, nor2). The final implementation of the preliminary Dot 4 test chip is illustrated in Figure 3. Altogether nine ABM cells and one TBIC were laid on an active area of 1980 x 1980 µ m 2 , featuring 39 pads connected to 5 control signals (mode1, mode2, update_DR, shift_DR and clock_DR), 2 digital test signals (TDI and TDO), 2 digital power supply lines (VDD and GND), 2 analog power supply lines (VDDA and GNDA), 2 analog test signals (AT1 and AT2) and 18 pairs (pin and core) of ABM functional signals.

Figure 3 :

 3 Figure 3: Preliminary test chip layout

Figure 4 :

 4 Figure 4: Simplified illustration of modified switching architecture in ABM cell

Figure 5 :

 5 Figure 5: Block diagram of the modified Dot 4 test chip

Figure 6 :

 6 Figure 6: Simulating core access from analog test bus during INTEST

Figure 7 :

 7 Figure 7: Bridging fault producing an intermediate voltage level during EXTEST

Table 3 :

 3 Proposed decode logic modification


Jie Wu
email: jie.wu@math.cnrs.fr

Qiang Wu
email: qiangwu@swu.edu.cn

MEAN VALUES FOR A CLASS OF ARITHMETIC FUNCTIONS IN SHORT INTERVALS

Keywords: Mathematics Subject Classification, 11N37, Key words and phrases, Asymptotic results on arithmetic functions

In this paper, we shall establish a rather general asymptotic formula in short intervals for a classe of arithmetic functions and announce two applications about the distribution of divisors of square-full numbers and integers representable as sums of two squares.

Introduction

This is the third paper of our series on the Selberg-Delange method for short intervals. Roughly speaking this method applies to evaluate mean values of arithmetic functions whose associated Dirichlet series are close to complex powers of the Riemann ζ-function. In the first part, by using a suitable contour (located to the left of the Korobov-Vinogradov zero-free region of ζ(s)) instead of Hankel's contour used in the original version of the Selberg-Delange method, Cui and Wu [START_REF] Cui | The Selberg-Delange method in short intervals with an application[END_REF] extended this method to handle mean values of arithmetic functions over a short interval, when their corresponding Dirichlet series is close to a positive power of ζ-function. In the second one, Cui, Lü and Wu [START_REF] Cui | The Selberg-Delange method in short intervals with some applications[END_REF] treated the complex power case with the help of the well-known Hooley-Huxley-Motohashi contour. Some similar results have were appeared in Ramachandra's paper [START_REF] Ramachandra | Some problems of analytic number theory[END_REF]. In this paper, we shall consider a more general case and give some arithmetic applications.

Assumptions.

Let us fix some notation: -ζ(s) is the Riemann ζ-function, -L(s, χ) is the Dirichlet L-function of χ, -ε is an arbitrarily small positive constant, -r ∈ N, α > 0, δ 0, A 0, M > 0 (constants), -z := (z 1 , . . . , z r ) ∈ C r and w := (w 1 , . . . , w r ) ∈ C r , -κ := (κ 1 , . . . , κ r ) ∈ (R + * ) r with 1 κ 1 < • • • < κ r 2κ 1 , -χ := (χ 1 , . . . , χ r ) with χ i non principal Dirichlet characters, -B := (B 1 , . . . , B r ) ∈ (R + * ) r and C := (C 1 , . . . , C r ) ∈ (R + * ) r , -The notation |z| B means that |z i | B i for 1 i r. Let f : N → C be an arithmetic function and its corresponding Dirichlet series is given by (1.1)

F(s) := ∞ n=1 f (n)n -s .
We say that this Dirichlet series F(s) is of type P(κ, z, w, B, C, α, δ, A, M ) if the following conditions are verified: (a) For any ε > 0 we have As usual, we denote by N (σ, T ) and N χ (σ, T ) the number of zeros of ζ(s) and L(s, χ) in the region e s σ and | m s| T , respectively. It is well known that there are two constants ψ and η such that (1.7) N (σ, T ), N χ (σ, T ) T ψ(1-σ) (log T ) η for 1 2 σ 1 and T 2. Huxley [START_REF] Huxley | The difference between consecutive primes[END_REF] showed that (1.8) ψ = 12 5 and η = 9 are admissible for ζ(s). It is not difficult to extend it for L(s, χ). The zero density hypothesis is stated as (1.9) ψ = 2.

(1.2) |f (n)| ε M n ε ( n
1.2. Set-up and main results.

Our main aim of this paper is to establish, under the previous assumptions, an asymptotic formula for the summatory function (1.10) x<n x+x 1-1/κ 1 y f (n) where y := x ϑ with ϑ ∈ (0, 1] as small as possible. In order to state our result, it is necessary to introduce some notation. Obviously the function Z(κs; z) := {(κs -1)ζ(κs)} z is holomorphic in the disc |s -1/κ| < 1/κ, and admits, in the same disc, the Taylor series expansion

Z(κs; z) = ∞ j=0 γ j (z, κ) j! (s -1/κ) j ,
where the γ j (z, κ)'s are entire functions of z satisfying the estimate

(1.11) γ j (z, κ) j! B,κ,ε (1 + ε) j (j 0, |z| B)
for all B > 0, κ and ε > 0. Write κ * = (κ 2 , . . . , κ r ) and z * = (z 2 , . . . , z r ). Under our hypothesis, the function

G(s)Z(κ 1 s; z 1 )ζ(κ * s) z * L(κs; χ) w is holomorphic in the disc |s -1/κ 1 | < 1/κ 1 -1/κ 2 and
(1.12)

|Z(κ 1 s; z 1 )ζ(κ * s) z * L(κs; χ) w G(s)| A,B,C,δ,ε M for |s -1/κ 1 | < 1/κ 1 -1/κ 2 -ε, |z| B and |w| C. Thus we can write (1.13) Z(κ 1 s; z 1 )ζ(κ * s) z * L(κs; χ) w G(s) = ∞ =0 g (κ, z, w, χ)(s -1/κ 1 ) for |s -1/κ 1 | 1 2 (1/κ 1 -1/κ 2 )
, where

(1.14) g (κ, z, w, χ) := j=0 γ j (z 1 , κ 1 ) ( -j)!j! • ∂ -j (ζ(κ * s) z * L(κs; χ) w G(s)) ∂s -j s=1/κ 1 .
The main result of this paper is as follows.

Theorem 1.1. Let r ∈ N, κ ∈ (R + * ) r , z ∈ C r , w ∈ C r , B ∈ (R + * ) r , C ∈ (R + * ) r , χ, α > 0, δ 0
, A 0, M > 0 be given as before. Suppose that the Dirichlet series F(s) defined as in (1.1) is of type P(κ, z, w, B, C, χ, α, δ, A, M ). Then for any ε > 0, we have

(1.15) x<n x+x 1-1/κ 1 y f (n) = y (log x) z-1 N =0 λ (κ, z, w, χ) (log x) + O M R N (x)
uniformly for

x 3, x (1-1/(ψ+δ))/κ 1 +ε y x 1/κ 1 , N 0, |z| B, |w| C,
where

y := κ 1 ((x + x 1-1/κ 1 y) 1/κ 1 -x 1/κ 1 ), λ (κ, z, w, χ) := κ -z 1 1 g (κ, z, w, χ)/Γ(z 1 -), R N (x) := c 1 N + 1 log x N +1 + (c 1 N + 1) N +1 e c 2 (log x/ log 2 x) 1/3 + y x 1/κ 1 log x
for some constants c 1 > 0 and c 2 > 0 depending only on B, C, δ and ε. The implied constant in the O-term depends only on κ, B, C, χ, A, α, δ and ε. In particular, ψ = 12 5 is admissible.

