domain: sdv sde

text:
ORCID Guillaume Borrel
email: guillaume.borrel@pasteur.fr

Biogeochemistry

A microbe that uses crude oil to make methane A microorganism that dwells in an underground oil reservoir has been found to degrade various petroleum compounds and use them to produce methane through a previously unreported biochemical pathway. See p.XXX Microbial communities tend to use the most energy-rich and easily metabolized compounds that they have at their disposition. This leads to a progressive enrichment of compounds that are difficult to break down and provide little energy, particularly in the absence of oxygen or other inorganic electron acceptors. Under these conditions, the use of hydrocarbonsmolecules made of carbon and hydrogen, such as alkaneshas been thought to rely entirely on a collaboration (known as syntrophy) between bacteria that break down these compounds into acetate and H 2 , and microorganisms called methanogenic archaea that use these molecules to produce methane (CH 4 ), the simplest hydrocarbon [START_REF] Zengler | Methane formation from long-chain alkanes by anaerobic microorganisms[END_REF][START_REF] Schink | Energetics of syntrophic cooperation in methanogenic degradation[END_REF] . Writing in Nature, Zhou et al. [START_REF] Zhou | Wide diversity of methane and short-chain alkane metabolisms in uncultured archaea[END_REF] overturn this long-standing account of a division of labour for the methanogenic degradation of hydrocarbons by reporting that a single type of microorganism can degrade various large hydrocarbons into methane (Fig. 1).

Long-chain alkanes

Hydrocarbonoclastic bacteria

Methanogenic archaea

MCR

Methyl-CoM

Whereas many microorganisms can use a large range of substrates to obtain energy, methanogenic microorganisms (methanogens) are very specialized. Most of them can only obtain energy by reducing carbon dioxide (CO 2 ) into CH 4 using molecular hydrogen (H 2 ) as an electron donor, and a few others can use acetate and methylated compounds (such as methanol and methylamines). Over the past five years, genomics studies have described several lineages of previously unknown and uncultured methanogenic microbes from various oxygen-free environments, including marine sediments, hot springs and subsurface oil reservoirs [4][START_REF] Nobu | Chasing the elusive Euryarchaeota class WSA2: genomes reveal a uniquely fastidious methyl-reducing methanogen[END_REF][START_REF] Vanwonterghem | Methylotrophic methanogenesis discovered in the novel archaeal phylum Verstraetearchaeota[END_REF][START_REF] Sorokin | Discovery of extremely halophilic, methyl-reducing euryarchaea provides insights into the evolutionary origin of methanogenesis[END_REF] .

Although most of these newly discovered methanogens were inferred to also rely on simple compounds, primarily methylated compounds reduced with H 2 4,8 , a great surprise was the prediction of a previously undescribed pathway for generating methane from multi-carbon alkanes and possibly long-chain fatty acids 4,9 . This metabolic pathway was initially predicted from genomes belonging to a class of uncultured archaea that was named Candidatus Methanoliparia 4 because its members mostly occur in oil reservoirs or environments contaminated with petroleum.

Cheng et al. collected crude oil from Shengli oilfield (northeast China) and found that Ca. Methanoliparia represented about half of the archaeal community in these samples. When these samples were incubated between 35°C and 65°C, without oxygen or other inorganic electron acceptors, long-chain alkanes (linear chains of 13 to 39 carbons) were completely depleted, and a large amount of methane was produced. The authors similarly observed reductions in the levels of hydrocarbon molecules composed of a carbon ring bond to a longchain alkyl, such as n-alkylcyclohexanes and n-alkylbenzenes.

Because Ca. Methanoliparia are far from being the only microorganisms in these cultures, one might ask at this stage whether the production of methane from petroleum compounds might have resulted from the activity of conventional bacteria-archaea associations. Indeed, bacteria known to break down hydrocarbons were present, even if they represented only 4% of the total microbial community in the original culture. However, after several transfers of the culture into fresh medium, interspersed by weeks of incubation, the proportion of these bacteria dropped to less than 0.1%. By contrast, the abundance of Ca. Methanoliparia (around 40% of the microbial community) and the rate of methanogenic degradation of long-chain alkane was maintained over these transfers.

Further supporting the absence of syntrophic associations involving Ca. Methanoliparia, microscopy revealed that these cells generally occurred alone and not in aggregates of multiple species [START_REF] Zhou | Wide diversity of methane and short-chain alkane metabolisms in uncultured archaea[END_REF][START_REF] Laso-Pérez | Anaerobic degradation of non-methane alkanes by "Candidatus Methanoliparia" in hydrocarbon seeps of the Gulf of Mexico[END_REF] . Moreover, Ca. Methanoliparia genomes lack genes encoding enzymes and nanowire structures that are involved in electron transfer between syntrophic partners 4,9 , corroborating the idea that Ca. Methanoliparia work alone.

The predicted metabolic route for methanogenic degradation of long-chain hydrocarbons by Ca. Methanoliparia involves a combination of enzymes that are otherwise partitioned between syntrophic archaea and bacteria, as well as enzymes specific to Ca. Methanoliparia (Fig. 1). Among the enzymes that are used by both Ca. Methanoliparia and the syntrophic archaea and bacteria association is methyl-coenzyme M reductase (MCR) complex, which performs the final step of methane production in all methanogens.

Ca. Methanoliparia species also carry the genes encoding an enzymic complex called alkyl-coenzyme M reductase (ACR) complex, which was recently described to mediate the first step of the degradation of short-chain alkanesforming alkyl-coenzyme M (alkyl-CoM)in non-methanogenic archaea [START_REF] Laso-Pérez | Thermophilic archaea activate butane via alkyl-coenzyme M formation[END_REF][START_REF] Chen | Anaerobic oxidation of ethane by archaea from a marine hydrocarbon seep[END_REF][START_REF] Hahn | Crystal structure of a key enzyme for anaerobic ethane activation[END_REF] . Ca. Methanoliparia members are the first microorganisms reported to have both an MCR and an ACR complex, an observation that initially led to propose that they may be capable of methanogenesis based on multi-carbon alkanes 4,9 .

Following the same principle as for short-chain alkanes, the activity of ACR complex on long-chain alkanes should also lead to the formation of alkyl-CoM. In cultures enriched with Ca. Methanoliparia and fed hexadecane (a long-chain alkane containing 16 carbons), Cheng et al. [START_REF] Zhou | Wide diversity of methane and short-chain alkane metabolisms in uncultured archaea[END_REF] detected the presence of hexadecyl-CoM, confirming that the ACR complex is involved in the first step of the degradation of long-chain alkanes. The authors made similar observations for long-chain monoaromatic hydrocarbons: addition of such molecules to the culture led to the formation of monoaromatic alkyl-CoM. The genes encoding MCR and ACR complexes in Ca. Methanoliparia were among the most highly expressed in the culture, supporting the proposal that these microorganisms couple hydrocarbon degradation with methane production. Altogether, these findings largely expand the range of substrates that can be used by the ACR complex, and methanogens in general.

