A Maurel-Pantel

M Fontaine

G Michel

S Thibaud

J C Gelin

Dr Ing Aurélien Maurel-Pantel

Experimental investigations from conventional to high speed milling on a 304L stainless steel

Keywords: Machinability, High-speed milling, Cutting forces, Tool run-out, 304L stainless steel. Nomenclature  Density c Calorific capacity T Temperature Cumulated plastic strain

Last years analytical or finite element models of milling become more efficient and focus on more physical aspects, nevertheless the milling process is still experimentally unknown on a wide range of use. This paper propose to analyse with accuracy milling operations by investigating the cutting forces values, shape of cutting forces curves obtained for different cutting speeds, and related phenomena as tool wear or tool run-out. These detailed experimental data in milling constitute a suitable experimental basis available to develop predictive machining modelling. All the tests have been conducted on the 304L stainless steel in many cutting configurations and for different tool geometries. The machinability of the 304L stainless steel with different tools geometries and configurations in shoulder milling is defined by three working zones: a conventional zone permitting stable cutting (low cutting speed; under 200-250 m.min -1 ), a dead zone (unfavourable for cutting forces level and cutting stability; between 250 and 450 m.min -1 ), and a high speed machining zone (high cutting speed; up to 450-500 m.min -1 ). All the used criteria (cutting forces, chips, wear) confirm the existence of these different zones and a correlation is proposed with cutting perturbations as run-out, cutting instability, ploughing and abrasive wear.

Introduction

Milling is one of the most frequently used machining processes in mechanical industries. But some essential physical phenomena are still difficult to understand. Indeed milling is a very complex process with physical interactions between shearing, compression, ploughing, friction, high strain rate, thermal effects, failure and forced vibrations. Nevertheless manufacturing industries own a good empirical knowledge of the process and search essentially to optimise productivity. Although, in academic research field, works on machining focus mainly on turning process and try to extrapolate issues to the milling process.

Many empirical, analytical or finite element models have been developed in orthogonal cutting, in oblique cutting or more recently in 3D turning and 3D milling. At the beginning, Merchant [START_REF] Merchant | Mechanics of the metal cutting process, i. orthogonal cutting[END_REF] has proposed the first basic analytical model in orthogonal cutting taking into account only shearing effects in a specific plane orthogonal to the cutting edge. Then, Oxley [START_REF] Oxley | Mechanics of metal cutting[END_REF] has improved the model with introduction of semi empirical relations and a behaviour law for the machined material. More recently, Ceretti et al. [START_REF] Ceretti | Application of 2D FEM to chip formation in orthogonal cutting[END_REF] have proposed a finite element model of a 2D orthogonal cutting to reproduce the chip formation with different cutting conditions. Then Baker [START_REF] Baker | Finite element investigation of the flow stress dependence of chip formation[END_REF] proposed a finite element model in orthogonal cutting and validate the cutting predictions with experimental cutting forces confrontation. The first milling models were developed with an empirical description of local cutting process based on cutting coefficients, and by using an axial decomposition of the mill to consider elementary straight edges in oblique cutting position by [START_REF] Kline | The prediction of cutting forces in end milling with application to cornering cuts[END_REF] and improved after by Yellowley [START_REF] Yellowley | Observations of the mean values of forces, torque and specific power in peripheral milling process[END_REF]. Many models based on the same approach were proposed during the twenty following years with some remarkable advances on ball-end milling by Yang and Park [START_REF] Yang | The prediction of cutting force in ball-end milling[END_REF], in milling process dynamics by Altintas and Lee [START_REF] Altintas | A general mechanics and dynamics model for helical end mills[END_REF], in an advanced cutters geometrical description by Engin and Altintas [START_REF] Engin | Mechanics and dynamics of general milling cutters. Part I: helical end mills[END_REF][START_REF] Engin | Mechanics and dynamics of general milling cutters. Part II: inserted cutters[END_REF], or even free-form machining by Zhu et al. [START_REF] Zhu | Mechanistic modelling of the ball end milling process for multi-axis machining of free-form surfaces[END_REF]. But very few more complete models of milling were proposed more complete including specifically an accurate description of different tool geometries, of tool complex pathes and engagement, and taking into consideration the tool run-out.

Only one model mixed these developments with a thermo-mechanical description of local cutting process with the purpose to enhance the understanding of milling cutting process, Fontaine et al. [START_REF] Fontaine | Predictive force model for ball end milling and experimental validation with a wavelike form machining test[END_REF] validate strong hypotheses associated to oblique cutting and propose a more predictive approach. This model uses the shear band model developed for high speed machining by Molinari et al. [START_REF] Molinari | Stationnary shear band in high speed machining[END_REF] and Dudzinski et al. [START_REF] Dudzinski | A modelling of cutting for viscoplastic materials[END_REF] and the analytical modelling of the 3D oblique cutting configuration developed by Moufki et al. [START_REF] Moufki | Thermomechanical modelling of oblique cutting and experimental validation[END_REF]. And now Zeroudi et al. [START_REF] Zeroudi | Prediction of cutting forces in 3-axes milling of sculptured surfaces directly from CAM tool path[END_REF] have integered in model the use of CAM tool paths and Zeroudi and Fontaine [START_REF] Zeroudi | Prediction of machined surface geometry based on analytical modelling of ball-end milling[END_REF] have also predicted the produced form and roughness. More recently finite elements models have been extended to milling process but only few detailed experimental research are available in literature : Shao et al. [START_REF] Shao | Machinability study on 3%Co-12%Cr stainless steel in milling[END_REF] proposed a machinability study on 3%Co-12%Cr stainless steel in milling and D'Errico et al. [START_REF] D'errico | Tool-life reliability of cermet inserts in milling tests[END_REF] studied the tool-life reliability of cermet inserts with milling tests. In the FEMTO-ST Institute, a 3D finite element model of milling has been developped with the commercial software LS-Dyna by Maurel-Pantel et al. [START_REF] Maurel-Pantel | 3D FEM simulations of shoulder milling operations on a 304L stainless steel[END_REF]. This milling simulation is able to model the chip formation, to predict the cutting forces curves and to furnish thermo-mechanical information during the cutting process. Some experimental research have been also conducted to determine the machinability of the material. Tekiner et al. [START_REF] Tekiner | Investigation of the cutting parameters depending on process sound during turning of AISI 304 austenitic stainless steel[END_REF] have developed measurement methods to compare process sound with the cutting parameters in turning of AISI 304 austenitic stainless steel. Sullivan et al. [START_REF] O'sullivan | Machinability of austenitic stainless steel SS303[END_REF] have proposed the on-line work hardening detection techniques to study the machinability of SS303 austenitic stainless steel. Salak et al. [START_REF] Salak | New short time face turning method for testing the machinability of PM steels[END_REF] have proposed an experimental method to compare the machinability of different powder metallurgy steels in face turning configuration. Other experimental research have been conducted to analyse some different aspects in stainless steel machining as tool wear in order to predict the tool life like in the works of Senthil Kumar et al. [START_REF] Kumar | The effect of tool wear on tool life of alumina-based ceramic cutting tools while machining hardened martensistic stainless steel[END_REF] or in order to compare free cutting additives influences on cutting conditions as in Akasawa et al. [START_REF] Akasawa | Effects of free cutting additives on the machinability of austenetic stainless steels[END_REF], or to study the surface integrity of machined workpiece as in Shao et al. [START_REF] Shao | Machinability study on 3%Co-12%Cr stainless steel in milling[END_REF], to identify and develop modelling of friction conditions as in Valiorgue et al. [START_REF] Valiorgue | Modelling of friction phenomena in material removal processes[END_REF][START_REF] Valiorgue | 3D modeling of residual stresses induced in finish turning of an AISI304L stainless steel[END_REF] and improve numerical model predictivity in turning simulation as in Bonnet et al. [START_REF] Bonnet | Identification of a friction model-Application to the context of dry cutting of an AISI 316L austenetic stainless steel with a TiN coated carbide tool[END_REF].

Last years analytical or finite element models of milling become more efficient and focus on more physical aspects, nevertheless the milling process is still experimentally unknown on a wide range of use in literature. Now, it seems interesting to improve the predictive capacity of these models on a wide machining range. Milling experiments in many conditions are necessary to analyse, to understand and to quantify the experimental phenomena with for example the help of cutting forces measurement directly during milling operations.

The works related in the present paper describe a set of milling tests lead on a common stainless steel (AISI 304L). The aim consist to analyse the material behaviour in a wide range of machining conditions, from low speed to high speed machining, in order to obtain a wide experimental knowledge of the machinability of this steel in milling. In this work, the analysis essentially leans on the direct measure of cutting forces during the milling operations and examination of the classical machining results (chips, wear, surface integrity…). The final aim consists to build an accurate experimental basis to develop and improve analytical and finite element models of milling on a wide range of use.