The admissible length of short intervals in Theorem 1.1 depends only on the zero density constant ψ of ζ(s) and δ in (1.6) (for which we take δ = 0 in most applications). Its independence from the power z of ζ(s) in the representation of F(s) seems interesting. Theorem 1.1 generalizes and improves [2, Theorem 1] to the case of complex powers and intervals of shorter length.

Taking N = 0 in Theorem 1.1, we obtain readily the following corollary.

Corollary 1.1. Under the conditions of Theorem 1.1, for any ε > 0, we have

(1.16
) In order to study the distribution of divisors of integers, Deshouillers, Dress & G. Tenenbaum [START_REF] Deshouillers | Lois de répartition des diviseurs[END_REF] introduced the random variable D n , which takes the value (log d)/ log n, as d runs through the set of the τ (n) divisors of n, with the uniform probability 1/τ (n), and consider its distribution function

x<n x+x 1-1/κ 1 y f (n) = y (log x) z-1 λ 0 (κ, z, w, χ) + O M log x uniformly for x 3, x (1-1/(ψ+δ))/κ 1 +ε y x 1/κ 1 , |z| B, |w| C, where λ 0 (κ, z, w, χ) := G(1/κ 1 ) κ z 1 1 Γ(z 1 ) 2 i r ζ(κ i /κ 1 ) z i 1 i r L(κ i /κ 1 , χ i ) w i
F n (t) = Prob(D n t) = 1 τ (n) d|n, d n t 1 (0 t 1).
It is clear that the sequence {F n } n 1 does not converge pointwisely on [0, 1]. However Deshouillers, Dress & Tenenbaum ( [START_REF] Deshouillers | Lois de répartition des diviseurs[END_REF] or [START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF]Theorem II.6.7]) proved that its Cesàro mean converges uniformly to the arcsin law. More precisely, they showed that the asymptotic formula

(1.17) 1 x n x F n (t) = 2 π arcsin √ t + O 1 √ log x
holds uniformly for x 2 and 0 t 1. Here we shall announce two applications of Corollary 1.1: the distribution of divisors of square-full numbers and of integers representable as sums of two squares. The proof will be given in another paper [START_REF] Wu | On the distribution of divisors de certains integers in short intervals[END_REF].

A. Beta law on divisors of square-full numbers in short intervals An integer n is called square-full if p | n ⇒ p 2 | n. Denote by 1 sf (n) the characteristic function of such integers and define

(1.18) U (x, y) := x<n x+x 1/2 y 1 sf (n).
Our first result is as follows.

Theorem 1.2. For any ε ∈ (0, 5 48 ), we have

(1.19) 1 U (x, y) x<n x+x 1/2 y 1 sf (n)F n (t) = B( 2 3 , 1 3 ) -1 t 0 w -1 3 (1 -w) -2 3 dw + O 1 3 √ log x
uniformly for x 3 and x 19/48+ε y x 1/2 , where B(u, v) is the beta function defined by

(1.20) B(u, v) := 1 0 w u-1 (1 -w) v-1 dw
for u > 0 and v > 0.

B. Beta law on divisors of integers representable as sums of two squares Define (1.21)

1 + (n) := 1 if n = + 0 otherwise and V (x) := n x 1 + (n).
They proved that the mean of F n (t) over integers representable as sum of two squares converges to the beta law [7, Theorem 1]: For x → ∞, we have

(1.22) 1 V (x) n x 1 + (n)F n (t) = B( 1 4 , 1 4 ) -1 t 0 w -3 4 (1 -w) -3 4 dw + O 1 4 √ log x .
Here we shall generalise (1.22) to the short interval case. Put Our result is as follows.

Theorem 1.3. For any ε ∈ (0, 5 12 ), we have

(1.24) 1 V (x, y) x<n x+y 1 + (n)F n (t) = B( 1 4 , 1 4 ) -1 t 0 w -3 4 (1 -w) -3 4 dw + O 1 4 √ log x
uniformly for x 3 and x 19/24+ε y x.

The Hooley-Huxley-Motohashi contour

The Hooley-Huxley-Motohashi contour appeared in Ramachandra [START_REF] Ramachandra | Some problems of analytic number theory[END_REF] and Motohashi [START_REF] Motohashi | On the sum of the Möbis function in a short segment[END_REF], independently. In [START_REF] Hooley | On intervals between numbers that are sums of two squares III[END_REF], Hooley stated, without proof, his joint result with Huxley:

x<n x+x θ 1 + (n) ∼ 2 -1/2 p≡1(mod 4) (1 -p -2 ) -1/2 x θ √ log x for x → ∞, provided θ > 7 12
. According to Ramachandra [14, page 314], Hooley and Huxley have educated him on their method and allowed him make some comments about their method. A key point of this method is to use an ingeneous contour, which is called the Huxley-Hooley contour by Ramachandra. In [START_REF] Motohashi | On the sum of the Möbis function in a short segment[END_REF], Motohashi constructed a essential same contour to prove

x<n x+x θ µ(n) = o(x θ ) (x → ∞) provided θ > 7 12 .
Here we shall follow Motohashi's argument and construct a Hooley-Huxley-Motohashi contour for our purpose.

Notation.

Let (κ, χ) be as before. First we write ζ(κs)L(κs, χ) :=

1 i r ζ(κ i s)L(κ i s, χ i ) =: n 1 τ κ (n; χ)n -s , (2.1) (ζ(κs)L(κs, χ)) -1 := 1 i r (ζ(κ i s)L(κ i s, χ i )) -1 =: n 1 τ -1 κ (n; χ)n -s (2.2)
for σ > 1/κ 1 , respectively. Here and in the sequel, we define implicitly the real numbers σ and τ by the relation s = σ + iτ . A simple computation shows that

τ κ (n; χ) = m κ 1 1 •••m κr r n κ 1 1 •••n κr r =n 1 i r χ i (n i ), (2.3) τ -1 κ (n; χ) = m κ 1 1 •••m κr r n κ 1 1 •••n κr r =n 1 i r µ(m i )µ(n i )χ i (n i ). (2.4)
From these, we deduce that (2.5)

|τ κ (n; χ)| τ κ,κ (n), |τ -1 κ (n; χ)| τ κ,κ (n), τ κ * τ -1 κ = 1 {1}
for all n 1, where

(2.6) τ κ,κ (n) := m κ 1 1 •••m κr r n κ 1 1 •••n κr r =n
1 and 1 {1} is the unit with respect to the Dirichlet convolution.

Upper bounds for ζ(s)

±1 and L(s, χ) ±1 in their zero-free regions.

It is well known that there is an absolute positive constant c such that ζ(s) = 0 for

(2.7) σ 1 -c(log |τ |) -2/3 (log 2 |τ |) -1/3 , |τ | 3
(the zero-free region, due to Korobov and Vinogradov) and in this region we have

(2.8) ζ(s) ±1 (log |τ |) 2/3 (log 2 |τ |) 1/3
(see [21, page 135] or [19, page 162]). For the Dirichlet L-functions, Richert [START_REF] Richert | Zur abschatzung der Riemannschen zetafunktion in der nähe der vertikalen σ = 1[END_REF] has established similar results.

Lemma 2.1. Let χ be a non-principal Dirichlet character module q and let L(s, χ) be the corresponding Dirichlet L-function. Then we have

(2.9) |L(s, χ)| |τ | 100(1-σ) 3/2 (log τ ) 4/3 if 1 2 σ 1 and |τ | 3, log |τ | if σ 1 and |τ | 3.
Further there is a positive constant c χ depending on χ such that

(2.10) |L(s, χ)| -1 (log |τ |) 2/3 (log 2 |τ |) 1/3 for σ 1 -500c χ (log |τ |) -2/3 (log 2 |τ |) -1/3 and |τ | 3.
Here the implied constants depend on χ only.