Could it be that short-chain alkanes are also targets of Ca. Methanoliparia ACR, as previously described in the non-methanogenic archaea? Apparently not, as no alkyl-CoM and no methane were produced in response to the addition of two to eigth carbon alkanes to the culture. This suggeststhat some ACR complexes might use short-chain hydrocarbons, whereas others use long-chain hydrocarbons. Now that cultures of Ca. Methanoliparia are available, it will be exciting to resolve the 3D structure of their ACR complex and compare it to that of evolutionary related MCR and ACR complexes [START_REF] Hahn | Crystal structure of a key enzyme for anaerobic ethane activation[END_REF][START_REF] Shima | Structure of a methyl-coenzyme M reductase from Black Sea mats that oxidize methane anaerobically[END_REF] . Such investigation will likely reveal the molecular mechanisms enabling the activity of these enzymes, including how they discriminate hydrocarbons depending on their length.

As for other archaea that use ACR to initiate the breakdown of hydrocarbons, such as Ca. Syntrophoarchaeum 10 , several steps of the process of hydrocarbon degradation in Ca. Methanoliparia are still unknown (Fig. 1). The authors suggest that one of these steps might involve a type of enzyme called long-chain fatty-acid CoA ligases. Intriguingly, although this study focused on the most enriched Ca. Methanoliparia species (Ca. Methanoliparum thermophilum), which contains both ACR and MCR, another Ca. Methanoliparia species whose genome was sequenced by the authors lacks the genes encoding the ACR complex, but encode long-chain fatty-acid CoA ligases. Thus, the diversity of methanogenic metabolic pathways in this class of archaea could be even greater than that proposed by the authors, as some species might perform methanogenic degradation of long-chain fatty acids directly, instead of starting from hydrocarbons.

Future studies on Ca. Methanoliparia members will probably uncover intriguing novel molecular mechanisms of methanogenesis and reveal even more of their already impressive range of substrates. Ca. Methanoliparia also represent a useful model to better understand which evolutionary and environmental factors favour the division of labour between two species or the 'do it yourself' approach of this class. Investigating these microbes further will also provide a better understanding of the role of microorganisms in the fate of petroleum in oil reservoirs deep underground.

Finally, the findings shed light on a previously unappreciated source of methane production. It will be crucial to consider the contribution of this natural process to methane emissions and, in turn, to global warming.

[Thank you for suggesting a figure. I have discussed it with our graphics editor, and we feel it needs to be simplified quite a bit in order to be visually clear, especially to nonspecialists. And, the figure also needs to be simplified for the figure legend to keep relatively short, since all technical terms need to be defined in the legend and the legend must keep within 150 words. Therefore, we suggest removing the red font parts (which didn't seem necessary), the different colours/style of arrows, and reducing the number of arrows in total.

I have included two options of what this could look like, below -with all labels in black and in the same size font, in line with our figure style.

We did have one question about part b, which is why I've included two options. We weren't sure whether in part b, acetyl-CoA and CO 2 could be shown together to feed into pathway, rather than separately (since technically, it seems that both acetyl-CoA and CO 2 are generated and are both involved inthe 2 pathway). Please could you let me know your thoughts on this? [In addition, I have redrafted the figure legend according to our house style. The legend must be <150 words long, must cite the discussed paper, should define any abbreviations or specialist terms used in the figure and address each visual element. Please check the legend carefully and revise as appropriate.]

Figure 1 | Two microbial processes of methane production from hydrocarbon molecules. a, Hydrocarbons, entirely made of hydrogen and carbon, are broken down and used to generate methane (CH 4 ) through a collaboration (syntrophy) between bacteria and archaea. In this process, long-chain alkanes are degraded into acetate and hydrogen (H 2 ) by hydrocarbonoclastic bacteria. One group of methanogenic archaea transforms acetate into carbon dioxide (CO 2 ) and CH 4 and another group of methanogenic archaea makes CH 4 using H 2 and CO 2 . b, Zhou et al. [START_REF] Zhou | Wide diversity of methane and short-chain alkane metabolisms in uncultured archaea[END_REF] identified that an archaeon belonging to Candidatus Methanoliparia can couple hydrocarbon degradation and methane generation. Several enzymes (MCR and those of steps 1, 2 and 3) and intermediates (acetyl-coenzyme A (acetyl-CoA) and methyl-coenzyme M (methyl-CoM)) are similar between Ca. Methanoliparia and the syntrophic microbes. Conversely, the initial step of alkane breakdown is mediated by different enzymes, ACR in Ca. Methanoliparia and AssA in hydrocarbonoclastic bacteria.

Long-chain alkanes

AssA Syntrophic n-alkane degradation N-alkane degradation in a single cell

  Methanoliparia and the syntrophic microbes. Conversely, the initial step of alkane breakdown is mediated by different enzymes, ACR in Ca. Methanoliparia and AssA in hydrocarbonoclastic bacteria.

	a
	CH 4	CH 4	Long-chain alkanes
		MCR
			Alkyl-CoM
	Methyl-CoM	Methyl-CoM
	1
	Figure 1 | Two microbial processes of methane production from hydrocarbon molecules. a, Hydrocarbons, entirely made of hydrogen and carbon, are broken down and used to generate methane (CH 4 ) through a collaboration (syntrophy) between bacteria and archaea. In this process, long-chain alkanes are degraded into acetate and hydrogen (H 2 ) by hydrocarbonoclastic bacteria. One group of methanogenic archaea transforms acetate into carbon dioxide (CO 2 ) and CH 4 and another group of methanogenic archaea makes CH 4 using H 2 and CO 2 . b, Zhou et al. Acetyl-CoA CO 2

[START_REF] Zhou | Wide diversity of methane and short-chain alkane metabolisms in uncultured archaea[END_REF]

identified that an archaeon belonging to Candidatus Methanoliparia can couple hydrocarbon degradation and methane generation. Several enzymes (MCR and those of steps 1 [Beta-oxidation], 2 [CODH] and 3 [methyl-branch of the Wood-Ljungdahl pathway + MTR]) and intermediates (acetyl-coenzyme A (acetyl-CoA) and methyl-coenzyme M (methyl-CoM)) are similar between Ca.