The 304L stainless steel

The 304L stainless steel is widely used in product manufacturing because it presents a very good forming capacity, an excellent resistance to corrosion and a high ductility. In last years, use of AISI 304L was widely expanded to other fields as chemical, biomedical and power plant industries. The steel grade 304 represents today 80 % of the world-wide stainless steel market but austenitic stainless steels are usually considered as quite difficult to machine. Actually stainless steel are commonly machined with a maximal cutting speed equal to 250 m.min-1 but this limit increases continually with improvements in machine-tools and tools design. The machinability of this steel is more difficult than other alloy steels due to low heat conductivity, high built-up edge tendency and high deformation hardening. The 304L stainless steel has a face centred cubic structure, and it is composed of purely austenitic phases but a mechanical solicitation creates carbide and martensite '. The chemical composition of the material used for experiments is related in Table 1. [START_REF] Johnson | Constitutive model and data for metals subjected to large strains, high strain rates, and high temperatures[END_REF] and a modified Merchant law [START_REF] Merchant | Mechanics of the metal cutting process, i. orthogonal cutting[END_REF]. The parameters related in Table 2 and Table 3 are identified with an inverse method based on an objective function build with the measured and numerical cutting forces [START_REF] Maurel-Pantel | 3D FEM simulations of milling on a 304L stainless steel[END_REF][START_REF] Maurel | Inverse Method for Identification of Material Parameters Directly from Milling Experiments[END_REF].

) (MPa A ) (MPa B n C m ) ( 1 0  s  ) ( K T f  ) ( 0 K T 
                                        m f n T T T T C B A R 0 0 0 1 ln 1      (1)
Where A corresponds to yield strength, B and n corresponds to the hardening coefficients, C defined the strain rate sensitivity, and m defined the temperature dependency coefficient. f T and 0 T are respectively the melting temperature and the ambient temperature.

    f f f n Arc λ λ A A    tan with 2 1     (2)
Where f  and f  are respectively the average friction angle and friction coefficient at the tool/chip interface, n  corresponds to the normal rake angle defined by the tool geometry and local direction of cutting velocity. 1 A and 2 A parameters depending of the machined material.

Milling experiments protocol

Micro-milling machine

All the milling tests described in this paper were performed at ENSMM (Engineering School in Mechanics and Microtechnics) in Besançon, on a three axes KERN© HSPC2525 micro-milling machine. This machine allows reaching high cutting speeds (spindle frequency up to 35000 rev/min) and guaranties very good repeatability and stability of the tests due to its high rigidity (concrete polymer frame) and high precision (+/-0.5 µm positioning).

Cutting forces measurement setup

The cutting forces measurement is performed with a Kistler dynamometer 9272.

Classically this product is used in drilling, but it presents a good polyvalence and especially a high frequency bandwidth. The 4 sensors of the device deliver an electric signal proportional to the loads in X, Y and Z directions and the cutting force on each axis are obtained by summation. The dynamometer is connected to a PC through a 5070 Kistler charge amplifier and a National Instruments input/output data acquisition card (sample frequency limit: 40 kHz). Then data is treated through a specific software, in which the sample frequency available is adjustable and always corresponds to an integer multiple of the tooth path frequency. The main device characteristics are exposed in Table 4.

Components Acquisition range

Frequency limits precision

Sample geometry

The samples in 304L stainless steel have been machined with a wire EDM process from extruded and hyperquenched bars furnished by Thyssen France. The samples length was chosen in order to have the centre of the tool path in a stabilised machining zone (geometry and setting are exposed on Figure 1). The sample is fixed on the dynamometer through an aluminium part in order to fix properly the sample and to reduce the weight to keep the frequency range. With these cautions and checking good positioning, the solicitations on sensors are balanced and a good operating is assured.

Acquisition limits in milling

The validity range of the cutting forces measurements is described by the cutting speed range available with our acquisition system. Figure 2 exposes the experimental system limits but also describe the actual technological difficulties to reach high cutting speed in milling. The case of a milling operation with a 6 mm diameter end mill is considered accordingly to the experiments.

Figure 2 relates the maximum number of sample points available to describe a tooth path and underlines the technological limits of the experimental setup.

 The first limit consists to keep at minimum 20 sample points by tooth to obtain a good description of the cutting forces curves. This first condition determined by the maximal frequency of the acquisition card, limit the experimental range to a maximal cutting speed of 1130 m.min-1.

 Then, two technological limits appear: the acquisition frequency maximal limit of the dynamometer and the limit of the machine spindle. It is very important to perform the tests below the frequency limit of the dynamometer, that's why we pay attention in our experimental test to keep a gap of 30 % in order to avoid measurement errors due to amplification.

The machine spindle impose the more severe criterion in this configuration and the maximal cutting speed reachable experimentally is 660 m.min -1 .

Device used for the analysis

A topo-microscope Alicona has been used for qualitative and quantitative analyses of machined samples. This device is designed to perform optical measures without contact in three dimensions. It is based on an optical microscope coupling with a camera. Topo-microscopy is combination of two types of measure: microscopy and micro-topography. It is possible from obtained high resolution juxtaposed 3D fields to perform measure of dimensions, topography, waviness profile, roughness, surfaces and volumes.

Tested tools

In this study, tools are solid cutters made in tungsten carbide coated or uncoated.

The mills have been specifically chosen for their good dynamic balancing and good symmetry of cutting edges. The both aspects permit to respect the tool engagement induced by the feed, to avoid the amplification of the radial tool runout and to reduce vibration phenomena induced by a high spindle frequency. All mills used in experiments have a 6 mm diameter and two teeth in order to simplify the interpretation of cutting forces curves in function of the tool engagement in matter. At the beginning, tests have been led on with coated and uncoated end mills commercially proposed for such applications. The uncoated ones present the benefits to homogenize and simplify the experimental observations. Then, bull nose end mills have been tested for their robustness and to reach higher feed values. The cylindrical end mills (noted mill A) and the bull nose end mills (noted mill B) have been provided by two different manufacturers. The mills characteristics are described and exposed in Figure 3.

Tested cutting conditions

Different feeds and cutting configurations, and a large range of cutting speeds were tested with the selected tools. The reference values of cutting speed and feed recommended by tools manufacturers were about 150 m.min -1 and 0.03 mm.tooth - 1 respectively. The minimum and maximum values were defined after preliminary milling tests in association with the tool manufacturers. The cutting speed of 650m.min -1 corresponds to the maximum value reachable with a 6 mm diameter mill and the used spindle. All the parameters configurations were tested with a cutting speed pitch of 30 m.min -1 in order to obtain a complete and pertinent data about machinability of the tested 304L steel. All tested parameters are summed up in Table 5. All tests have been performed in slotting or shoulder milling in down and up cutting configurations. In all milling tests the tool engagement is defined with an axial depth of cut of a d = 1 mm and the radial depth of cut of r d = 3 mm (semi immersion) for the shoulder milling tests.

Analysis of milling experiments

All machining tests were led on with micro-lubrication in order to limit difficulties to control and to repeat the cooling conditions. Moreover, lubrication could add a complexity in modelling of the interface between tool and material.

This difficulty would be taken into account in numerical models developed, and could add incertitude in models predictivity. Finally, some cutting conditions show that machining stainless steels without lubricant can be efficient and it is interesting to investigate this more earth and health friendly configuration.

All the cutting forces curves measured for milling operations have the characteristic shape exposed on Figure 4. An unstable zone appears at the beginning and at the end of the acquisition. These zones correspond to entry and exit of the tool in sample. In the centre of acquisition curves appears a steady zone that is defined as reference for the different analyses. The vertical milling machine frame (X, Y, Z) is defined as reference for the cutting forces for all figures and discussions. The positive sense on these axes corresponds to the tool displacement in relation to the workpiece in accordance with conventions in machining. As a convention, all forces are considered like acting on the tool and the following legend is always respected in the paper:

 The blue colour is used for transversal forces x F .

 The red colour is used for feed forces y F .

 The green colour is used for vertical forces z F .

Cutting configurations in milling

In this section, the main tool engagement conditions in three axes milling are analysed from the cutting forces curves measured and calculated with our analytical model using a rigid description of the tool engagement and firstly proposed in Fontaine et al. [12] [32]. Discussions in this section focus mainly on the transverse force x F and feed force y F . Indeed, it is difficult to confront the vertical force z F with the tool engagement because z F force it shows lesser cutting force levels and is directly perturbed by non shearing cutting phenomena as ploughing mainly due to the cutting zone at tool end where cutting speed and chip thickness tend to zero. For instance Fontaine et al. [START_REF] Fontaine | Parametric geometry for modelling of milling operations[END_REF] described in slotting a confrontation between the negative values on z F obtained with an end mill and the positive values obtained with a bull nose end mill for which the projection of local forces on the tool change and the non shearing phenomena at tool end increase. The case considered corresponds to a mill with two teeth, but conclusions can be easily generalised for a mill designed with one, three or more teeth. A spectral analysis has been conducted on each test in order to verify that measures are not disturbed by unexpected frequencies. We are able with this protocol to have a good correlation and interpretation of cutting forces curves with the tool engagement in matter.