Proof. Let s = σ + iτ . Without loss of generality, we can suppose that τ 2. For σ := e s > 1 and 0 < w 1, the Hurwitz ζ-function is defined by

ζ(s, w) := ∞ n=0 (n + w) -s .
This function can be extended to a meromorphic function over C {1}. According to [START_REF] Richert | Zur abschatzung der Riemannschen zetafunktion in der nähe der vertikalen σ = 1[END_REF]Satz], there is a absolute constant c > 0 such that we have

(2.11) ζ(s, w) -w -s c|τ | 100(1-σ) 3/2 log |τ | 2/3 uniformly for 0 < w 1, 1 2 σ 1 and |τ | 3. Since χ(n) is of period q, we can write, for σ > 1, L(s, χ) = q a=1 ∞ n=0 χ(a + nq)(nq + a) -s = q -s q a=1 χ(a) ζ(s, a/q) -(a/q) -s + q a=1 χ(a)a -s .
This relation also holds for all s ∈ C {1} by analytic continuation. Inserting (2.11), we immediately get the first inequality in (2.9). The second one is classical.

In view of (2.9), we can prove (2.10) exactly as [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF]Theorem 3.11] with the choice of

θ(τ ) = log 2 τ 100 log τ 2/3 and φ(τ ) = log 2 τ.
This completes the proof of Lemma 2.1.

Definition of the Hooley-Huxley-Motohashi contour L T .

Let (ε, κ, χ) be as before and let T 0 = T 0 (ε, κ, χ), c 0 = c 0 (κ, χ) be two large constants and let C 0 = C 0 (κ, χ) be a suitable positive constant. For T T 0 , put (2.12)

δ T := C 0 (log T ) -2/3 (log 2 T ) -1/3 .
According to (2.8), (2.9) and (2.10) where such that

(2.13) (log |τ |) -c 0 r |ζ(κs)L(κs, χ)| (log |τ |) c 0 r
for σ (1 -100δ T )/κ 1 and 3 |τ | 100T , where the implied constants depend on (κ, χ). For T T 0 , write (2.14)

J T := [( 1 2 -δ T ) log T ] and K T := [T (log T ) -1 ].
For each pair of integers (j, k) with 0 j J T and 0 k K T , we define (2.15)

σ j := ( 1 2 + j(log T ) -1 )/κ 1 , τ k := (1 + k log T )/κ 1 and (2.16) ∆ j,k := {s = σ + iτ : σ j σ < σ j+1 and τ k τ < τ k+1 }. Define M x (s) = M x (s; κ, χ) := n x τ -1 κ (n; χ) n s , (2.17) M(ς, T ) = M(ς, T ; κ) := max σ ς 1 |τ | T |ζ(κs)| 2 . (2.

18)

Let A be a fix large integer, and put

(2.19) N j := A (log T ) 5 M(4σ j -3/κ 1 , 8T ) 1/2(1/κ 1 -σ j ) .
We divide ∆ j,k into two classes (W ) and (Y ) as follows.

• The case of

σ j (1 -ε)/κ 1 . ∆ j,k ∈ (W ) if ∆ j,k contains at least one zero of ζ(κs)L(κs, χ), and ∆ j,k ∈ (Y ) otherwise. • The case of (1 -ε)/κ 1 < σ j (1 -δ T )/κ 1 .
We write

(2.20) ∆ j,k ∈ (W ) ⇔ ∃ s ∈ ∆ j,k such that |ζ(κs)L(κs, χ)M N j (s)| < 1 2 and (2.21) ∆ j,k ∈ (Y ) ⇔ ζ(κs)L(κs, χ)M N j (s) 1 2 for all s ∈ ∆ j,k .
For each k, we define j k := max{j : ∆ j,k ∈ (W )} and put

(2.22) D := ∪ 0 k K T ∪ 0 j j k ∆ j,k and D 0 := ∪ 0 k K T ∪ j k <j J T ∆ j,k . Clearly D 0 consists of ∆ j,
d h := (log 2 T )/κ 1 , d v := ε 2 /κ 1 if σ (1 -ε)/κ 1 , (log T ) -1 /κ 1 if (1 -ε)/κ 1 < σ < (1 -δ T )/κ 1 .
Clearly L T is symmetric about the real axis. The following figure shows its upper part The aim of this section is to prove the following proposition.

[from the point ((1 -ε)/κ 1 , 0) to the point ((1 -δ T )/κ 1 , T )]. T ∆ j,k σ j σ j+1 τ k τ k+1 ∆ j 0 ,0 ∆ j k ,k ∆ j K ,K M T r dv d h D D 0 (1 -δ T )/κ 1 (1 -ε)/κ 1 1/2κ 1 b 1/κ 1 σ τ O
Proposition 3.1. Under the previous notation, we have

(3.1) T -544 √ 2ε(1-κ 1 σ) (log T ) -4 |ζ(κs)L(κs, χ)| T 136 √ 2ε(1-κ 1 σ) (log T ) 4 .
for all s ∈ L T , where the implied constants depend on (ε, κ, χ).

First we establish two preliminary lemmas.

Lemma 3.1. Under the previous notation, we have

(3.2) e -(log T ) 1-ε 2 |ζ(κs)L(κs, χ)| e (log T ) 1-ε 2 for s ∈ L T with σ (1 -ε)/κ 1 , or s with (1 -ε)/κ 1 < σ 1 κ 1 (1 -ε + ε 2 )
on the horizontal segments in L T that intersect the vertical line e s = (1 -ε)/κ 1 . Here the implied constant depends only on ε.

Proof. Let s = σ + iτ satisfy the conditions in this lemma. Without loss of generality, we can suppose that τ T 0 (ε, κ, χ). Let us consider the four circles C 1 , C 2 , C 3 and C 4 , all centered at s 0 := log 2 τ + iτ , with radii

r 1 := log 2 τ -(1 + η)/κ 1 , r 2 := log 2 τ -σ, r 3 := log 2 τ -σ + ε 2 /(2κ 1 ), r 4 := log 2 τ -σ + ε 2 /κ 1 ,
respectively. Here η > 0 is a parameter to be chosen later. We note that these four circles pass through the points (1 + η)/κ 1 + iτ , σ + iτ , σ -ε 2 /(2κ 1 ) + iτ and σ -ε 2 /κ 1 + iτ , respectively.

Clearly ζ(κs)L(κs, χ) = 0 in a region containing the disc |s -s 0 | r 4 . Thus we can unambiguously define log(ζ(κs)L(κs, χ)) in this region. We fix a branch of the logarithm throughout the remaining discussion.

Let M i denote the maximum of | log(ζ(κs)L(κs, χ))| on C i relative to this branch. By Hadamard's three-circle theorem and the fact that s = σ + iτ is on C 2 , we have

(3.3) | log(ζ(κs)L(κs, χ))| M 2 M 1-φ 1 M φ 3 , where φ = log(r 2 /r 1 ) log(r 3 /r 1 ) = log(1 + (1 + η -κ 1 σ)/(κ 1 log 2 τ -1 -η)) log(1 + (1 + η -κ 1 σ + ε 2 /2)/(κ 1 log 2 τ -1 -η)) = 1 + η -κ 1 σ 1 + η -κ 1 σ + ε 2 /2 + O 1 log 2 τ . On taking η = κ 1 σ -1 2 1 + ε 2 + ε 2 1+ε 2 (η ε 4 2(1+ε 2 ) , since σ 1 κ 1 ( 1 2 + ε 2 )), we have (3.4) φ = 1 -ε 2 -ε 4 + O((log 2 τ ) -1
).

On the circle C 1 , we have

(3.5) M 1 max e s (1+η)/κ 1 ∞ n=2 Λ(n) n κ 1 s log n ∞ n=2 1 n 1+η 1 η ,
where Λ(n) is the von Mangoldt function.