ACR Unknown steps Ca. Methanoliparia Alkane degradation in a single cell

	CH 4
	1


Sébastien Rimbert

David Trocellier

Fabien Lotte

Impact of the baseline temporal selection on the ERD/ERS analysis for Motor Imagery-based BCI

Motor Imagery-based Brain-Computer Interfaces (MI-BCIs) are neurotechnologies that exploit the modulation of sensorimotor rhythms over the motor cortices, respectively known as Event-Related Desynchronization (ERD) and Synchronization (ERS). The interpretation of ERD/ERS is directly related to the selection of the baseline used to estimate them, and might result in a misleading ERD/ERS visualization. In fact, in BCI paradigms, if two trials are separated by a few seconds, taking a baseline close to the end of the previous trial could result in an over-estimation of the ERD, while taking a baseline too close to the upcoming trial could result in an under-estimation of the ERD. This phenomenon may cause a functional misinterpretation of the ERD/ERS phenomena in MI-BCI studies. This may also impair BCI performances for MI vs Rest classification, since such baselines are often used as resting states. In this paper, we propose to investigate the effect of several baseline time window selections on ERD/ERS modulations and BCI performances. Our results show that considering the selected temporal baseline effect is essential to analyze the modulations of ERD/ERS during MI-BCI use.

I. INTRODUCTION

Motor Imagery-based Brain-Computer Interfaces (MI-BCIs) are communication and control systems that enable end-users to interact with a computer by performing motor imagery tasks only, e.g., by imagining hand movements, usually measured with electroencephalography (EEG) signals [START_REF] Clerc | Brain-computer interfaces 2: technology and applications[END_REF]. Indeed, movements and imagination of movements activate similar neural networks [START_REF] Filgueiras | The neural basis of kinesthetic and visual imagery in sports: an ale meta-analysis[END_REF], enabling MI-based BCIs to exploit the modulations of sensorimotor rhythms (SMR) over the motor cortices [START_REF] Pfurtscheller | Event-related EEG/MEG synchronization and desynchronization: basic principles[END_REF] for control purposes. Such modulations are respectively known as Event-Related Desynchronization (ERD) and Event-Related Synchronization (ERS), and are coming from the mu (7-13 Hz) and beta [START_REF] Benaroch | When should MI-BCI feature optimization include prior knowledge, and which one?[END_REF][START_REF] Rimbert | Is event-related desynchronization variability correlated with bci performance?[END_REF](17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30) frequency bands [START_REF] Pfurtscheller | Event-related EEG/MEG synchronization and desynchronization: basic principles[END_REF]. MI-BCIs are very promising neurotechnologies for many applications, particularly to control assistive technologies, for sport training or even for poststroke motor rehabilitation [START_REF] Clerc | Brain-computer interfaces 2: technology and applications[END_REF].

Many articles have described the robustness of the ERD and its relevance to design efficient BCIs. Typically, most MI-BCI designs are based on the ERD motor pattern generated during MI. Indeed, it is well known that MI generates a contralateral [START_REF] Pfurtscheller | Event-related EEG/MEG synchronization and desynchronization: basic principles[END_REF] ERD during the task followed by a contralateral ERS afterwards with respect to the requested limb, thereby enabling MI to be detected. Motor imagery generates a gradual decrease in power which starts in the *This work was supported by the ERC with project BrainConquest (grant ERC-2016-STG-714567)

1 S. Rimbert, D. Trocellier and F. Lotte are with the Inria Center at the University of Bordeaux / LaBRI (CNRS / Univ. Bordeaux / Bordeaux INP), 33405, Talence, France sebastien.rimbert@inria.fr preparatory phase of the motor task in the mu and beta rhythms and reaches a maximum during its execution. In general, the ERD has a larger amplitude in the mu-band than in the beta-band, and seems to be modulated according to the experimental criteria (e.g., type of MI performed [START_REF] Rimbert | Modulation of beta power in EEG during discrete and continuous motor imageries[END_REF], level of consciousness [START_REF] Rimbert | Innovative Brain-Computer Interface based on motor cortex activity to detect accidental awareness during general anesthesia[END_REF], type of feedback received [START_REF] Kilavik | The ups and downs of beta oscillations in sensorimotor cortex[END_REF] or user's traits [START_REF] Rimbert | How ERD modulations during motor imageries relate to users' traits and bci performances[END_REF]). We believe that a better understanding of the ERD/ERS modulations during MI-BCI tasks is crucial for developing effective interfaces.

ERD/ERS modulations are often very well described in the literature but appearances can be deceptive, particularly because of the baseline used in ERD estimations. Indeed, ERD/ERS are computed as a change of band power during MI with respect to such a baseline without MI, using a ratio of band powers (e.g., [START_REF] Pfurtscheller | Event-related EEG/MEG synchronization and desynchronization: basic principles[END_REF]). The term 'baseline' thus usually refers to a resting condition of a few seconds before MI that would maximize the ERD phase occurring during MI. Since ERD is a ratio, the estimation of ERD is directly related to the selection of the baseline used to compute it [START_REF] Pfurtscheller | Event-related EEG/MEG synchronization and desynchronization: basic principles[END_REF], [START_REF] Tangwiriyasakul | Importance of baseline in event-related desynchronization during a combination task of motor imagery and motor observation[END_REF]. Several studies have already shown that the selection of the nature of the baseline is significant and could affect the visualization of ERD/ERS [START_REF] Piazza | The effect of baseline on toddler event-related mu-rhythm modulation[END_REF], [START_REF] Tangwiriyasakul | Importance of baseline in event-related desynchronization during a combination task of motor imagery and motor observation[END_REF]. In particular, Tangwiriyasakul et al., used five different baseline movies in a motion imagery task and showed that the majority of the subjects had significantly higher mu power for particular baseline conditions, thus resulting in more pronounced mu ERD [START_REF] Tangwiriyasakul | Importance of baseline in event-related desynchronization during a combination task of motor imagery and motor observation[END_REF]. Another study showed that a baseline taken during a static image seems better (i.e., lead to a larger ERD) compared to a baseline taken during a period of stillness [START_REF] Piazza | The effect of baseline on toddler event-related mu-rhythm modulation[END_REF]. While the selection of the nature of the baseline seems to affect the visualized ERD, the effect of the time at which the baseline is selected is not well-documented.