Slotting configuration

In slotting configuration, the radial depth of cut corresponds to the mill diameter ( r d = 6 mm). Classically, a d corresponds to the vertical depth of cut, the configuration is described in Figure 5. If a cross section in (X, Y) plane for an altitude z is considered, the tooth path related in Figure 5 corresponds to the section of material removed by a tooth on a height a d and a maximum width corresponds to the feed t f . The mill has a helix angle and the cutting edge follows this helix, that's why the cutting edge is not in the same place for different altitudes z considered. The tool engagement is totally three-dimensional with a trochoïdal geometry and imposes the chip motion and formation. These essential remarks put in evidence limits to make comparisons between turning process and milling process. It is proposed here to analyse cutting forces curves in slotting milling conditions with the assumptions of a rigid tool. The cutting forces curves for a tooth path obtained analytically are related in Figure 5 and the numbers used to locate the extremity of the mill cutting edge along the track in the matter are reported on the cutting forces graphic. The main steps during each tooth path are described in order to understand the curve shapes of a milling operation in slotting configuration. So, five stages are considered. Each one put in evidence a characteristic evolution of the cutting forces curve in the rigid case:

 The stage 1 corresponds to the entrance of the tooth in matter, and in slotting configuration it begins with a chip thickness close to 0 mm. On cutting forces curves, y F becomes negative when x F becomes positive and come back quickly to negative values. This phenomenon is due to projection of the forces on the relief face of the tool in contact with the machined surface, and this is characteristic to the tool entrance in matter.

 The stage 2 is the critical place for the feed force y F . Indeed this step is characterised by the maximal negative value on y F and a decrease to negative values on x F .

 The stage 3 corresponds to the maximal loading zone, in this zone the tooth removes a maximal thickness of matter equal to the feed t f , and the deformed volume of material is maximal. This step is characterised by the maximal negative value on transverse force x F and by the annulment of the feed force y F . The stage 3 is the critical place for the force x F .

Usually, this step corresponds also to the maximal value of the vertical force z F .

 The stage 4 is symmetric to the stage 2. This step is characterised by the maximal positive value on y F and a force x F tending to zero.

 The virtual stage 5 corresponds to the outgoing of the tooth in matter. Both forces y F and x F are close to zero corresponding to the end of the cutting cycle for the first tooth. In case of slotting with a two teeth end mill, the curves begin immediately another identical cycle with the second tooth as related in Figure 6 (a). An angular difference appears between the tool position located and the different stages on the curves. This is due to the fact that the maximal loads are located on a point in the median immersion plane of the mill, and it depends on the depth of cut a d and the value of the helix angle. The tooth angular position is considered at the extremity of the tool cutting edge and this choice induced the angular difference observed on the curves. For a mill with more than two teeth, the interpretation would not be easy but the same tendencies will be observed on the resultant force on the tool. This analysis allows having a better understanding to analyse the experimental curves.

Shoulder milling in up cutting configuration

In up cutting configurations in shoulder milling, the radial depth of cut is strictly inferior to the mill diameter (

  r d
). The operation is a contouring one and consists to remove the matter only on a side of the tool with an axial depth of cut a d . Up cutting configuration is described on figure 7 and corresponds to the case where the matter comes on a mill tooth globally in the opposite way than the tool displacement in relation to the workpiece. The teeth begin to remove matter with a thickness equal to 0 mm and finish cutting with a varying thickness (maximal in case of radial semi-immersion and equal to feed per tooth t f ). The shoulder milling case in up cutting is considered in this paper for a radial semi-immersion

(i.e. 2   r d
). If a cross section in (X, Y) plane is considered as previously, the tooth path related in Figure 7 corresponds to the section of matter removed by a tooth. Figure 7 In this step no tooth is not in contact with matter and the three cutting forces are equal to 0 as related in Figure 7. For a mill with more than two teeth, the interpretation would not be easy but the same tendencies will be observed on the resultant force on the tool. It is interesting to compare the cutting forces curves of shoulder milling in up cutting configurations for three cutting speeds with the analytically calculated curves. describe the same curves shape than described previously. There is a good correlation for low cutting speed with the assumption of an undeformable tool classically used in machining models. In this case between 0 and 180 m.min -1 , analytical modelling gives a good description of cutting forces. For higher cutting speeds, some stages disappear. For instance on chart (c) there is no continuous zero value and on (d) the nickpoint has disappeared, the curves are sinusoidal and the cutting process seems to be continuous. Cutting forces curves (d) seems to present the same curve shapes than in slotting operations. Actually, rigid machining models are not able to be predictive in this machining zone. Vibrations phenomena and flexible behaviour of the tool have to be considered in modelling in order to reach high speed milling zones.

Shoulder milling in down cutting configuration

In down cutting configurations in shoulder milling, the radial depth of cut is strictly inferior to the mill diameter (

  r d
). The operation is the symmetric of up cutting. Down cutting configuration corresponds to the case where the matter comes on a mill tooth globally in the same way than the tool displacement in relation to the workpiece. The teeth begin to remove matter with a maximal thickness (equal to t f in semi-immersion case) and finish cutting with thickness equal to 0 mm. This configuration is described in Figure 9. The shoulder milling case in down cutting is considered in this paper for a radial semi-immersion (i.e.

2   r d
). If a cross section in (X, Y) plane is considered as previously, the tooth path related in Figure 9 corresponds to the section of material removed by a tooth. In this step no tooth is in contact with matter.

As previously, Figure 10 describes four sets of cutting forces curves in down cutting configuration. There is a good correlation for low cutting speeds with the assumption of an undeformable tool. Between 0 and 230 m.min -1 in this case, the analytical model predict well the cutting forces. Between 0 and 230 m.min -1 in this case, the analytical model predict well the cutting forces.

Machinability of the 304L stainless steel

This analysis focus on the machinability of 304L stainless steel and its behaviour for different tools and in a wide range of cutting speeds. The main criterion is the influence of tools and cutting conditions on the measured cutting forces. In the following section, the average peak-to-peak amplitude values of transverse forces

Results obtained with mill A

In the experiments with the end mill A, analysis focus on differences observed between coated and uncoated end mills as related in Figure 11, and following conclusions appeared:

 In both cases (coated and uncoated mill), cutting forces values increase globally proportionally to the feed.

 For the two tools machinability curves are similar, but a difference has to be noted, particularly for high feed values: the levels of cutting forces for the coated mills are appeared slightly lower than for the uncoated ones.

The TiN based coatings are well known to significantly reduce the friction at the tool-material interface [START_REF] More | Tool wear and machining performance of cBN-TiN coated carbide inserts and PCBN compact inserts in turning AISI 4340 hardened steel[END_REF] and it is particularly sensitive in this case because of high load associated to high feed rate and of the sticking behaviour of stainless steels in machining [START_REF] Chandrasekaran | Chip Flow and Notch Wear Mechanisms during the Machining of High Austenitic Stainless Steels[END_REF].

 On all curves, three working zones are observed: a first zone for small cutting speeds where cutting forces are constant or presenting a light decrease; then a critical second zone where the cutting forces increase strongly appears; and finally a third zone with high cutting speeds where cutting forces decrease strongly and reach values lower than in first zone.

 On all curves, the secondary zone is less favourable and is more critical for high feed values.

Results obtained with mill B

The results obtained with the bull nose end mill B are here analysed. In this case, the both configurations in shoulder milling have been tested and compared. The curves are related in function of cutting speeds and feeds in up cutting in Figure 12 and in down cutting in Figure 13. The following conclusions appeared:

 Classically and as seen before, in both cases cutting forces values increase proportionally to the feed. But for the both highest feeds, difference is not significant.

 As seen before and in both cases, criticity of the secondary zone directly depends on feed value.

 In down cutting configuration, the lower feeds seem to be unfavourable for the tool, because the cutting forces in first zone reach the same values than for the maximal feeds.