In order to bound M 3 , we shall apply the Borel-Carathéodory theorem to the function log

M 3 2r 3 r 4 -r 3 max |s-s 0 | r 4 log |ζ(κs)L(κs, χ)| + r 4 + r 3 r 4 -r 3 | log(ζ(κs 0 )L(κs 0 , χ))| 2(log 2 τ -σ + ε 2 /(2κ 1 )) ε 2 /(2κ 1 ) log τ + 2(log 2 τ -σ + ε 2 /(2κ 1 )) ε 2 /(2κ 1 ) (log 2 τ ) log τ.
From (3.3), (3.4), (3.5) and (3.6), we deduce that

| log(ζ(κs)L(κs, χ))| (η -1 ) 1-φ ((log 2 τ ) log τ ) φ ε ((log 2 τ ) log τ ) 1-ε 2 -ε 4 ε (log τ ) 1-ε 2 .
This leads to the required estimates. Lemma 3.2. Under the previous notation, we have

(3.7) T -544r(1-κ 1 σ j ) 3/2 (log T ) -c 0 r |ζ(κs)L(κs, χ)| T 136r(1-κ 1 σ j ) 3/2 (log T ) c 0 r for s ∈ L T with (1 -ε)/κ 1 < σ j σ < σ j+1 .
Here the implied constants depend on (κ, χ).

In particular we have

(3.8) T -544r √ 2ε(1-κ 1 σ j ) (log T ) -c 0 r |ζ(κs)L(κs, χ)| T 136r √ 2ε(1-κ 1 σ j ) (log T ) c 0 r for s ∈ L T with (1 -ε)/κ 1 < σ j σ < σ j+1 .
All the implied constants are absolute.

Proof. By (2.11) with w = 1 and Lemma 2.1, we have

(3.9) |ζ(κs)L(κs, χ)| τ 100r(1-κ 1 σ) 3/2 (log τ ) c 0 r ( 1 2κ 1 σ 1 κ 1 , τ 2
). This immediately implies the second inequality in (3.7).

Next we consider the first inequality in (3.7). Let s ∈ L T with (1-ε)/κ 1 < σ j σ < σ j+1 . Since 1/κ 1 4σ j -3/κ 1 (1 -4ε)/κ 1 1/2κ 1 , the inequality (2.11) with w = 1 allows us to derive that (3.10)

N 2(1/κ 1 -σ j ) j = A (log T ) 5 max σ 4σ j -3/κ 1 1 |τ | 8T |ζ(κs)| 2 T 200 √ 2r(1-κ 1 σ j ) 3/2 (log T ) c 0 r .
According to the definition of L T , there is an integer k such that s ∈ ∆ j,k and this ∆ j,k must be in (Y ) and (2.21) holds for all s of this ∆ j,k . On the other hand, (2.5) and (3.10) imply that for σ j σ < σ j+1 ,

|M N j (s)| n N j τ κ,κ (n)n -σ j (1 -κ 1 σ j ) -2 N 1/κ 1 -σ j j T 544(1-κ 1 σ j ) 3/2 (log T ) c 0 r .
Combining this with (2.21) immediately yields

|ζ(κs)L(κs, χ)| (2|M N j (s)|) -1 T -544(1-κ 1 σ j ) 3/2 (log T ) -c 0 r for s ∈ L T with (1 -ε)/κ 1 < σ j σ < σ j+1 . Finally we note (3.8) is a simple consequence of (3.7) since (1 -ε)/κ 1 < σ j implies that (1 -κ 1 σ j ) 1/2 √ ε.
Now we are ready to prove Proposition 3.1.

Proof. When s ∈ L T with |τ | 1, the estimations (3.1) are trivial. Next suppose that s ∈ L T with |τ | > 1.
Then there is a j such that σ j σ < σ j+1 . We consider the three possibilities.

• The case of (1 -ε)/κ 1 < σ j .

The inequality (3.1) follows immediately from (3.8) of Lemma 3.2.

• The case of σ j σ (1 -ε)/κ 1 .

In this case, the first part of Lemma 3.1 shows that (3.1) holds again since √ ε(1 -κ j σ) ε 3/2 (log T ) -ε 2 for T T 0 (ε, κ, χ).

• The case of σ j (1 -ε)/κ 1 < σ.

In this case, s must be on the horizontal segment in L T , because the vertical segment keeps the distance ε 2 from the line e s = σ j and σ j < σ < σ j+1 . Thus we can apply the second part of Lemma 3.1 to get (3.1) as before.

4. Montgomery's method 4.1. Montgomery's mean value theorem.

Let τ κ,κ (n) and M(ς, T ) be defined as in (2.6) and (2.18), respectively. The following proposition is a variant of [START_REF] Montgomery | Topics in multiplicative number theory[END_REF]Theorem 8.4] for our purpose, which will play a key role in the proof of Proposition 5.1 below. Proposition 4.1. Let σ 0 > 0, > 0 and T 2 three real numbers. Let r ∈ N and κ be as before. Let S = S T (σ 0 , ) be a finite set of complex numbers s = σ + iτ such that for any distinct points s = σ + iτ and s = σ + iτ in S. For any sequence of complex numbers {a n } verifying

(4.3) a n = 0 ⇒ τ κ,κ (n) 1, real numbers θ ∈ (1/κ 2 , 1/κ 1 )
and N 1, we have

(4.4) s∈S n N a n n -s 2 (1 + -1 )N 1/κ 1 + M(θ, 4T ) + 1 (1 -κ 1 θ)(κ 2 θ -1) N θ |S| n N |a n | 2 n 2σ 0 ,
where the implied constant depends on κ only.

First we prove two preliminary lemmas. The first one is due to Bombieri (see also [START_REF] Montgomery | Topics in multiplicative number theory[END_REF]Lemma 5.1]). For the convenience of reader, we give a direct proof. Lemma 4.1. Let S be a finite set of complex numbers s. For any {a n } 1 n N ⊂ C, we have

(4.5) s∈S n N a n n -s 2 n N |a n | 2 b -1 n max s∈S s ∈S |B(s + s )| where B(s) := n 1 b n n -s
is absolutely convergent at the points s + s , and the b n are non-negative real numbers for which b n > 0 whenever a n = 0. Here the implied constant is absolute.

Proof. According our hypothesis on a n and b n , we can write

a n = a n b -1/2 n b 1/2
n for all n N . Introducing the notation

A(s) := n N a n n -s , then we can write s∈S |A(s)| 2 = s∈S n N a n b -1/2 n b 1/2 n n -s m N a m m -s = n N (a n b -1/2 n ) b 1/2 n s∈S n -s m N a m m -s .
By the Cauchy inequality, it follows that

(4.6) s∈S |A(s)| 2 2 n N |a n | 2 b -1 n Υ,
where

Υ := n N b n s∈S n -s m N a m m -s 2 .
Since B(s) is absolutely convergent at the points s + s , we can deduce

Υ n 1 b n s∈S n -s m N a m m -s 2 = n 1 b n s ∈S n -s d N a d d -s s∈S n -s m N a m m -s = s∈S s ∈S d N a d d -s m N
a m m -s B(s + s ).

On the other hand, the trivial inequality |ab| 1 2 (|a| 2 + |b| 2 ) allows us to write

d N a d d -s m N a m m -s 1 2 |A(s)| 2 + |A(s )| 2 . Thus Υ s∈S s ∈S 1 2 |A(s)| 2 + |A(s )| 2 |B(s + s )| 1 2 s∈S s ∈S |A(s)| 2 |B(s + s )| + 1 2 s∈S s ∈S |A(s )| 2 |B(s + s )|
n 1 τ κ,κ (n) n s e -n/(2N ) -e -n/N N 1/κ 1 e -|τ | + M(θ, 2|τ |) + (1 -κ 1 θ) -1 (κ 2 θ -1) -1 N θ ,
where the implied constant depends on κ only.