There are several possible choices of baselines to visualize the ERD during a motor task: before each trial, at the beginning of a run or a session, or by averaging the complete EEG signal over a session [START_REF] Cuevas | The infant EEG mu rhythm: Methodological considerations and best practices[END_REF]. The most common practice in BCI is to use a baseline taken before each trial, thus providing a baseline that corresponds to the user's state at that moment (with the same level of fatigue, workload or concentration which are known to affect the EEG signal [START_REF] Roy | Mental fatigue and working memory load estimation: Interaction and implications for eeg based passive bci[END_REF], [START_REF] Cuevas | The infant EEG mu rhythm: Methodological considerations and best practices[END_REF]). However, BCI paradigms are typically designed in such a way that each trial follows the other very quickly, i.e., there are only a few seconds between the moment when an MI is completed and when the next MI begins. Consequently, the time window for choosing which baseline to use is often limited and might result in a misleading ERD visualization due to the ERD amplitude variation associated with specific periods of the trial [START_REF] Kilavik | The ups and downs of beta oscillations in sensorimotor cortex[END_REF]. For example, choosing a baseline close to the end of the previous MI could result in a selected baseline occurring during the ERS phase (e.g., which occurs one to two seconds after the end of the previous MI), thus significantly increasing the probability of visualizing an ERD, even if there is no absolute power decrease during MI.

Therefore, we believe that the inter-subject variability of ERD/ERS previously described by numerous studies [START_REF] Wriessnegger | Inter-and intra-individual variability in brain oscillations during sports motor imagery[END_REF], [START_REF] Rimbert | Modulation of beta power in EEG during discrete and continuous motor imageries[END_REF] could be partly explained by the temporal baseline selection used. Such temporal baseline selection would influence the ERD visualization and understanding but could also be crucial in terms of BCI classification performances of resting state versus MI, since the resting state class is often selected similarly to the baseline. Consequently, a better understanding of the effect of temporal baseline selection may also lead to better BCI classification performance. In this paper, we thus propose to investigate and quantify the effects of several baseline time window selections on both ERD/ERS modulations and BCI performances. Our results show that the choice of the baseline is essential to understand and analyze the ERD/ERS modulations during MI-BCI use.

II. MATERIAL AND METHODS

A. Participants

Our analyses were performed on a data base of 71 righthanded healthy and MI-BCI-naive subjects (35 women; 25.83 ± 9.52 y.o.), coming from two previous experiments [START_REF] Pillette | Experimenters Influence on Mental-Imagery based Brain-Computer Interface User Training[END_REF], [START_REF] Benaroch | When should MI-BCI feature optimization include prior knowledge, and which one?[END_REF]. These experiments followed the statements of the WMA declaration of Helsinki on ethical principles for medical research involving human subjects, and were approved by Inria ethical committee (COERLE, approval number: 2018-13). All participants signed an informed consent form.

B. Experimental BCI protocol

Each participant completed one MI-BCI session which comprised six 7-minutes runs during which they had to learn to perform left and right hand MI tasks with the BCI. The Graz training protocol was used and was divided into two steps: first, the training of the system (i.e., calibration phase, collecting data to train the EEG machine learning classifier) and second, the training of the user with online feedback (see Figure 1). The EEG data of the first two runs were used as calibration data for the MI-BCI machine learning algorithms. More precisely, EEG signals were first band-pass filtered in a data-driven user-specific frequency band (5-35 Hz), then spatially filtered using Common Spatial Patterns (CSP) filters. Finally, the resulting features were classified using a Linear Discriminant Analysis (LDA) classifier (see [START_REF] Pillette | Experimenters Influence on Mental-Imagery based Brain-Computer Interface User Training[END_REF], [START_REF] Benaroch | When should MI-BCI feature optimization include prior knowledge, and which one?[END_REF] for details). The trained MI-BCI was used to provide online feedback based on right-hand vs. left-hand MI discrimination during the four subsequent runs.

During the first two runs (calibration phase), users were provided with a sham feedback, i.e., a blue bar randomly appearing and varying in length. Such feedback (whether sham here, or real during the next phase) represents the BCI classifier decision (left or right hand MI when the bar goes left or right) and its associated confidence in such decision (i.e., the longer the bar the more confident). During each run (see Figure 1), users had to perform 40 trials (20 per MI-task, presented in a random order), each trial lasted 8s. At t = -3s, a cross was displayed on the screen. At t = -1s, an acoustic signal announced the appearance of a red arrow, which appeared one second later (at t = 0s) and remained displayed for 1.25s. The arrow pointed in the direction of the task to be performed. Participants were instructed to start performing the corresponding MI-task as soon as the arrow appeared, and to keep doing so until the cross disappeared. Finally, from t = 1.25s, a visual feedback was continuously provided in the shape of a blue bar, the length of which varied proportionally to the BCI classifier output, i.e., the distance to the LDA hyperplane (for run 3 to 6) or randomly (for the first two runs). The feedback was provided for 3.75s and was updated at 16Hz (1s sliding window). After 8 seconds, the screen turned black again until the beginning of the next trial. The participant could then rest for a few seconds: from 1.5s to 3.5s, randomly chosen.

C. Electrophysiological recordings

EEG signals were recorded with two g.USBAmp amplifiers (g.tec, Austria), sampled at 512 Hz, from 27 electrodes around the primary motor, the motor and the somatosensory cortices (for more details see [START_REF] Rimbert | How ERD modulations during motor imageries relate to users' traits and bci performances[END_REF]).

1) Topographic ERD/ERS map: All offline analyses of ERD/ERS modulations were performed using the EEGLAB toolbox 22.1 and MATLAB 2016a. The EEG data was divided into 13 second long epochs corresponding to 5 seconds before the MI-cue (t= 0s; see Figure 1) until 8 seconds after, for each trial. Then, we have selected 4 different baseline time windows ([-5;-2]s, [-4;-1]s, [-3;-0]s, [-2;+1]s) all of the same length (3 seconds) in accordance with our hypotheses to compute the ERD/ERS modulations (see Figure 1).

Brain topographies allow us to display the possible changes over different electrodes on the scalp in order to localize which part of the brain was involved when the subject performed the MI. In this article, we computed the topographic Event-Related Spectral Perturbations (ERSP; which is equivalent to ERD and ERS) in the mu + beta ( 8

III. RESULTS

A. ERD modulations according to baseline time selection

As expected, and as previously described [START_REF] Rimbert | How ERD modulations during motor imageries relate to users' traits and bci performances[END_REF], [START_REF] Rimbert | Is event-related desynchronization variability correlated with bci performance?[END_REF], our results showed an activation of the motor cortex with a contralateral ERD, i.e., covering either right and left-hand motor areas depending on the task, during the 5s when the MI task was performed (Figure 2). This result is in harmony with the literature: the ERD phase is mainly observed during the MI task and represents an activation of the motor cortex.