 On these curves, three working zones are observed: a first zone with quite low cutting forces; a critical second zone where these forces increase strongly; and a third zone with high cutting speeds where cutting forces decrease rapidly. In both cases the critical zone has not been observed for the transverse forces x F . The critical zone exists for the feed forces y F only in up cutting configuration. Transverse forces describe in second machining zone a stable zone slightly decreasing. This effect is probably due to the bull nose end mill geometry that gives a better resistance to high loading on teeth. With bull nose end mill the critical machining zone is not a problem for the transverse forces x F .  In return, on up cutting curves, the feed force y F is greater than the transverse force x F in critical zone. The transverse forces are theoretically upper than the feed forces in end milling for slotting configuration. It is commonly still the case for semi or even quarter immersion configuration [START_REF] Fontaine | Parametric geometry for modelling of milling operations[END_REF]. Only inferior radial immersion can produce an inverse repartition, especially in up-cutting mode. This tendency to inverse the cutting forces repartition in up-cutting is due to two major phenomena: cutting efficiency of the cutting edge due to its sharpness in regard of the underfomed chip thickness; and cutting stability due to dynamic tool's behaviour linked to its stiffness. In fact, the cutting force generated at the beginning of a new chip (see figure 7) is amplified when the sharpness of the tool is reduced (i.e. the cutting edge radius/chip thickness ratio becomes unfavourable), and during the whole tooth path in the part's material, the radial component of the generated force tends to grow and the tangential one tends to decrease. These local modifications of cutting force amplitude and direction can change the repartition in force value between feed direction and transverse direction in milling. This observation is particularly evident in micro-milling where the sharpness of the tool is critical [START_REF] Gilbin | Capability of tungsten carbide micro-mills to machine hardened tool steel[END_REF]. The previous results in up-cutting mode (figures 11 & 12) show that mill B tends to present this behaviour and not the mill A, it can be explained by the difference in tungsten carbide structure between these two uncoated mills. The micro grain tungsten carbide structure of the mill A gives to the tool a better sharpness and a better stiffness, that generate less cutting force amplitude and a repartition on reference axes closer to the theoretical rigid and perfectly sharp case, especially for intermediary cutting speeds (critical zone).

To conclude, three zones seem clearly appear from the experiments: two favourable machining zones and one unfavourable machining zone. Machining zones with stable machining conditions and with reduced cutting forces are described by a conventional machining zone between 0 and 200 m.min -1 and a high speed machining zones beginning from 550 m.min -1 . The critical zone with severe cutting forces is described by a dead zone between 200 and 500 m.min -1 .

Many phenomena have to be noted. Firstly, in up cutting configuration with the bull nose end mills B the dead zone is wider than for the end mills A. And in down cutting configuration with the bull nose end mills B the dead zone does not exist. Nevertheless the high speed machining zone is observed on all the experimental results.

The tool run-out in milling

The tool run-out is a very important phenomenon in milling with strong influences on the tool engagement in matter. Indeed, the tool run-out modifies the theoretical position of mill cutting edges during machining. In this way, it is an important factor of cutting instabilities and tool wear. It particularly modifies the cutting edges loading repartition and then unbalances cutting forces and wear between the teeth. The tool run-out can be defined as a position gap between the spindle revolution axis and the tool axis as related in Figure 14. Classically, this phenomenon is divided in two different reference defaults:

 The axial tool run-out (angular gap between the tool axis and the spindle axis)

 The radial tool run-out (eccentricity of the tool axis in relation to the spindle revolution axis) This default depends mainly on the qualities of spindle, tool holder and tool used.

It has to be considered as one of the most severe defaults in machining and particularly in high speed machining. Its effects are critical in milling and dramatic in high speed milling or in micro milling because of the higher rotation frequencies used. The defects resulting to a critical run-out are form errors, amplified roughness, cutting instability, accelerated tool wear and breakage and accelerated wear of the spindle bearings The radial run-out is most of the time more pronounced and more difficult to compensate dynamically by the flexibility of the cutting system (tool + tool-holder) than the axial run-out. For example, modelling the radial run-out only usually permits to predict accurately the cutting forces unbalance [START_REF] Fontaine | Predictive force model for ball end milling and experimental validation with a wavelike form machining test[END_REF][32] [START_REF] Fontaine | Modelling of cutting forces in ballend milling with tool-surface inclination. Part II: Influence of cutting conditions, run-out, ploughning and inclination angle[END_REF]. That is the reason why we focus specifically on the radial default in the following works. For the experiments led on with the uncoated end mills A, a measure of the tool run-out has been performed before each milling operation in order to quantify the values of the radial defect and to observe its influence on cutting forces in function of cutting speeds. Measurement has been performed with a charge amplifier and two position sensors fixed on the machine table and touching the cylindrical part of the tool along X and Y axis respectively. The maximal accuracy reached with this device is about 0.1 µm and the measures are really repeatable. The measurement protocol consists to slowly spin manually the mill in order to quantify the defect between the extreme positions which corresponds to two times the value of eccentricity. The setting up is related in Figure 3. The tool run-out is here located by an eccentricity value e and an angular position of the maximal defect on the tool cylinder, like on the sketch related in Figure 14. For all the experimental tests in milling, the measured tool run-out took values between 2 µm and 5 µm. In case of conventional milling, this defect represents at maximum 30 % of tested feed per tooth. The tool run-out is not critical in milling but possesses a real influence on cutting stability, dynamics phenomena and tool wear. The tool runout effect is described with two cutting forces curves and measured with two different tool run-out values, Figure 15. Each curves represents a mill rotation or in this case two tooth paths. On these curves, the tool run-out modifies the cutting forces repartition between each tooth of the mill. The additional loading on a tooth is a proposed value to quantify tool run-out effect. This value is a percentage of the cutting forces average value. It is added positively or negatively on each tooth.

So with this definition, overloading and discharge are equal and opposite, and during our experiments the maximal value of measured overloading exceeds 50 % of the average cutting forces. After experimental analysis, the behaviour of this additional loading has been observed in function of tool run-out values and cutting speeds, and the results are related in Figure 16. On these curves, additional loading increases in an important way with cutting speed, then the tool run-out effect seems to increase with cutting speed. Moreover the additional loading increases linearly with the tool run-out values and increases globally with cutting speed. For high cutting speeds and due to dynamic effects, the tool run-out phenomenon can become a critical point for tool behaviour, machined surface quality and tool wear.

Chip formation

Chips give essential information about milling process. They are the results of the matter deformation and also the witness zone of the most intense mechanical or thermal solicitations occurring in cutting. In literature, a very few works focused on chip observations in up cutting configuration in milling. That is right that it is very difficult to make a connection between the chip geometry and its way of formation, because the final shape of a chip can be very different from the theoretical shape which can be deduced from a rigid and elementary case. In turning, this difference is reduced, so it is easier to conclude. In this analysis, the aim not consists to compare a real chip with a chip in a theoretical case, but rather to observe differences and evolutions in chip morphology in function of machining parameters. A first qualitative analysis is here proposed, and then completed in analysing the evolutions of chip length and thickness. The qualitative analysis related in Figure 17 shows that the obtained chips inherit some characteristics of the mill flute, and in particular a striated surface due to the ploughing of the chip by irregularities of the rake face of the flutes obtained after tool grinding. The metallic aspect of chips is close to the initial material but for a cutting speed equal to 450 m.min -1 , an important changing appears on the chip surface. Oxide colours (red, blue, orange) appeared on chip surface, it is probably due to the thermal effects induced by high cutting and friction speeds. A quantitative analysis has been performed with measurements of chip length and thickness with a topo-microscope as related in Figure 18. The chip length is measured in taking into account the 3D curve shape, and we choosed to determine length with the best curve passing along the chip. Concerning, the chip thickness, we measured the maximal and the minimal thickness in orthogonal direction of the curve chip. Each measurement are based on 5 same cutting condition chips, then the average maximal and minimal thickness is calculated, and the global average thickness is deduced. Results are related in Figure 19 and the following observations have been made:

 Firstly, the chip length is always smaller than the theoretical length which can be deduced from a rigid case as related in Figure 5, Figure 7 and  Secondly, on all curves, the same tendency is observed, the chip length decreases with the increase of the cutting speed and three stages appear : a first zone of fast decrease followed by two stationary zones. The stages cutting speed limit values are different in function of the feed, but the discrepancy is relatively small.  About the chip thickness, for each feed, three curves are related in Figure 19. Because the chips have commonly a characteristic shape as related in Figure 18, minimal and maximal thicknesses have to be measured. Indeed, in a shoulder milling test in up cutting configuration, the tool engagement begin with a chip thickness close to 0 mm and finish with a theoretical chip thickness equal to t f (Figure 7). The real geometry is very far of the rigid case, because the shearing due to the cutting process induces a lot of compression in length direction and of spreading in thickness direction.

The average thickness has been calculated and added to describe the evolution in global way. On these curves, the same behaviour than for the chip length is observed: a first zone of decrease followed by two stationary zones.

It is easy to compare these results with the results obtained on machinability curves. In both cases, three stages are observed with the same limit in terms of cutting speed values. Moreover this analysis put in evidence a high cutting speed zone where the chip formation seems to be characterized by low chip length and thickness and then a high material compression occurring during chip formation.