Proof. Denote by S N (s) = S N (s; κ, χ) the series on the left-hand side of (4.7). By the Perron formula [21, page 151, Lemma], we can write

(4.8) S N (s) = 1 2πi 2/κ 1 +i∞ 2/κ 1 -i∞ ζ(κ(w + s)) 2 Γ(w) (2N ) w -N w dw,
where ζ(κs) is defined as in (2.1) above. We take the contour to the line e w = θ -σ with 1/κ 2 -σ < θ -σ < 1/κ 1 -σ, and in doing so we pass a simple pole at w = 1/κ 1 -s. Put G(w) :=

2 j r ζ(κ j (w + s)) 2 Γ(w) (2N ) w -N w .
The residue of the integrand at this pole is, in view of the hypothesis σ 0,

Res w=1/κ 1 -s ζ(κ(w + s)) 2 Γ(w) (2N ) w -N w = γG(1/κ 1 -s) + G (1/κ 1 -s)/κ 1 /κ 1 N 1/κ 1 e -|τ | ,
where we have used the Stirling formula [20, page 151]:

(4.9) |Γ(s)| = √ 2π e -(π/2)|τ | |τ | σ-1/2 1 + O | tan( arg s 2 )| |τ | + |a| 2 + |b| 2 |τ | 2 + |a| 3 + |b| 3 |τ | 3
valable uniformly for a, b ∈ R with a < b, a σ b and |τ | 1, where the implied O-constant is absolute.

On the other hand, by using the Stirling formula again it is easy to see that for e w = θ-σ and | m w| |τ |, the integrand in (4.8) is

M(θ, 2|τ |) + (1 -κ 1 θ) -1 (κ 2 θ -1) -1 N θ-σ e -| m w| ,
while otherwise the Stirling formula and the convexity bounds of ζ-function imply that

N θ-σ e -| m w| M(θ, 2|τ |) + (1 -κ 1 θ) -1 (κ 2 θ -1) -1 N θ-σ e -| m w| .
The required result follows from this. Now we are ready to prove Proposition 4.1.

Proof. We shall apply Lemma 4.1 with the choice of

b n := τ κ,κ (n)n 2σ 0 e -n/(2N ) -e -n/N .
In view of the simple fact that e -n/(2N ) -e -n/N 1 for N n 2N and the hypothesis (4.3), we have (4.10)

s∈S N n 2N a n n -s 2 N n 2N |a n | 2 n -2σ 0 max s∈S s ∈S |B(s + s )|, where
B(s) := n 1 τ κ,κ (n) n s-2σ 0 e -n/(2N ) -e -n/N .
In view of (4.1) and (4.2), we can apply Lemma 4.2 to deduce

|B(s + s )| = n 1 τ κ,κ (n) n σ+σ -2σ 0 +i(τ -τ ) e -n/(2N ) -e -n/N N 1/κ 1 e -|τ -τ | + M(θ, 2|τ -τ |) + (1 -κ 1 θ) -1 (κ 2 θ -1) -1 N θ
for any distinct points s = σ + iτ ∈ S and s = σ + iτ ∈ S. By (4.2), the contribution of the term N 1/κ 1 e |τ -τ | to the sum s ∈S

s =s |B(s + s )| is N 1/κ 1 s ∈S |τ -τ | e -|τ -τ | N 1/κ 1 n 0 e -n s ∈S n<|τ -τ | n+1 1 (1 + -1 )N 1/κ 1 .

Thus (4.11)

s ∈S, s =s

|B(s + s )| (1 + -1 )N 1/κ 1 + M(θ, 2|τ -τ |) + (1 -κ 1 θ) -1 (κ 2 θ -1) -1 N θ |S|.
When s = s , we have (4.12)

B(s + s ) = B(2σ) n 1 τ κ,κ (n) e -n/(2N ) -e -n/N N 1/κ 1 .
Now the required result follows from (4.10), (4.11) and (4.12).

Density estimation of small value points

In [START_REF] Montgomery | Topics in multiplicative number theory[END_REF], Montgomery developed a new method for studying zero-densities of the Riemann ζ-function and of the Dirichlet L-functions. Subsequently by modifying this method, Huxley [START_REF] Huxley | The difference between consecutive primes[END_REF] established his zero-density estimation (1.8). In [START_REF] Motohashi | On the sum of the Möbis function in a short segment[END_REF], Motohashi noted that Montgomery's method can be adapted to estimate the density of "small value points" of ζ(s) (see [START_REF] Cui | The Selberg-Delange method in short intervals with some applications[END_REF]Section 2.3] for a detail description). Here we shall adapt this method to prove the following proposition.

Proposition 5.1. Under the previous notation, for j = 0, 1, . . . , J T we have

(5.1) k K T : ∆ j,k ∈ (W ) T ψ(1-κ 1 σ j ) (log T ) η if κ 1 σ j 1 -ε, and
(5.2) k K T : ∆ j,k ∈ (W ) T 500r(1-κ 1 σ j ) 3/2 (log T ) 10r if 1 -ε κ 1 σ j 1 -δ T .
Here the implied constants depend on κ and ε only.

Proof. When κ 1 σ j 1 -ε, the number of ∆ j,k of type (W ) does not exceed the number of non-trivial zeros of ζ(κs)L(κs, χ). Thus

|{k K T : ∆ j,k ∈ (W )}| 1 i r N (κ i σ j , 2T ) + N χ i (κ i σ j , 3T ) .
Now the required bound follows from (1.7).

Next we suppose 1 -ε κ 1 σ j 1 -δ T . Let K j (T ) be a subset of the set {log T k K T : ∆ j,k ∈ (W )} such that the difference of two distinct integers of K j (T ) is at least 3A , where A is the large integer specified in (2.17). Obviously

|{(log T ) 2 k K T : ∆ j,k ∈ (W )}| 3A |K j (T )|.
Therefore it suffices to show that (5.3) |K j (T )| ε T 170(1-κ 1 σ j ) 3/2 (log T ) 13 for T T 0 (ε, κ, χ).

Let M x (s) be defined as in (2.17 Let φ x (n) be the nth coefficient of the Dirichlet series Φ x (s), then

(5.5)

φ x (n) = d|n, d x τ -1 κ (d; χ)τ κ (n/d; χ).
By the Perron formula [21, Lemma, page 151], we can write

n 1 φ x (n) n s e -n/y = 1 2πi 2/κ 1 +i∞ 2/κ 1 -i∞ Φ x (w + s)Γ(w)y w dw
for y > x 3 and s = σ + iτ ∈ C with σ j σ < σ j+1 . We take the contour to the line e w = α j -σ < 0 with α j := 4σ j -3/κ 1 < σ j < 1/κ 1 , and in doing so we pass two simple poles at w = 0 and w = 1/κ 1 -s. Our equation becomes

n 1 φ x (n)
n s e -n/y = Φ x (s) + Ψ x,y (s) + I(s; x, y), where Ψ x,y (s) :=

2 j r ζ(κ j /κ 1 )L(κ/κ 1 , χ)M x (1/κ 1 )Γ(1/κ 1 -s)y 1/κ 1 -s , I(s; x, y) := 1 2π +∞ -∞ Φ x (α j + iτ + iu)Γ(α j -σ + iu)y α j -σ+iu du.
In view of (5.5) and (2.5), we have

(5.6) φ x (n) = 1 if n = 1, 0 if 1 < n x,

and

(5.7)

|φ x (n)| (τ κ,κ * τ κ,κ )(n) (n > x),
where τ κ,κ (n) is defined as in (2.6) above. It is easy to see that

n t (τ κ,κ * τ κ,κ )(n) t 1/κ 1 (log t) 3 .
By a simple partial integration, we can deduce that

n>y 2 φ x (n) n s e -n/y ∞ y 2 t -σ e -t/y d n t (τ κ,κ * τ κ,κ )(n) e -y y 1-2σ (log y) 3 + y -1 ∞ y 2 e -t/y t 1/κ 1 -σ (log t) 3 dt e -y/2
for σ 1/(2κ 1 ). Inserting it into the precedent relation, we find that (5.8) e -1/y + x<n y 2 φ x (n) n s e -n/y + O(e -y/2 ) = Φ x (s) + Ψ x,y (s) + I(s; x, y)

for s ∈ C with σ j σ < σ j+1 and y > x 3.