The comparison of ERDs across baselines shows that the closer the baseline used is to the preceding trial (e.g., Baseline 1 or Baseline 2), i.e. after the end of the previous MI, the more global the ERD is (more electrodes show an ERD) and the stronger its amplitude (p < 0.01). This is true both for right and left-hand MIs, but also for both calibration and user training phases (Figure 2).

B. BCI performances according to baseline time selection

Regarding classification performances for MI vs Rest for various baseline/rest time windows, Figure 3 shows that the closer the selected baseline/rest time window to the preceding trial, i.e. to the end of the previous MI, the higher the BCI performance. In contrast, if the selected resting time is close to the MI cue, the BCI performance is worse. BCI performance is significantly better for Baseline 1 compared to Baseline 3 (+4%, p < 0.05) or Baseline 4 (+8%, p < 0.001) for both right-hand MI and left-hand MI tasks. Interestingly, Baseline 2 is an interesting alternative since it improves the classification comparing to Baseline 4 (+7%, p < 0.001). Although the use of baseline 4 in a practical case is rare since the person has already started to perform the MI, it provides a perspective on the impact of having a BCI paradigm designed in such a way that the participant anticipates task performance or does not follow instructions.

IV. DISCUSSION AND CONCLUSION

A. Functional interpretation of the ERD Our results show that the baseline selected can have an impact on the interpretation of the ERD. If a baseline is selected closely to the cue of the following trial, it is possible that the subject's concentration and anticipation will decrease the SMR power, resulting in an under-representation of the ERD. In contrast, our results suggest that the selection of a baseline close to the end of the preceding MI, probably in the middle of the ERS phase, results in an over-representation of a global ERD on many electrodes (Figure 2). This suggests that it is necessary to be careful in interpreting results related to SMR activity, especially because the baseline temporal selection can possibly affect the identification of task-related SMR modulations. This may be an issue for studies preselecting participants or channels based on ERD modulations or simply to enable comparisons between different studies.

B. Implications for MI-BCIs

When designing an experiment to measure SMR, it is important to know what the ideal baseline measurement is and when it should be presented, which is rarely a priority in the BCI field. Indeed, BCI paradigms are typically designed in such a way that each trial follows the other very quickly, i.e., there are only a few seconds between the moment when an MI task is completed and when the next one begins. Our result suggested that this situation can make the baseline choice having a strong impact on the overall results, both in terms of neurophysiological interpretation and BCI performance (for rest VS MI classification). This mostly depends on whether the post-MI ERS from the previous trial affects the baseline measurement of the current trial. Thus, we would recommend designing MI-BCI protocols that either comprise dedicated resting state trials (i.e., with no MI) or that have an inter-trial duration that is long enough to ensure that baselines taken in that period are not contaminated by post-MI ERS from the previous trial. For the later point, since post-movement ERS typically peak 1.5s after the movement end [START_REF] Kilavik | The ups and downs of beta oscillations in sensorimotor cortex[END_REF], we recommend at least 3s in addition to the baseline duration for the inter-trial length, e.g., 6s between trials for a 3s long baseline/rest. For the same reasons, to enable between-studies comparisons between ERD/ERS, we encourage authors to carefully report the timing and length of the baselines they use for ERD/ERS estimation, notably with respect to previous trials.

C. Baseline selection variability

In previous studies [START_REF] Piazza | The effect of baseline on toddler event-related mu-rhythm modulation[END_REF], [START_REF] Tangwiriyasakul | Importance of baseline in event-related desynchronization during a combination task of motor imagery and motor observation[END_REF], the optimal baseline was found to be subject-dependent, with no common preferred baseline for all subjects. Such a time-specific analysis of the ERD should be performed on our data and could confirm these inter-subject variability findings. Additional analysis showed that a majority of subjects has the best BCI performance with either Baseline 1 (for 55% of subjects for left-hand MI vs rest and for 52% of subjects for right-hand MI vs rest) or Baseline 2 (for 30% of subjects for left-hand MI vs rest and for 22% of subjects for right-hand MI vs rest). However, using a subject specific baseline only improved the classification performance (+2%) non significantly.

Overall, we hope this study showed the need to carefully study and report the ERD/ERS baselines for MI-BCI design.

Fig. 1 :

 1 Fig. 1: Detailed steps of two MIs trial with the corresponding visual feedback. 4 different baseline time windows were selected ([-5;-2]s, [-4;-1]s, [-3;-0]s, [-2;+1]s), all of the same length (3 seconds).

  -30 Hz) band during the left-hand and right-hand MI tasks for both the calibration and user training phases in the time window [0;+5]s according to 4 different baseline time windows ([-5;-2]s, [-4;-1]s, [-3;-0]s, [-2;+1]s (Figure 2). 2) Offline BCI performances: We computed offline BCI performances for classifying resting state EEG signals from a single hand MI task EEG (either left-hand or right-hand MI): rest VS left hand MI and rest VS right hand MI classification accuracies. All offline analyses of classification were performed using the MNE and Scikit-learn package in Python 3.9. These performance metrics were estimated offline by first band-pass filtering EEG signals in 8-30Hz and then extracting 3 second long MI and rest epochs from each trial. The MI corresponding to [0.5;3.5]s and the rest phase is identical to the different baseline time windows i.e. ([-5;-2]s, [-4;-1]s, [-3;-0]s, [-2;+1]s (Figure 1). For each baseline/rest paradigm, a left hand MI vs rest and a right MI vs rest classifiers were built using CSP spatial filters and an LDA classifier, as done online for left-hand VS right-hand MI classification. Such classifiers were also trained on the first two calibration runs and tested on the four subsequent runs to obtain (offline) classification accuracies.

Fig. 2 :

 2 Fig. 2: Topographic map of ERD/ERS% (grand average, n=71) in the alpha/mu+beta band during the right-hand and left-hand MIs for both calibration and user training phases.

Fig. 3 :

 3 Fig. 3: Boxplots showing the distribution of classification accuracies (N=71) for the Left-hand MI vs. Rest and Righthand MI vs. Rest classes according to 4 different resting state times selected ([-5;-2]s, [-4;-1]s, [-3;-0]s, [-2;+1]s). *corresponds to a p-value <0.05, ** to a p-value <0.01 and *** to a p-value <0.001.


Bernard Brighi

A propos de la fonction zêta

Keywords: Fonction zêta, série de Taylor, formule d'Euler-Maclaurin, prolongement analytique

Nous montrons que la série de Taylor de la fonction zêta, en un point z ∈ C dont la partie réelle est strictement supérieure à 1, a un rayon de convergence égal à |z -1|. Nous revenons également sur le prolongement analytique de la fonction zêta à C \ {1}.