To conclude, the milling process implies a favourable high speed machining zone which is characterised by lower cutting forces and a different way of chip formation. It corresponds to remarks often made in literature but it deserves to be shown experimentally for milling [37][38].

Mill geometry evolution

On all the experiments, the wear of mills has been analysed in a qualitative way by comparing the mill after the test with a new one, It was choosen in this section to relate the measurements for the mills A in Figure 20 and Figure 21 for a cutting speed of 500 m.min -1 and different feed rates. All observations have been performed with the topo-microscope Alicona. The results are related in Figure 20 and the following conclusions can be done:

 For low feed rates, an important abrasive wear is observed on the relief face of the mill.

 For higher feeds, abrasive wear is reduced but a radius tends to appear at the extremity of the end mills A for high cutting speeds. Moreover, the relief face presents an irregular wear with cracks and craters.

 For the highest feed (i.e 0.06 mm.tooth -1 ) and moderate cutting speeds, some teeth of end mills broke suddenly.

 For high cutting speeds, the coating of mills is usually fast deteriorated (abrasion and delamination).

For these experiments which induce a short machining time, the rake faces and mill flutes do not showed a real wear. Indeed, for high cutting speeds, the decrease of chip length tends to limit the contact between chips and mill flutes or rake faces. In these cutting conditions, the chip seems to be ejected.  Some cracks and craters are generated during machining on the relief face at tool extremity and this phenomenon becomes very important for high cutting speeds. Then cutting edge has a sharpness of 45 µm, but at a height near to 1 mm the value is about 38 µm and in this zone the wear seems to be still restrained. From these measurements no significant wear is observed on the rake face of the tool, even for high cutting speeds.

These analyses put in evidence different wear phenomena and allow quantifying the quick reduction of the cutting edges sharpness in many feeds and for the high speed machining zone. Results have shown that dominant wear phenomena depend to feeds and cutting speeds values. These investigations can provide a relevant sharpness value to finite element or analytical model developers.

Conclusions

These works furnish a wide experimental basis in milling and high speed milling of a 304L stainless steel. These data are proposed to help engineers and researchers to understand the milling process specificities and key points, but also to develop or to improve modelling of machining. A wide range of experiments has been performed with a large panel of cutting conditions, tool geometries or cutting configurations in slotting and shouldering. The set up acquisition system allows obtaining a good accuracy and repeatability on cutting forces curves. These milling curves are used as main data to analyse the cutting edges engagement in machined material, the appearing of dynamic effects with the increase of cutting speed, the influence of tool run-out and to identify proper cutting conditions in terms of cutting speed and feed rate. These results were also used for our parallel works on numerical simulation [START_REF] Maurel-Pantel | 3D FEM simulations of shoulder milling operations on a 304L stainless steel[END_REF] or identification [START_REF] Maurel | Inverse Method for Identification of Material Parameters Directly from Milling Experiments[END_REF]. The cutting forces data are completed with information about chips morphology and tools wear. In this context, the choice of cutting coefficients or laws used to model empirically the milling process needs to consider a tight panel of cutting conditions to stay representative. In the same way predictive modelling of machining needs to use constitutive laws to describe material behaviour and friction conditions, so the huge variations in the process response are a limitation.

For example, in our modelling works a Johnson & Cook constitutive law was tested and permitted to obtain very good results in the first zone (low to medium cutting speeds). In the other working zones, this law is not really efficient, especially when the strain rate and temperature influences for our stainless steel are identified on classical mechanical test. We have the same considerations when friction is considered through a constant coefficient. So it is necessary to find an appropriate constitutive law relevant on a wide range of thermo-mechanical solicitations. In the same way, the friction conditions and their modelling at the tool chip interface have to be analysed further. Moreover classical mechanical characterisation tests are inefficient for extreme strain rates coupling to high temperatures. As a first step, the development of constitutive laws and identification by inverse method using measured data obtained directly from milling tests (as cutting forces curves) seems to be an interesting way to improve milling models as demonstrated in previous work [START_REF] Maurel-Pantel | 3D FEM simulations of milling on a 304L stainless steel[END_REF]) [START_REF] Maurel | Inverse Method for Identification of Material Parameters Directly from Milling Experiments[END_REF]. In analytical or finite element models developed by our team and using rigid tool assumption have shown their limits to reproduce the cutting forces curves measured at high cutting speeds. In machining models actually, the dynamics and thermo-mechanical coupling introduced by high cutting speed are not taken into account. These detailed experimental data in milling constitute a suitable experimental basis available to develop predictive machining modelling. Now the characterisation of cutting dynamics become a major experimental challenge in high speed milling or in micro milling. And It becomes also a challenge in modelling to develop predictive analytical of finite element models at high cutting speeds in milling.

Fig . 1 .

 1 Fig .1. Assembly of the samples used for the experiments and pictures of the milling acquisition systems.

Fig . 2 .

 2 Fig .2. Measure validity range with a 6 mm diameter mill function of cutting speed

Fig . 3 .

 3 Fig .3. Characteristics of tools used in the experiments.

Fig . 4 .

 4 Fig .4. Measured cutting forces curves with a zoom on the steady zone. (a) slotting configuration and (b) down cutting configuration (semi immersion)

Fig . 5 .

 5 Fig .5. Theoretical tool engagement, tooth path, and cutting forces curves characteristics in slotting configuration

Fig . 6 .

 6 Fig .6. Qualitative comparison of cutting forces curves in slotting configuration with mill A. (a) rigid case obtained analytically. (b) cutting speed equal to 70 m.min -1 . (c) cutting speed equal to 310 m.min -1 . (d) cutting speed equal to 590 m.min -1 . (f t = 0.015 mm.tooth -1 , d r = 6 mm, d a = 1 mm)

Figure 6

 6 exposes four sets of cutting forces curves for a mill rotation (two tooth paths in this case). The first cutting forces curves noted (a) are calculated analytically with the material parameters previously exposed and with the same cutting conditions than the cutting forces curves (b) that is to say for a low cutting speed. Cutting forces curves (c) and (d) are chosen respectively for a middle cutting speed and for the higher cutting speed tested in slotting. The curve shapes obtained analytically in a rigid case are equivalent for the three cutting speeds and only the forces level slightly decreases.The best correlation is obtained for the cutting forces curves (b) with the lower cutting speed. In slotting configuration, the tool motion is balanced by the guiding of the slot, and the tool globally respects a motion close to the rigid case considered in machining models. Only some medium cutting speeds imply cutting instabilities and then cutting forces amplification, for example for 310 m.min -1 .

Fig . 7 .

 7 Fig .7. Tool engagement, tooth path, cutting forces curves characteristics in up cutting configuration.



  described the cutting forces curves obtained in up cutting configuration analytically in a rigid case and like previously the four main stages are described: The stage 1 corresponds to the entrance of the tooth in matter, and in this case begins by a thickness equal to 0 mm. For this step same interpretations than in slotting configurations can be considered. The stage 2 is the critical place for the feed force like in slotting configuration. One more time the stage 3 corresponds to the maximal loading zone. In this position the tooth removes a maximal thickness of material equal to the feed t f (only in case of semi-immersion). The stage 3 is the critical place for the transverse force x F and the split-up of the chip. Just after this stage, the chip is removed, and the tooth comes outside the matter, a nickpoint appears on curves, the cutting forces increase quickly to value zero. In this zone the chip separation mode and the tool geometry have a high influence. And lots of phenomena (wear, vibrations…) are initiated in this zone due to the fast change of loading on the tool. The stage 4 is characterised by the annulment of the three cutting forces.

Fig . 8 .

 8 Fig .8. Qualitative comparison of cutting forces curves in shouldering configuration (up cutting) with mill A. (a) rigid case obtained analytically. (b) cutting speed equal to 70 m.min -1 . (c) cutting speed equal to 370 m.min -1 . (d) cutting speed equal to 650 m.min -1 . (f t = 0.03 mm.tooth -1 , d r = 3 mm, d a = 1 mm).

Figure 8

 8 Figure 8 describe four sets of cutting forces curves, (a) and (b) corresponding to

Fig. 9 .

 9 Fig.9. Tool engagement, tooth path, cutting forces curves characteristics in down cutting configuration.

Figure 9 FF

 9 Figure 9 described the cutting forces curves obtained in down cutting

Fig. 10 .

 10 Fig.10. Qualitative comparison of cutting forces curves in shouldering configuration (down milling) with mill B. (a) rigid case obtained analytically. (b) cutting speed equal to 70 m.min -1 . (c) cutting speed equal to 310 m.min -1 . (d) cutting speed equal to 590 m.min -1 . (ft=0.045mm.tooth -1 , dr= 3 mm, da= 1 mm).

xF

  and feed forces y F are identified in the stabilised zone and are related in function of cutting speed and feed tested in different cutting configurations as shown in Figure 4.

Fig. 11 .