If k ∈ K j (T ), then there is at least a s k := v k + it k ∈ ∆ j,k such that (5.9)

|Φ N j (s k )| = |ζ(κs k )L(κs k ; χ)M N j (s k )| 1 2
• By the definition of K j (T ), we have

σ j v k σ j+1 , (log T ) 2 t k T and |t k 1 -t k 2 | 3A log T (k 1 = k 2 ).
Since |t k | (log T ) 2 , the Stirling formula (4.9) allows us to deduce (5.10)

|Ψ x,y (s k )| = 2 j r ζ(κ j /κ 1 )L(κ/κ 1 , χ)M x (1/κ 1 )Γ(1/κ 1 -s k )y 1/κ 1 -s k (log x)y 1/2-v k e -(π/2)|t k | |t k | 1/2-v k 1 10
for all 3 x y T 100 .

Similarly, using the estimates

ζ(κ(α j + it k + iu)) (T + |u|) r , L(κ(α j + it k + iu), χ) (T + |u|) r , M x (α j + it k + iu) x 1/κ 1 -α j log x T 100
and the Stirling formula (4.9), we derive that (5.11)

|u| A log T Φ x (α j + it k + iu)Γ(α j -v k + iu) y α j -v k du 1 10
for all 3 x y T 100 . Taking (s, x) = (s k , N j ) in (5.8) and combining with (5.9), (5.10) and (5.11), we easily see that (5.12)

N j <n y 2 φ N j (n) n s k e -n/y 1 6
or (5.13)

A log T -A log T Φ N j (α j + it k + iu)Γ(α j -v k + iu)y α-v k +iu du 1 6
or both. Let K j (T ) and K j (T ) be the subsets of K j (T ) for which (5.12) and (5.13) hold respectively. Then (5.14)

|K j (T )| |K j (T )| + |K j (T )|.
First we bound |K j (T )|. By a dyadic argument, there is a U ∈ [N j , y 2 ] such that (5.15)

U <n 2U φ N j (n) n s k e -n/y 1 18 log y holds for |K j (T )|(log y) -1 integers k ∈ K j (T ).
Let S be the set of corresponding points s k . With the help of (5.6), it is easy to see that φ N j (n) = 0 ⇒ τ κ,κ (n) 1. Thus we can apply Proposition 4.1 with a n = φ N j (n) and θ = α j := 4σ j -3/κ 1 . In view of the bound

U <n 2U (τ κ,κ * τ κ,κ )(n) 2 n 2σ j e -2n/y e -2U/y 2U U t -2σ j d U <n t (τ κ,κ * τ κ,κ )(n) 2
U 1/κ 1 -2σ j (log T ) 3 e -2U/y , it follows that (5.16)

s k ∈S U <n 2U φ N j (n) n s k e -n/y 2 U 2(1/κ 1 -σ j ) + |S |U -2(1/κ 1 -σ j ) M(α j , 4T ) e -2U/y (log T ) 3 .
Since U N j , we have

U -2(1/κ 1 -σ j ) (log T ) 3 M(α j , 4T ) A -1 (log T ) -2 .
On the other hand, the inequality (5.15) implies that the member on the left-hand side of (5.16) is |S |(18 log y) -2 |S |(1800 log T ) -2 . Since A is a fixed large integer, the last term on the right-hand side of (5.16) is smaller than this lower bound. Thus it can be simplified as

|S |(log T ) -2 U 2(1/κ 1 -σ j ) (log T ) 3 e -2U/y
for all N j y T 100 and some U ∈ [N j , y 2 ]. Noticing that

|S | |K j (T )|(log T ) -1 ,
we obtain

(5.17)

|K j (T )| y 2(1/κ 1 -σ j ) (log T ) 6
(for all N j y T 100 )

N (10/3)(1/κ 1 -σ j ) j (log T ) 7/3
(for y given by (5.19)).

Next we bound

|K j (T )|. Let u k ∈ [-A log T, A log T ] such that Φ N J (s k ) = max |u| A log T |Φ N j (α j + i(t k + u))|
where s k := α j + it k and t k := t k + u k . Thus from (5.13) we deduce that 1 6

A log T -A log T Φ N j (α j + i(t k + u))Γ(α j -v k + iu)y α j -v k +iu du y α j -v k Φ N j (s k ) A log T -A log T Γ(α j -v k + iu) du.
Since Γ(s) has a simple pole at s = 0 and |α j -v k | (log T ) -1 , we can derive, via (4.9), that

A log T -A log T Γ(α j -v k + iu) du log T
and thus 1 y α j -σ j M N j (s k ) M(α j , 8T ) log T, or equivalently

M N j (s k ) y σ j -α M(α j , 8T ) log T -1 .
Hence there is a V ∈ [1, N j ] such that

V <n 2V τ -1 κ (n; χ)n -s k y σ j -α M(α j , 8T ) -1 (log T ) -2
holds for |K j (T )|(log T ) -1 integers k ∈ K j (T ). Let S be the corresponding set of points s k . We note |t k | 2T and

|t k 1 -t k 2 | |t k 1 -t k 2 | -|u k 1 -u k 2 | A log T.
Using Proposition 4.1 with θ = α j := 4σ j -3/κ 1 and a n = τ -1 κ (n; χ) and in view of the bound

V <n 2V τ κ,κ (n) 2 n -2α j V 1/κ 1 -2α j (log V ) 3 V 7/κ 1 -8σ j (log V ) 3 ,
it follows that (5.18)

s k ∈S V <n 2V τ -1 κ (n; χ)n -s k 2 V 8(1/κ 1 -σ j ) + |S |M(α j , 8T )V 4(1/κ 1 -σ j ) (log V ) 3 .
Take y such that

(5.19) y 2(σ j -α j ) = A 3 N 4(1/κ 1 -σ j ) j M(α j , 8T ) 3 (log T ) 4 = N 10(1/κ 1 -σ j ) j (log T ) -11 .
The left-hand side of (5.18) is |S |y 2(σ j -α j ) M(α j , 8T ) -2 (log T ) -4 .

Hence the inequality (5.18) can be simplified as

|S |y 2(σ j -α j ) M(α j , 8T ) -2 (log T ) -4 N 8(1/κ 1 -σ j ) j . With |S | |K j (T )|(log T ) -1 ,
we deduce that (5.20)

|K j (T )| N 8(1/κ 1 -σ j ) j y 2(α j -σ j ) M(α j , 8T ) 2 (log T ) 5 N 2(1/κ 1 -σ j ) j (log T ) 7 .
On combining (5.14), (5.17) and (5.20), it follows that

|K j (T )| N (10/3)(1/κ 1 -σ j ) j (log T ) 3 .
Now the required inequality follows from (3.10). This completes the proof.