Introduction

La fonction zêta de Rieman, notée ζ, est définie dans le demi-plan Π 1 = {z ∈ C ; Re z > 1} par l'égalité :

ζ(z) = +∞ n=1 1 n z •
Elle est bien définie, puisque la série au second membre est absolument convergente pour tout z ∈ Π 1 ; en effet, pour tout z = x + iy avec x, y ∈ R et x > 1, on a |n z | = |n x exp(iy ln n)| = n x . La fonction ζ est analytique dans Π 1 , c'est-à-dire développable en série entière en tout point de Π 1 , et on a : alors le rayon de convergence de la série entière a n (z) v n est égal au rayon du plus grand disque ouvert de centre z inclus dans C \ {1}, c'est-à-dire à |z -1|. Le résultat qui affirme qu'une fonction analytique dans un ouvert U de C est égale à sa série de Taylor en tout point z de U et ceci dans le plus grand disque ouvert de centre z inclus dans U, n'est nullement évident ; on le montre le plus souvent à l'aide des formules de Cauchy.

Dans ce qui suit, nous nous proposons de montrer directement (i.e. sans faire appel à un quelconque résultat de la théorie des fonctions analytiques) que le rayon de convergence de la série entière a n (z) v n est égal à |z -1|. Pour cela, à l'aide de la formule d'Euler-Maclaurin, nous allons montrer que, pour tout z ∈ Π 1 , on a :

S n (z) := +∞ k=1 (ln k) n k z ∼ n! (z -1) n+1 quand n → +∞,
ce qui donnera le résultat attendu. Les estimations utilisées pour obtenir cet équivalent vont aussi permettre d'écrire facilement une expression du prolongement analytique de la fonction ζ sur C \ {1}, que l'on reliera à celle donnée par Thomas Stieltjes à la fin du dix-neuvième siècle.

B 0 = 1, B 1 = X -1 2 et B 2 = X 2 -X + 1 6 . Exercice 1 . -Montrer que B n = nB n-1 pour tout n 1 ; en déduire que B n (0) = B n (1) pour tout n 2.

Les formules d'Euler-Maclaurin

Soit m et q des entiers strictement positifs et ψ une fonction de classe C q sur [1, m] et à valeurs dans C. Pour cette fonction ψ, la formule d'Euler-Maclaurin d'ordre q s'écrit :

m k=1 ψ(k) = m 1 ψ(t) dt + 1 2 ψ(1) + ψ(m) + q p=2 (-1) p b p p! ψ (p-1) (m) -ψ (p-1) (1) + r q (ψ) (2)
où b p est la valeur en 0 du polynôme de Bernoulli B p de degré p, et où le reste r q (ψ) est donné par l'égalité :

r q (ψ) = (-1) q-1 q! m 1 B q ({t}) ψ (q) (t) dt, (3)
où {t} = t -t désigne la partie fractionnaire de t. La démonstration de la formule (2), bien qu'un peu calculatoire, n'est pas très difficile (on pourra, par exemple, consulter [START_REF] Demailly | Analyse numérique et équations différentielles[END_REF], chapitre III, paragraphe 4).

Exercice 2 . -Montrer que B n (1 -x) = (-1) n B n (x) pour tout n 0 et tout x ∈ R, puis que b 2j+1 = 0 pour tout j 1. En déduire une simplification de la somme au second membre de l'égalité (2) et une réécriture de cette égalité dans le cas où q = 2l + 1.

Remarque 1 . -Les nombres b p = B p (0), pour p 0, sont les nombres de Bernoulli.

Remarque 2 . -C'est au cours du dix-huitième siècle que Léonard Euler et Colin Maclaurin mettent au point, indépendamment l'un de l'autre, la formule qu'on appelle aujourd'hui formule d'Euler-Maclaurin ou formule sommatoire d'Euler-Maclaurin. Comme c'était souvent le cas à cette époque, ils la présentent sous la forme d'un développement en série, sans réellement se soucier de sa convergence. Lorsque Léonard Euler considère cette série pour du calcul numérique, il n'en prend que la somme des termes qui vont en décroissant, et à partir de l'indice où les termes commencent à croître, il les remplace par un reste dont il n'indique pas l'origine ; voir [START_REF] Nicolas | Eléments de Mathématique : Livre IV, Fonctions d'une variable réelle[END_REF], Note historique, page 157.

Dans un article de W. Wirtinger, datant de 1902, on trouve une démonstration moderne de la formule d'Euler-Maclaurin, obtenue par des intégrations par parties successives, avec une expression du reste semblable à (3) ; voir [START_REF] Wilhelm | Einige anwendungen der Euler-MacLaurin summenformel, insbesondere auf eine aufgabe von Abel[END_REF], paragraphe II.

Les applications de la formule d'Euler-Maclaurin sont nombreuses. La plus évidente concerne les polynômes, puisque le reste est nul dès que l'ordre dépasse le degré du polynôme ; en particulier, elle permet d'avoir une expression de la somme des puissances p-ièmes des n premiers entiers. La formule d'Euler-Maclaurin se révèle très efficace pour obtenir des développements asymptotiques ayant un nombre de termes arbitrairement grand ; à l'ordre q + 1 elle permet d'obtenir le prolongement méromorphe de la fonction zêta dans le demi-plan de C des nombres complexes dont la partie réelle est strictement supérieure à -q ; etc.

Nous l'appliquerons ici pour calculer des rayons de convergence et toute l'information dont nous aurons besoin proviendra du reste.

Premières remarques

Pour z ∈ C et n ∈ N, on note f z,n la fonction définie sur [1, +∞[ par : f z,n (t) = (ln t) n t z . Lemme 3 . -Pour tout z ∈ Π 1 et tout entier n 0, l'intégrale impropre I n (z) = +∞ 1 f z,n (t) dt est absolument convergente et on a : I n (z) = n! (z -1) n+1 . D ÉMONSTRATION. -Soit z = x + iy avec x > 1 et y ∈ R. Comme |t z | = t x pour tout t 1, et que l'intégrale impropre +∞ 1 (ln t) n t x dt
est une intégrale de Bertrand qui converge, la convergence absolue de I n (z) s'en déduit. L'égalité donnant I n (z) est évidente pour n = 0. Ensuite, pour tout n 1, à l'aide d'une intégration par parties, on a :

I n (z) = +∞ 1 (ln t) n t z dt = t 1-z (ln t) n 1 -z +∞ 1 - +∞ 1 t 1-z 1 -z × n(ln t) n-1 t dt = n z -1 I n-1 (z).
Par récurrence, on en déduit que l'égalité souhaitée a lieu pour tout n ∈ N.