 11 Fig.11. Cutting forces average amplitude evolutions function of cutting speed in up cutting configuration for coated and uncoated end mills A.

Fig. 12 .

 12 Fig.12. Cutting forces average amplitude evolutions function of cutting speed in up cutting configuration for end mills B.  In down cutting, the critical zone becomes a stable zone with values

Fig. 13 .

 13 Fig.13. Cutting forces average amplitude evolutions function of cutting speed in down cutting configuration for end mills B.

Fig. 14 .

 14 Fig.14. Parametric description of the tool run-out. (Fontaine et al., 2007)[START_REF] Fontaine | Predictive force model for ball end milling and experimental validation with a wavelike form machining test[END_REF]

Fig. 15 .

 15 Fig.15. Influence of the tool run-out on cutting forces in milling

Fig . 16 .

 16 Fig .16. Additional loading evolution function of tool run-out and cutting speed

Fig . 17 .

 17 Fig .17. Chip evolution function of cutting speed in milling of a 304L steel

Fig . 18 .

 18 Fig .18. Characteristic shape of a chip obtained after semi-immersion milling tests.

Figure 9 .

 9 Figure 9.

Fig . 19 .

 19 Fig .19. Evolution of chip thickness and length function of cutting speed for three values of feed.

Fig . 20 .

 20 Fig .20. Characteristic wear of mills for different feed rates.

  That's why during our experiments, we have observed unidirectional chip flows with a regular rate from 500 m.min -1 . Then, a more accurate analysis has been performed on mills used for two different feeds and on a new mill for a set of experiments from 70 m.min-1 to 650 m.min-1. With the topo-microscope Alicona, the teeth of each tool have been rebuilded in three dimensions, in order to quantify the wear phenomena and to focus on the evolution of cutting edges geometry. Measures of tool edge sharpness at different level on tool axis have been carried out with the optical 3D InfiniteFocus measurement device. Figure 21 (a), (b) and (c) describe respectively results obtained with a new mill, mills used with a feed of 0.015 mm.tooth-1 and with mills used with a feed of 0.03 mm.tooth-1. The optical pictures of mills are proposed to locate the measurement sections along the cutting edges. The local profiles are shown to put in evidence the real shape of cutting edges in different places. The following results have been obtained:

Fig . 21 .

 21 Fig .21. Evolution of cutting edges sharpness for different feed rates.

Fig . 22 .

 22 Fig .22. Representative machinability of the tested 304L stainless steel in milling (feed rate 0.045 mm.tooth -1 , end mill A 2 teeth micro grain tungsten carbide, diameter 6 mm, d a 1 mm, d r 3 mm) .

  The experiments led on with mills A and B for different cutting speeds and feeds have furnished a very important experimental database, and all available data has been analysed in order to put in evidence the physical phenomena induced by milling operations. The analysis of cutting forces curves for many cutting speeds has shown that dynamic phenomena appear during machining particularly in shoulder milling configuration. With these dynamic effects, the measured cutting forces curves are far away from the curves obtained analytically in a rigid case or for low cutting speeds. Until now modelling of milling from analytical or finite element models used the assumption of undeformable tools. But this assumption limits drastically the model predictivity until the low conventional cutting speeds are overpassed. The machinability of the 304L stainless steel with different tools geometries and configurations in shoulder milling is defined by three working zones: a conventional zone permitting stable cutting (low cutting speed; under 200-250 m.min -1 ), a dead zone (unfavourable for cutting forces level and cutting stability; between 250 and 450 m.min -1 ), and a high speed machining zone (high cutting speed; up to 450-500 m.min -1 ). All the used criteria (cutting forces, chips, wear) confirm the existence of these different zones and a correlation is proposed with cutting perturbations as run-out, cutting instability, ploughing and abrasive wear.Then the tool run-out influence has been described and measured for many experiments. With increase of cutting speed, tool run-out accentuates vibrations and unbalances cutting forces repartition and wear between the teeth. The analysis led on tool wear has permitted to set up a methodology to measure cutting edges sharpness on mills in order to introduce a real description of the tool geometry in finite element or analytical modelling. All this conclusions are useful to improve models predictivity and to extend modelling to high speed machining zone. These results confirm the assumption of the changing of thermo-mechanical conditions from the cutting speed equal approximately to 450 m.min -1 . It underlines the difficulty to choose and identify a constitutive law to describe the material behaviour, because high strain rate and high temperatures corresponding to the high speed zone are non accessible with classic mechanical tests.

Table . 1

 . . Chemical composition of the machined 304L stainless steel.

	% C	% Si % Mn % P	% S % Cr % Mo % Ni % Ti % Cu % Co % N
	0.028 0.26	1.5	0.033 0.03 18.25	-	8.25	-	0.53	0.14 0.0895

Table . 4

 . . Characteristics of the Dynamometer Kistler 9272.

Table .

 .

	Tools	Type Coating Cutting speed(m.min -1 )	Feed (mm.tooth -1 )	Configurations
		Mill A	-	70 to 650	0.03	Slotting
		Mill A	-	70 to 650	0.015	Up cutting
		Mill A	-	70 to 650	0.03	Up cutting
	Tool 1	Mill A	-	70 to 650	0.045	Up cutting
		Mill A	-	70 to 370	0.06	Up cutting
		Mill A	-	70 to 650	0.03	Down cutting
		Mill A	TiCN	70 to 650	0.015	Up cutting
		Mill A	TiCN	70 to 650	0.03	Up cutting
	Tool 2	Mill A	TiCN	70 to 650	0.045	Up cutting
		Mill A	TiCN	70 to 650	0.03	Down cutting
		Mill B	-	70 to 590	0.03	Slotting
		Mill B	-	70 to 590	0.01	Up cutting
		Mill B	-	70 to 590	0.015	Up cutting
		Mill B	-	70 to 590	0.03	Up cutting
	Tool 3	Mill B	-	70 to 590	0.045	Up cutting
		Mill B	-	70 to 590	0.06	Up cutting
		Mill B	-	70 to 590	0.015	Down cutting
		Mill B	-	70 to 590	0.03	Down cutting
		Mill B	-	70 to 590	0.045	Down cutting

5. Milling tests performed.

Acknowledgements

The CETIM foundation is greatly acknowledged for providing the PhD grant support to Dr A. Maurel-Pantel in the PGV project. These works were carried out in the context of the working group Manufacturing 21 which including 18 French laboratories. Targeted issues are the modelling and the optimization of manufacturing processes and the innovation in the field of mechanical manufacturing. Thanks to F. Gaillard from DIAGER Industry® for his help in choosing and obtaining tools.


Adel Alamadhi

Michel Planat
email: michel.planat@femto-st.fr

Patrick Sol É

CHEBYSHEV'S BIAS AND GENERALIZED RIEMANN HYPOTHESIS

Keywords: Mathematics Subject Classification. Primary 11N13, 11N05; Secondary 11N37 Prime counting, Chebyshev functions, Riemann hypothesis

come

Introduction

In the following, we denote π(x) the prime counting function and π(x; q, a) the number of primes not exceeding x and congruent to a mod q. The asymptotic law for the distribution of primes is the prime number theorem π(x) ∼ x log x . Correspondingly, one gets [4, eq. ( 14), p. 125] (1.1) π(x; q, a) ∼ π(x) φ(q)

that is, one expects the same number of primes in each residue class a mod q, if (a, q) = 1. Chebyshev's bias is the observation that, contrarily to expectations, π(x; q, N ) > π(x; q, R) most of the times, when N is a non-square modulo q, but R is.

Let us start with the bias (1.2) δ(x, 4) := π(x; 4, 3) -π(x; 4, 1)

found between the number of primes in the non-quadratic residue class N = 3 mod 4 and the number of primes in the quadratic one R = 3 mod 4. The values δ(10 n , 4), n ≤ 1, form the increasing sequence A091295 = {1, 2, 7, 10, 25, 147, 218, 446, 551, 5960, . . .}.

The bias is found to be negative in thin zones of size {2, 410, 15 358, 41346, 42 233 786, 416 889 978, . . .} spread over the location of primes of maximum negative bias [START_REF] Bays | Numerical and graphical description of all axis crossing regions for the moduli 4 and 8 which occurs before 10 12[END_REF] {26861, 623 681, 12 366 589, 951 867 937, 6 345 026 833, 18 699 356 321 . . .}.

It has been proved that there are are infinitely many sign changes in the Chebyshev's bias (1.2). This follows from the Littlewood's oscillation theorem [START_REF] Deléglise | Counting primes in residue classes[END_REF][START_REF] Ingham | The distribution of prime numbers[END_REF] (

1.3) δ(x, 4) := Ω ± x 1/2 log x log 3 x .
A useful measure of the Chebyshev's bias is the logarithmic density [START_REF] Rubinstein | Chebyshev's bias[END_REF][START_REF] Deléglise | Counting primes in residue classes[END_REF][START_REF] Fiorilli | Inequalities in the Shanks-Rényi prime number-race: an asymptotic formula for the densities[END_REF] (1.4)

d(A) = lim x→∞ 1 log x a∈A,a≤x 1 a
for the positive ∆ + and negative ∆ -regions calculated as d(∆ + ) = 0.9959 and d(∆ -) = 0.0041.