6. Proof of Theorem 1.1

We shall conserve the notation in Section 2. First we prove a lemma. Lemma 6.1. Let r ∈ N, κ := (κ 1 , . . . , κ r ) ∈ N r with 1 κ 1 < • • • < κ r 2κ 1 , z := (z 1 , . . . , z r ) ∈ C r , B := (B 1 , . . . , B r ) ∈ (R + * ) r , and let α > 0, δ 0, A 0, M > 0 be some constants. Suppose that the Dirichlet series

F(s) := ∞ n=1 f (n)n -s
is of type P(κ, z, w, B, C, α, δ, A, M ). Then there is an absolute positive constant D and a constant

B = B 1 + • • • + B r + C 1 + • • • + C r such that we have (6.1) F(s) M D B T (δ+B √ ε)(1-κ 1 σ) (log T ) A+B
for all s ∈ L T , where the implied constant depends only on ε.

Proof. Since we have chosen the principal value of complex logarithm, we can write

ζ(κs) z L(κs, χ) w = 1 i r |ζ(κ i s)| e z i |L(κ i s, χ i )| e w i e -( m z i ) arg ζ(κ i s)-( m w i ) arg L(κ i s,χ i ) e π(B 1 +•••+Br+C 1 +•••+Cr) 1 i r |ζ(κ i s)| e z i |L(κ i s, χ i )| e w i
for all s ∈ C verifying 1 i r ζ(κ i s)L(κ i s, χ i ) = 0. Invoking Proposition 3.1, we see that there is a suitable absolute constant D and a constant B = B(B, C) depending on (B, C) such that (6.2)

ζ(κs) z L(κs, χ) w ε,χ D B T B √ ε(1-κ 1 σ) (log T ) B
for all s ∈ L T , where the implied constant depends only on (ε, χ).

Finally the required bound (6.1) follows from (6.2) and the hypothesis (1.6).

Now we are ready to prove Theorem 1.1. Since the Dirichlet series F(s) is of type P(κ, z, w, B, C, α, δ, A, M ), we can apply [19, Corollary II.2.2.1] with the choice of parameters σ a = 1/κ 1 , α = α, σ = 0 to write

x<n x+x 1-1/κ 1 y f (n) = 1 2πi b+iT b-iT F(s) (x + x 1-1/κ 1 y) s -x s s ds + O ε M x 1/κ 1 +ε T ,
where b = 1/κ 1 + 1/ log x and e

√ log x
T x is a parameter to be chosen later. Denote by Γ T the path formed from the circle |s -1/κ 1 | = r 0 := 1/(2κ 1 log x) excluding the point s = 1/κ 1 -r 0 , together with the segment [(1 -δ T )/κ 1 , 1/κ 1 -r 0 ] traced out twice with respective arguments +π and -π. By the residue theorem, the path

[b -iT, b + iT ] is deformed into Γ T ∪ [(1 -δ T )/κ 1 -iT, (1 -δ T )/κ 1 + iT ] ∪ [(1 -δ T )/κ 1 ± iT, b ± iT ].
In view of Lemma 6.1, for any a ∈ (1/(2κ 1 ), 1/κ 1 ), the integral over the horizontal segments

[a ± iT, b ± iT ] is b±iT a±iT F(s) (x + x 1-1/κ 1 y) s -x s s | ds| M D C (log T ) A+B T b a T max{(δ+B √ ε)(1-κ 1 σ), 0} x σ dσ M D C x 1/κ 1 T (log T ) A+B 1/κ 1 a x 1/κ 1 T δ+B √ ε κ 1 σ-1 dσ + 1 M D C x 1/κ 1 T (log T ) A+B , provided (6.3) T δ+B √ ε x 1/κ 1 .
Thus (6.4)

x<n x+x 1-1/κ 1 y f (n) = I + O M D C x 1/κ 1 +ε T ,
where

I := 1 2πi Γ T ∪[(1-δ T )/κ 1 -iT, (1-δ T )/κ 1 +iT ] F(s) (x + x 1-1/κ 1 y) s -x s s ds.
and the implied constant depends on (ε, χ) only. Let L T be the Motohashi contour defined as in Section 2. Consider the two symmetric simply connected regions bounded by L T , the segment [(1 -δ T )/κ 1 -iT, (1 -δ T )/κ 1 + iT ] and the two line segments [σ j 0 +1 + d v , (1 -δ T )/κ 1 ] with respective arguments +π and -π measured from the real axis on the right of 1 -δ T . It is clear that F(s) is analytic in these two simply connected regions. Denote by Γ * T the path joining (the two end-points of) Γ T with the two line segments [σ j 0 +1 + d v , (1 -δ T )/κ 1 ] of the symmetric regions. Thanks to the residue theorem, we can write (6.5)

I = I 1 + I 2 ,
with

I 1 := 1 2πi Γ * T F(s) (x + x 1-1/κ 1 y) s -x s s ds, I 2 := 1 2πi L T F(s) (x + x 1-1/κ 1 y) s -x s s ds.
A. Evaluation of I 1 According to our hypothesis, the function s → Z(κ

1 s; z 1 )ζ(κ * s) z * L(κs, χ) w G(s) is holo- morphic in the disc |s -1/κ 1 | < 1/κ 1 -1/κ 2 .
In view of (1.12), the Cauchy integral formula implies that

(6.6) g (κ, z, w, χ) M c - ( 0, |z| |B|, |w| |C|),
where g (κ, z, w, χ) is defined as in (1.14) and c := 2 3 (1/κ 1 -1/κ 2 ). From this and (1.13), we deduce that for any integer N 0 and |s -

1/κ 1 | 1 2 (1/κ 1 -1/κ 2 ), Z(κ 1 s; z 1 )ζ(κ * s) z * L(κs, χ) w G(s) = N =0 g (κ, z, w, χ) s -1 κ 1 + O M (|s -1 κ 1 |/c) N +1 .
Thus we have (6.7)

I 1 = N =0 κ -z 1 1 g (κ, z, w, χ)M (x, y) + O M c -N E N (x, y) ,
where

M (x, y) := 1 2πi Γ * T (s -1/κ 1 ) -z 1 (x + x 1-1/κ 1 y) s -x s s ds, E N (x, y) := Γ * T (s -1/κ 1 ) N +1-z 1 (x + x 1-1/κ 1 y) s -x s s | ds|.
Firstly we evaluate M (x, y). Using the formula (6.8)

(x + x 1-1/κ 1 y) s -x s s = x+x 1-1/κ 1 y x t s-1 dt
and Corollary II.5.2.1 of [START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF], we write

M (x, y) = x+x 1-1/κ 1 y x 1 2πi Γ * T (s -1/κ 1 ) -z 1 t s-1 ds dt = x+x 1-1/κ 1 y x t 1/κ 1 -1 (log t) z 1 -1- 1 Γ(κ 1 -) + O (c 1 + 1) t δ T /2 dt,
where we have used the following inequality

47 |z 1 -| Γ(1 + |z 1 -|) B 1 (c 1 + 1) ( 0, |z 1 | B 1 ).
The constant c 1 and the implied constant depend at most on B 1 . Besides for |z 1 | B 1 , an elementary computation shows that

x+x 1-1/κ 1 y x t 1/κ 1 -1 (log t) z 1 -1-dt = x 1-1/κ 1 y 0 (x + t) 1/κ 1 -1 (log(x + t)) z 1 -1-dt = y (log x) z 1 -1-1 + O B 1 ( + 1)y x 1/κ 1 log x .
Inserting this into the preceeding formula, we obtain (6.9) M (x, y) = y (log x)

z 1 -1- 1 Γ(z 1 -) + O B 1 ( + 1)y Γ(z 1 -)x 1/κ 1 log x + (c 1 + 1) x δ T /2
for 0 and |z 1 | B 1 . Next we estimate E N (x, y). In view of the trivial inequality (6.10)

(x + x 1-1/κ 1 y) s -x s s yx σ-1/κ 1 ,
we deduce that (6.11)

E N (x, y)

1/κ 1 -1/ log x 1/2κ 1 +ε 2

(1/κ 1 -σ) N +1-e z 1 x σ-1/κ 1 y dσ + y (log x) N +2-e z 1 y (log x) N +2-e z 1 ∞ 1 t N +1-e z 1 e -t dt + 1 y(log x) e z 1 -1 c 1 N + 1 log x N +1

uniformly for x y 2, N 0 and |z 1 | B 1 , where the constant c 1 > 0 and the implied constant depends only on B 1 . Inserting (6.9) and (6.11) into (6.7) and using (6.6) and the fact that y y, we find that (6.12)

I
I 2 M D B x 1/2κ 1 +ε 2 T (δ+B √ ε)(1/2-κ 1 ε 2 ) (log T ) A+4B+1
M x (1/2+δ/2(ψ+δ))/κ 1 + √ ε M x (1-1/(ψ+δ))/κ 1 + √ ε with the value of T given by (6.15) below and ψ 2.