Compte tenu du lemme 3, notre objectif consiste donc à montrer que S n (z) ∼ I n (z) quand n → +∞. Dans le cas particulier où z ∈ ]1, +∞[ la fonction f z,n est à valeurs réelles et, en étudiant ses variations, il n'est pas très difficile, par comparaison série-intégrale, d'obtenir cet équivalent.

Exercice 3 . -Montrer que si n ∈ N * et x ∈ ]1, +∞[, alors f x,n est croissante de 0 à un certain point t n et décroissante de t n à +∞. En déduire que |S n (x) -I n (x)| f x,n (t n ), puis que S n (x) ∼ I n (x) quand n → +∞. Si z ∈ Π 1 , on peut utiliser la formule d'Euler-Maclaurin d'ordre 1. Soit z = x + iy avec x > 1 et n 1. Ecrivons (2) avec ψ = f z,n et q = 1, puis faisons tendre m vers +∞, on obtient : S n (z) -I n (z) = +∞ 1 (n -z ln t)(ln t) n-1 t z+1 B 1 ({t}) dt.
Comme B 1 (s) = s -1 2 , on en déduit que :

|S n (z) -I n (z)| +∞ 1 (n + |z| ln t)(ln t) n-1 2t x+1 dt = nI n-1 (x + 1) + |z|I n (x + 1) 2 = n! 2x n+1 (x + |z|)
(la dernière égalité résultant du lemme 3). Par conséquent,

S n (z) I n (z) -1 x + |z| 2 |z -1| x n+1 et donc S n (z) ∼ I n (z) quand n → +∞ pour tout z = x + iy tel que |z -1| < x, i.e. tel que |y| < √ 2x -1.
En utilisant la formule d'Euler-Maclaurin d'ordre 2, on obtient un peu mieux, à savoir que S n (z) ∼ I n (z) quand n → +∞ pour tout z = x + iy tel que |y| < 2 √ x.

On pressent alors que, pour z fixé dans Π 1 , on doit pouvoir conclure en utilisant une formule d'Euler-Maclaurin d'ordre suffisamment élevé. Mais pour cela, il va nous falloir connaître suffisamment précisément les dérivées successives de f z,n . C'est l'objet lemme qui suit.

Lemme 4 . -Soit z ∈ C et n ∈ N. Posons g z,n = f -z,n . Pour tout p ∈ N tel que p < n et tout t > 0, on a : g (p) z,n (t) = t z-p p j=0 n j A (j) p (z)(ln t) n-j (4)
où (A p ) p 0 est la suite de polynômes définie, par récurrence, par

A 0 = 1 et A p+1 = (X -p)A p si p 0. Autrement dit : A p = X(X -1) • • • (X -p + 1). D ÉMONSTRATION.
-On montre ce résultat par récurrence sur p. Il n'y a rien à montrer pour p = 0. Supposons ensuite que l'égalité donnant g (p) z,n est vraie pour un entier p 0 quelconque. Par conséquent, pour tout t > 0, on a :

g (p+1) z,n (t) = (z -p) t z-p-1 p j=0 n j A (j) p (z)(ln t) n-j + t z-p-1 p j=0 (n -j) n j A (j) p (z)(ln t) n-j-1 = t z-(p+1) p j=0 n j (z -p)A (j) p (z)(ln t) n-j + p j=0 (j + 1) n j + 1 A (j) p (z)(ln t) n-(j+1) = t z-(p+1) p j=0 n j (z -p)A (j) p (z)(ln t) n-j + p+1 j=1 j n j A (j-1) p (z)(ln t) n-j = t z-(p+1) (z -p)A p (z)(ln t) n + p j=1 n j (z -p)A (j) p (z) + jA (j-1) p (z) (ln t) n-j + (p + 1) n p + 1 A (p) p (z)(ln t) n-(p+1) . Or, on a (z -p)A p (z) = A p+1 (z) et (p + 1)A (p) p (z) = (p + 1)p! = (p + 1)! = A (p+1)
p+1 (z), et aussi :

A (j) p+1 = (X -p)A p (j) = (X -p)A (j) p + jA (j-1)
p pour tout j ∈ {1, . . . , p}.

L'égalité souhaitée pour g (p+1) z,n en résulte. D'où la conclusion.

Remarque 5 . -Si p = n, l'égalité (4) est encore vraie (en convenant, pour t = 1, que 0 0 = 1). Si p > n, elle est encore vraie pour t = 1.

Calcul du rayon de convergence

Soit z ∈ Π 1 et n ∈ N ; d'après le lemme 4, pour tout entier p tel que 0 p < n, on a :

∀t 1, f (p) z,n (t) = 1 t z+p p j=0 n j A (j) p (-z)(ln t) n-j et donc : f (p) z,n (1) = 0 et lim t→+∞ f (p) z,n (t) = 0.
Par conséquent, en écrivant la formule d'Euler-Maclaurin (2) pour ψ = f n,z et q n, puis en faisant tendre m vers +∞, on obtient :

I n (z) -S n (z) = (-1) q q! +∞ 1 B q ({t}) f (q) z,n (t) dt. (5)
Le nombre complexe z = x + iy étant fixé dans Π 1 , considérons un entier q 1, que nous choisirons plus loin.

Pour tout n q, on a l'égalité (5) et donc :

|S n (z) -I n (z)| c q +∞ 1 |f (q) z,n (t)| dt où c q = 1 q! max s∈[0,1] |B q (s)|.
Par conséquent, compte tenu des lemmes 3 et 4, pour tout n q, il vient :

|S n (z) -I n (z)| c q q j=0 n j A (j) q (-z) +∞ 1 (ln t) n-j t x+q dt = n! c q q j=0 |A (j) q (-z)| j! (n -j)! I n-j (x + q) = n! c q q j=0 |A (j)
q (-z)| j! (x + q -1) n-j+1 puis :

S n (z) I n (z) -1 c q |z -1| n+1 q j=0 |A (j) q (-z)| j! (x + q -1) n-j+1 = c q q j=0 |A (j) q (-z)| j! (x + q -1) -j |z -1| x + q -1 n+1 (6) ce qui montre qu'en choisissant q > |z -1| -x + 1, on obtient finalement que S n (z) ∼ I n (z) quand n → +∞, autrement dit que S n (z) ∼ n! (z -1) n+1 quand n → +∞. Il s'ensuit que lim n→+∞ |a n (z)| 1/n = lim n→+∞ 1 |z -1| 1+ 1 n = 1 |z -1|
et donc que le rayon de convergence de la série entière a n (z) v n est égal à |z -1|.