In essence, Chebyshev's bias δ(x, 4) is similar to the bias

(1.5) δ(x) := Li(x) -π(x).
It is known that δ(x) > 0 up to the (very large) Skewes' number x 1 ∼ 1.40 × 10 316 but, according to Littlewood's theorem, there also are infinitely many sign changes of δ(x) [START_REF] Ingham | The distribution of prime numbers[END_REF].

The reason why the asymmetry in (1.5) is so much pronounced is encoded in the following approximation of the bias [START_REF] Rubinstein | Chebyshev's bias[END_REF][START_REF] Bays | Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias[END_REF]

1 (1.6) δ(x) ∼ √ x log x 1 + 2 γ sin(γ log x + α γ ) 1/4 + γ 2 ,
where α γ = cot -1 (2γ) and γ is the imaginary part of the non-trivial zeros of the Riemann zeta function ζ(s). The smallest value of γ is quite large, γ 1 ∼ 14.134, and leads to a large asymmetry in (1.5).

Under the assumption that the generalized Riemann hypothesis (GRH) holds that is, if the Dirichlet L-function with non trivial real character κ 4

(1.7) L(s, κ 4 ) = n≥0 (-1) n (2n + 1) s ,
has all its non-trivial zeros located on the vertical axis ℜ(s) = 1 2 , then the formula (1.6) also holds for the Chebyshev's bias δ(x, 4). The smallest non-trivial zero of L(s, κ 4 ) is at γ 1 ∼ 6.02, a much smaller value than than the one corresponding to ζ(s), so that the bias is also much smaller. 1 The bias may also be approached in a different way by relating it to the second order Landau-Ramanujan constant [START_REF] Moree | Chebyshev's bias for composite numbers with restricted prime divisors[END_REF].

A second factor controls the aforementionned assymmetry of a L-function of real non-trivial character κ, it is the variance [START_REF] Martin | Asymmetries in the Shanks-Rényi prime number race, Number theory for the millenium[END_REF] (1.8)

V (κ) = γ>0 2 1/4 + γ 2 .
For the function ζ(s) and L(s, κ 4 ) one gets V = 0.045 and V=0.155, respectively.

Our main goal. In a groundbreaking paper, Robin reformulated the unconditional bias (1.5) as a conditional one involving the second Chebyshev function ψ(x) = p k ≤x log p (1.9)

The equality δ ′ (x

) := li[ψ(x)] -π(x) > 0 is equivalent to RH.
This statement is given as Corollary 1.2 in [START_REF] Planat | Efficient prime counting and the Chebyshev primes[END_REF] and led the second and third author of the present work to derive a good prime counting function

(1.10) π(x) = 3 n=1 µ(n)li[ψ(x) 1/n ].
Here, we are interested in a similar method to regularize the Chebyshev's bias in a conditional way similar to (1.9). In [START_REF] Robin | Sur la difference Li(θ(x)) -π(x)[END_REF], Robin introduced the function

(1.11) B(x; q, a) = li[φ(q)ψ(x; q, a)] -φ(q)π(x; q, a),
that generalizes (1.9) and applies it to the residue class a mod q, with ψ(x, q, a) the generalized second Chebyshev's function. Under GRH he proved that [2, Lemma

(1.12)

B(x; q, a) = Ω ± √ x log 2 x , that is (1.13)
The inequality B(x; q, a) > 0 is equivalent to GRH.

For the Chebyshev's bias, we now need a proposition taking into account two residue classes such that a = N (a non-quadratic residue) and a = R (a quadratic one).

Proposition 1.1. Let B(x; q, a) be the Robin B-function defined in (1.11), and R (resp. N ) be a quadratic residue modulo q (resp. a non-quadratic residue), then the statement δ ′ (x, q) := B(x; q, R) -B(x; q, N ) > 0, ∀x (i), is equivalent to GRH for the modulus q.

The present paper deals about the numerical justification of proposition 1.1 in Sec. 2 and its proof in Sec. 3. The calculations are performed with the software Magma [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF] available on a 96 MB segment of the cluster at the University of Franche-Comté.

The regularized Chebyshev's bias

All over this section, we are interested in the prime champions of the Chebyshev's bias δ(x, q) (as defined in (1.2) or (2.3), depending on the context). We separate the prime champions leading to a positive/negative bias. Thus, the n-th prime champion satisfies (2.1) x versus the Chebyshev's bias δ(x, 4) at the prime champions of δ(x, 4) (when δ(x, 4) > 0) and at the prime champions of -δ(x, 4) (when δ(x, 4) < 0). The extremal prime champions in the plot are x = 359327 (with δ = 105) and x = 951867937 (with δ = -48). The curve is asymmetric around the vertical axis, a fact that reflects the asymmetry of the Chebyshev's bias. As explained in the text, a violation of GRH would imply a negative value of the regularized bias δ ′ (x, 4). The small dot curve corresponds to the fit of δ ′ (x, 4)/ √ x by 2/ log x calculated in Sec. 3.

δ (ǫ) (x n , q) = ǫn, ǫ = ±1.
We also introduce a new measure of the overall bias b(q), dedicated to our plots, as follows

(2.2) b(q) = n,ǫ δ (ǫ) (x n , q) x n .
Indeed, smaller is the bias lower is the value of b(q). Anticipating over the results presented below, Table 1 summarize the calculations. Chebyshev's bias for the modulus q = 4. As explained in the introduction, our goal in this paper is to reexpress a standard Chebyshev's bias δ(x, q) into a regularized one δ ′ (x, q), that is always positive under the condition that GRH holds. Indeed we do not discover any numerical violation of GRH and we always obtains a positive δ ′ (x, q). The asymmetry of Chebyshev's bias arises in the plot δ vs δ ′ , where the fall of the normalized bias δ

√

x is faster for negative values of δ than for positive ones. Fig. 1 clarifies this effect for the historic modulus q = 4. We restricted our plot to the champions of the bias δ and separated positive and negative champions.

Chebyshev's bias for a prime modulus p. For a prime modulus p, we define the bias so as to obtain an averaging over all differences π(x; p, N ) -π(x; p, R), where as above N and R denote a non-quadratic and a quadratic residue, respectively

(2.3) δ(x, p) = - a a p π(x; p, a),
where a p is the Legendre symbol. Correspondingly, we define the regularized bias as

(2.4) δ ′ (x, p) = 1 ⌊p/2⌋ a a p B(x; p, a).
Proposition 2.1. Let p be a selected prime modulus and δ ′ (x, p) as in (2.4) then the statement δ ′ (x, p) > 0, ∀x, is equivalent to GRH for the modulus p.

As mentioned in the introduction, the Chebyshev's bias is much influenced by the location of the first non-trivial zero of the function L(s, κ q ), κ q being the real non-principal character modulo q. This is especially true for L(s, κ 163 ) with its smaller non-trivial zero at γ ∼ 0.2029 [START_REF] Bays | Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias[END_REF]. The first negative values occur at {15073, 15077, 15083, . . .}.

Fig. 2 represents the Chebyshev's bias δ ′ for the modulus q = 163 versus the standard one δ (thick dots). Tha asymmetry of the Chebyshev's bias is revealed at small values of |δ| where the the fit of the regularized bias by the curve 2/ log x is not good (thin dots).

For the modulus q = 13, the imaginary part of the first zero is not especially small, γ 1 ∼ 3.119, but the variance (1.8) is quite high, V (κ -13 ) ∼ 0.396. The first negative values of δ(x, 13) at primes occur when {2083, 2089, 10531, . . .}. Fig. 3 represents the Chebyshev's bias δ ′ for the modulus q = 13 versus the standard one δ (thick dots) as compared to the fit by 2/ log x (thin dots).

Finally, for the modulus q = 11, the imaginary part of the first zero is quite small, γ 1 ∼ 0.209, and the variance is high, V (κ -11 ) ∼ 0.507. In such a case, as shown in Fig. 4, the approximation of the regularized bias by 2/ log x is good in the whole range of values of x.

Proof of proposition 1.1

For approaching the proposition 1.1 we reformulate it in a simpler way as Proposition 3.1. One introduces the regularized couting function π ′ (x; q, l) := π(x; q, l)-ψ(x; q, l)/ log x. The statement π ′ (x; q, N ) > π ′ (x; q, R), ∀x (ii), is equivalent to GRH for the modulus q. Proof. First observe that proposition 1.1 follows from proposition 3.1. This is straightforward because according to [2, p. 260], the prime number theorem for arithmetic progressions leads to the approximation (3.1) li[φ(q)ψ(x; q, l)] ∼ li(x) + φ(q)ψ(x; q, l) -x log x .