Next we bound I 2 . In view of (6.10), we can write that T (δ+B √ ε)(1-κ 1 (σ j +dv)) x σ j +dv-1/κ 1 • T ψ(1-κ 1 σ j ) , I 2, † := J T,0 <j J T T (δ+B √ ε)(1-κ 1 (σ j +dv)) x σ j +dv-1/κ 1 • T 100 √ ε(1-σ j ) .

Taking (6.15)

T := x (1-κ 1 √ ε)/κ 1 (ψ+δ+B √ ε)
and in view of (2.23), it is easy to check that I 2, * x ε 2 /κ 1 0 j J T,0

x 1/κ 1 /T ψ+δ+B √ ε -(1-κ 1 σ j ) log x x 2ε 2 /κ 1 -ε 3/2 x -ε 2 and I 2, † J T,0 <j J T x/T δ+100(B+1) √ ε -(1-κ 1 σ j )
e -2c 2 (log x) 1/3 (log 2 x) -1/3 .

Inserting it into the preceeding estimate for I 2 , we conclude that (6.16) I 2 B M ye -c 2 (log x) 1/3 (log 2 x) -1/3 . Now from (6.4), (6.5), (6.12), (6.13) and (6.16). we deduce that + (c 1 N + 1) N +1 e c 2 (log x) 1/3 (log 2 x) -1/3 for some constants c 1 > 0 and c 2 > 0 depending only on B, C, δ and ε.

It remains to prove that the first term on the right-hand side can be absorbed by the third. In view of (1.12), the Cauchy formula allows us to write g (κ, z, w, χ) M y x 1/κ 1 log x holds uniformly for x 3, x (1-1/(ψ+δ))/κ 1 +ε y x 1/κ 1 , N 0, |z| B and |w| C. This completes the proof.

  [START_REF] Bateman | On a theorem of Erdős and Szekeres[END_REF], where the implied constant depends only on ε;(b) We have∞ n=1 |f (n)|n -σ M (σ -1/κ 1 ) -α (σ > 1/κ 1 );(c) The Dirichlet series F(s) has the expression(1.3) F(s) = ζ(κs) z L(κs; χ) w G(s),whereζ(κs) z := 1 i r ζ(κ i s) z i , (1.4) L(κs; χ) w := 1 i r L(κ i s, χ i ) w i (1.5) and the Dirichlet series G(s) is a holomorphic function in (some open set containing) σ (2κ 1 ) -1 and, in this region, G(s) satisfies the bound (1.6) |G(s)| M (|τ | + 1) max{δ(1-κ 1 σ),0} log A (|τ | + 3) uniformly for |z| B and |w| C, where and in the sequel we implicitly define the real numbers σ and τ by the relation s = σ + iτ and choose the principal value of the complex logarithm.

1 . 3 .

 13 and the implied constant in the O-term depends only on A, B, C, α, δ and ε. Note that ψ =12 5 is admissible. Takning r = 2, κ = (1, 2), z = (z, w), w = (0, 0) in Theorem 1.1 and Corollary 1.1, we can obtain Theorem 1.1 and Corollary 1.2 of Cui, Lü & Wu[START_REF] Cui | The Selberg-Delange method in short intervals with some applications[END_REF]. Distribution of divisors de certains integers.

  k of class (Y ) only. The Hooley-Huxley-Motohashi contour L T = L T (ε, κ, χ) is sysmmetric about the real axe. Its supérieur part is the path in D 0 consisting of horizontal and vertical line segments whose distances away from D are respectively d h and d v , where d h and d v are defined by (2.23)

Figure 1 - 3 .

 13 Figure 1 -Superieur part of the contour L T

  (ζ(κs)L(κs, χ)) on the circles C 3 , C 4 . On the circle C 4 , it is well known that e log(ζ(κs)L(κs, χ)) = log |ζ(κs)L(κs, χ)| log τ thanks to the convexity bounds of ζ(s) and of L(s, χ). Hence the Borel-Carathéodory theorem gives (3.6)

( 4 . 1 )

 41 σ σ 0 , 1 |τ | T for all s ∈ S and (4.2) |τ -τ |

Lemma 4 . 2 .

 42 + s )|. Noticing that |B(s + s )| = |B(s + s )| = |B(s + s )| = |B(s + s )|, the precede inequality becomes Υ s∈S |A(s)| 2 max s∈S s ∈S |B(s + s )|. Inserting it into (4.6), we get the required result. The next lemma is an analogue of [12, page 157, Theorem II.2]. Let r, κ, τ κ,κ (n) and M(θ, T )be as before. For any s = σ + iτ with σ 0 and |τ | 1, real numbers θ ∈ (1/κ 2 , 1/κ 1 ) and N 1, we have

  ) and write (5.4) Φ x (s) := ζ(κs)L(κs, χ)M x (s).

1 2κ 1 +

 11 1 = y (log x) z-1 N =0 λ (κ, z, w, χ) (log x) + O B 1 E * N (x, y), Evaluation of I 2 Let L T be the union of those vertical line segments of L T whose real part is equal to ε 2 and L T := L T L T . Denote by I 2 and I 2 the contribution of L T and L T to I 2 , respectively. Using the trivial inequality(x + x 1-1/κ 1 y) s -x s s x 1/2κ 1 +ε 2 |τ | + 1 (s ∈ L T )and Lemma 6.1, we can deduce(6.13)

  |x σ-1/κ 1 | ds| y 0 j J T 0 k K T ∆ j,k ∈(W ) L [j,k] T |F(s)|x σ-1/κ 1 | ds|,where L [j,k] T is the vertical line segment of L T around ∆ j,k and the horizontal line segments with σ σ j + d v . Clearly the length of L [j,k] T is log T . Thus by Lemma 6.1, it followsL [j,k] T |F(s)|x σ-1 | ds| M D B (log T ) A+4B+1 T (δ+B √ ε)(1-κ 1 (σ j +dv)) x σ j +dv-1/κ 1for all 0 k K T . Inserting it into (6.14) and using Proposition 5.1, we can deduce, with the notation J T,0 := [( 1 2 -ε) log T ], that I 2 M D B y(log T ) A+4B+18+η (I 2, * + I 2, † ), where I 2, * := 0 j J T,0

x<n x+x 1

 1 -1/κ 1 y f (n) = y (log x) z-1 N =0 λ (κ, z, w, χ) (log x) + O R * N (x, y)uniformly for x 3, x (1-1/(ψ+δ))/κ 1 +ε y x 1/κ 1 , N 0, |z| B and |w| C, where R * N (x, y)

  A,B,C,δ M 3 for |z| B, |w| C and 1. Combining this with the Stirling formula, we easily derive λ (κ, z, w, χ) A,B,C,δ M (9/ ) for |z| B, |w| C and 1. This implies that y x 1/κ 1 N +1 =1 |λ -1 (κ, z, w, χ)| (log x) A,B,C,δ

Acknowledgement. This work is supported in part by NSFC (Grant Nos. 11771121 and 11471265).