As a result

δ ′ (x, q) = B(x; q, R) -B(x; q, N ) = li[φ(q)ψ(x; q, R)] -li[φ(q)ψ(x; q, N )] + φ(q)δ(x, q) ∼ φ(q)[π ′ (x; q, N ) -π ′ (x; q, R)].
The asymtotic equivalence in (3.1) holds up to the error term [2, p. 260] O( R(x) x log x ), with R(x) = min x θq log 2 x, xe -a √ log x , a > 0, θ q = max κ mod q (sup ℜ(ρ), ρ a zero of L(s, κ)).

Let us now look at the statement GRH ⇒ (i). Following [3, p 178-179], one has ψ(x; q, a) = 1 φ(q) κ mod q κ(a)ψ(x, κ) x versus the Chebyshev's bias δ(x, 13) at the prime champions of δ(x, 13) (when δ(x, 13) > 0), and the curve at the prime champions of -δ(x, 13) (when δ(x, 13) < 0). The extremal prime champions in the plot are x = 263881 (with δ = 123) and x = 905761 (with δ = -40). The small dot curve corresponds to the fit of δ ′ (x, 13)/ √ x by 2/ log x calculated in Sec. and under GRH π(x; q, a) = π(x) φ(q) -c(q, a) φ(q)

√ x log x + 1 φ(q) log x κ =κ0 κ(a)ψ(x, κ) + O( √ x log 2 x ),
where κ 0 is the principal character modulo q and c(q, a) = -1 + #{1 ≤ b ≤ q : b 2 = a mod q} for coprimes integers a and q. Note that for an odd prime q = p, one has c(p, a) = a p . Thus, under GRH π(x; q, N ) -π(x; q, R)

= 1 φ(q) log x [ √ x(c(q, R) -c(q, N )) + κ mod q (κ(N ) -κ(R))ψ(x, κ) + O( √ x log 2 x ]. (3.2)
The sum could be taken over all characters because κ0 (N ) = κ0 (R). In addition, we have

(3.3) ψ(x; q, N ) -ψ(x; q, R) = 1 φ(q) κ mod q [κ(N ) -κ(R)]ψ(x, κ).
Using (3.2) and (3.3) the regularized bias reads

δ ′ (x, q) ∼ π ′ (x; q, N ) -π ′ (x; q, R) = √ x log x [c(q, R) -c(q, N )] + O √ x log 2 x . (3.4)
For the modulus q = 4, we have c(q, 1) = -1 + 2 = 1 and c(q, 3) = -1 so that δ ′ (x, 4) = 2 √ x log x The same result is obtained for a prime modulus q = p since c(p, N ) = -1 and c(p, R) = c(p, 1) = 1 p = 1. This finalizes the proof that under GRH, one has the inequality π ′ (x; q, N ) > π ′ (x; q, R).

If GRH does not hold, then using [2, lemma 2], one has B(x; q, a) = Ω ± (x ξ ) for any ξ < θ q .

Applying this assymptotic result to the residue classes a = R and a = N , there exist infinitely many values x = x 1 and x = x 2 satisfying B(x 1 ; q, R) < -x ξ 1 and B(x 2 ; q, N ) > x ξ 2 for any ξ < θ q , so that one obtains (3.5) B(x1; q, R) -B(x 2 ; q, N ) < -x ξ 1 -x ξ 2 < 0. Selecting a pair (x 1 , x 2 ) either B(x 1 ; q, R) > B(x 2 ; q, R) so that B(x 2 ; q, R) -B(x 2 ; q, N ) < 0 and (i) is violated at x 2 , or (3.6) B(x 1 ; q, R) < B(x 2 ; q, R).

In the last case, either B(x 1 ; q, N ) > B(x 2 ; q, N ), so that B(x 1 ; q, R)-B(x 1 ; q, N ) < 0 and the inequality (i) is violated at x 1 , or simultaneously B(x 1 ; q, N ) < B(x 2 ; q, N ) and B(x 1 ; q, R) < B(x 2 ; q, R), which implies (3.5) and the violation of (i) at x = x 1 = x 2 .

To finalize the proof of 3.1, and simultaneously that of 1.1, one makes use of the asymptotic equivalence of (i) and (ii), that is if GRH is true ⇒ (ii) ⇒ (i), and if GRH is wrong, (i) may be violated and (ii) as well.

Then, proposition 2.1 also follows as a straigthforward consequence of proposition 1.1. n

Summary

We have found that the asymmetry in the prime counting function π(x; q, a) between the quadratic residues a = R and the non-quadradic residues a = N for the modulus q can be encoded in the function B(x; q, a) [defined in (1.11)] introduced by Robin the context of GRH [START_REF] Robin | Sur la difference Li(θ(x)) -π(x)[END_REF], or into the regularized prime counting function π ′ (x; q, a), as in Proposition 3.1. The bias in π ′ reflects the bias in π conditionaly under GRH for the modulus q. Our conjecture has been initiated by detailed computer calculations presented in Sec. 2 and proved in Sec. 3. Further work could follow the work about the connection of π, and thus of π ′ , to the sum of squares function r 2 (n) [START_REF] Moree | Chebyshev's bias for composite numbers with restricted prime divisors[END_REF].

Figure 1 .

 1 Figure 1. The normalized regularized bias δ ′ (x, 4)/ √x versus the Chebyshev's bias δ(x, 4) at the prime champions of δ(x, 4) (when δ(x, 4) > 0) and at the prime champions of -δ(x, 4) (when δ(x, 4) < 0). The extremal prime champions in the plot are x = 359327 (with δ = 105) and x = 951867937 (with δ = -48). The curve is asymmetric around the vertical axis, a fact that reflects the asymmetry of the Chebyshev's bias. As explained in the text, a violation of GRH would imply a negative value of the regularized bias δ ′ (x, 4). The small dot curve corresponds to the fit of δ ′ (x, 4)/ √ x by 2/ log x calculated in Sec. 3.

Figure 2 .

 2 Figure 2. The normalized regularized bias δ ′ (x, 163)/ √ x versus the Chebyshev's bias δ(x, 163) at all the prime champions of |δ(x, 163)| [from |δ(x, 163)| > 74 the bias is δ(x, 163) < 0 negative], superimposed to the curve at the prime champions of -δ(x, 163) (when δ(x, 163) < 0). The extremal prime champions in the plot are x = 68491 (with δ = 74) and x = 174637 (with δ = -86). The asymmetry is still clearly visible in the range of small values of |δ| but tends to disappear in the range of high values of |δ|. The small dot curve corresponds to the fit of δ ′ (x, 163)/ √ x by 2/ log x calculated in Sec. 3.

Figure 3 .

 3 Figure 3. The normalized regularized bias δ ′ (x, 13)/ √x versus the Chebyshev's bias δ(x, 13) at the prime champions of δ(x, 13) (when δ(x, 13) > 0), and the curve at the prime champions of -δ(x, 13) (when δ(x, 13) < 0). The extremal prime champions in the plot are x = 263881 (with δ = 123) and x = 905761 (with δ = -40). The small dot curve corresponds to the fit of δ ′ (x, 13)/ √ x by 2/ log x calculated in Sec. 3.

  Figure 3. The normalized regularized bias δ ′ (x, 13)/ √x versus the Chebyshev's bias δ(x, 13) at the prime champions of δ(x, 13) (when δ(x, 13) > 0), and the curve at the prime champions of -δ(x, 13) (when δ(x, 13) < 0). The extremal prime champions in the plot are x = 263881 (with δ = 123) and x = 905761 (with δ = -40). The small dot curve corresponds to the fit of δ ′ (x, 13)/ √ x by 2/ log x calculated in Sec. 3.

Figure 4 .

 4 Figure 4. The normalized regularized bias δ ′ (x, 11)/ √x versus the Chebyshev's bias δ(x, 11) at the prime champions of δ(x, 11) (when δ(x, 11) > 0), and the curve at the prime champions of -δ(x, 11) (when δ(x, 11) < 0). The extremal prime champions in the plot are x = 638567 (with δ = 158) and x = 1867321 (with δ = -32).The small dot curve corresponds to the (very good) fit of δ ′ (x, 11)/ √ x by 2/ log x calculated in Sec. 3.

Table 1 .

 1 The new bias (2.2) (column 2) and the standard logarithmic density (1.4) (column 3).

	modulus q bias b(q) log density d(∆ + ) first zero γ 1
	4	0.7926	0.9959 [3]	14.134
	11	0.1841	0.9167 [3]	0.2029
	13	0.2803	0.9443 [3]	3.119
	163	0.0809	0.55 [9]	2.477