Prolongement méromorphe de la fonction ζ

Dans le paragraphe 4 ci-dessus, c'est grâce à l'estimation (6) que nous avons pu conclure. Cette estimation permet aussi de montrer que la fonction F définie sur Π 1 par :

F (u) = ζ(u) - 1 u -1
se prolonge analytiquement sur tout C. Avec les notations des paragraphes précédents, pour tout u ∈ D(2, 1), on a :

F (u) = +∞ n=0 a n (2)(u -2) n - +∞ n=0 (-1) n (u -2) n = +∞ n=0 (-1) n S n (2) n! -1 (u -2) n .
En utilisant l'inégalité (6) avec z = 2 et en notant que n! = I n (2), il vient :

∀q 1, ∀n q, S n (2) n! -1 K q (q + 1) n où K q = c q q j=0 |A (j) q (-2)| j! (q + 1) -j+1 .
Par conséquent, le rayon de convergence de la série entière

G(v) = +∞ n=0 (-1) n S n (2) n! -1 v n
est égal à +∞ (car plus grand que tout nombre réel q 1). Il en résulte que la fonction zêta admet un prolongement analytique sur C \ {1}, encore noté ζ et donné par :

ζ(u) = 1 u -1 + G(u -2). (7)
Remarque 7 . -Dans nombre d'articles sur la fonction zêta, on trouve l'égalité suivante :

ζ(u) = 1 u -1 + +∞ k=0 (-1) k k! γ k (u -1) k avec γ k = lim m→+∞ m j=1 (ln j) k j - (ln m) k+1 k + 1 . (8)
Il s'agit là du développement en série de Laurent au point 1 de la fonction ζ. Ce développement est déjà évoqué en 1885 par Thomas Stieltjes dans sa correspondance avec Charles Hermite ; cf. [START_REF]Correspondance d'Hermite et de Stieltjes[END_REF], lettres 71 (page 45) et suivantes. Pour faire le lien entre (7) et (8), il suffit de remarquer que, pour tout u ∈ C, on a :

G(u -2) = +∞ n=0 S n (2) n! -1 (2 -u) n = +∞ n=0 S n (2) n! -1 1 + (1 -u) n = +∞ n=0 S n (2) n! -1 n k=0 n k (1 -u) k = +∞ k=0 (-1) k +∞ n=k n k S n (2) n! -1 (u -1) k = +∞ k=0 (-1) k k! +∞ n=k S n (2) -n! (n -k)! (u -1) k ce qui montre que : γ k = +∞ n=k S n (2) -n! (n -k)! .
Ce que l'on a fait ci-dessus avec 2, on peut le faire avec n'importe quel autre point z ∈ Π 1 . En effet, si z = x + iy avec x > 1, alors pour tout u ∈ D(z, x -1), on a :

F (u) = G(u -z) avec G(v) = +∞ n=0 (-1) n S n (z) -I n (z) n! v n
et le rayon de convergence de cette série entière est égal à +∞ ; en écrivant le développement de F au point 1, on obtient alors :

∀k 0, γ k = +∞ n=k S n (z) -I n (z) (n -k)! (z -1) n-k et ceci pour tout z ∈ Π 1 .
Les nombres γ k sont les constantes de Stieltjes ou constantes généralisées d'Euler, cette dernière dénomination provenant (certainement) du fait que γ 0 est égal à la constante d'Euler γ ≈ 0.5772.

La littérature est abondante à propos de ces constantes. Le résultat le plus marquant a été obtenu par Y. Matsuoka en 1984 (voir [START_REF] Yasushi | Generalized Euler constants associated with the Riemann zeta function[END_REF], Theorem 6). Il s'agit de l'estimation suivante :

∀n 10, |γ k | 10 -4 (ln k) k (9)
qui, entre autre, assure que le rayon de convergence de la série entière présente dans (8) est égal à +∞. La démonstration de (9) est assez technique et l'appréhender dans tous ses détails nécessite quelques efforts1 .

6 Appendice

= x + iy ∈ Π 1 et u ∈ D(z, x -1). Pour k ∈ N * et n ∈ N, posons : w k,n = (-1) n (ln k) n (u -z) n n! k z et
ζ(u) = +∞ k=1 1 k z+(u-z) = +∞ k=1 1 k z e -(u-z) ln k = +∞ k=1 1 k z +∞ n=0 (-1) n (ln k) n (u -z) n n! = +∞ n=0 (-1) n n! +∞ k=1 (ln k) n k z (u -z) n
(la dernière égalité s'obtenant en intervertissant les sommations, ce qui est possible grâce à la convergence absolue montrée plus haut). D'où (1).

Remarque 8 . -On peut aussi parler de famille sommable, mais on peut éviter ces notions (série double ou famille sommable) et obtenir l'analyticité de ζ sur Π 1 en montrant un peu moins que [START_REF]Correspondance d'Hermite et de Stieltjes[END_REF]. Précisément, pour Remarque 9 . -La preuve ci-dessus se généralise facilement si, à la place de C \ {1}, on considère un ouvert U dans lequel deux points quelconques sont joignables par une ligne brisée. On pourra, par exemple, consulter [START_REF] Jean | Calcul infinitésimal[END_REF], chapitre VI, paragraphe 7.

z = x+iy ∈ Π 1 , si u ∈ D(z, x-1 2 ) alors, en utilisant que S N (t) ∼ I N (t) quand N → +∞ avec t = x-|u-z|, on peut montrer que +∞ k=1 1 k z +∞ n=N (-1) n (ln k) n (u -z) n n! → 0 quand N → +∞.

∀z ∈ Π 1 ,( 1 )

 11 ∀u ∈ D(z, x -1), ζ(u) = ln k) n k z (u -z) n (où x = Re z). (Voir paragraphe 6.1 plus loin. Par ailleurs, on peut montrer de différentes fac ¸on que la fonction ζ se prolonge analytiquement sur C \ {1} et que le point 1 est un pôle simple de ζ. Par conséquent, si pour tout z ∈ Π 1 et tout n 0, on pose a n (z) = (-1) n n! +∞ k=1 (ln k) n k z

Remarque 6 .

 6 -Ce qui précède montre que les fonctions u → a n (z) (u -z) n pour z ∈ Π 1 , fournissent un prolongement analytique de ζ à C\ ] -∞, 1].

1 k

 1 montrons que la série double de terme général w k,n est absolument convergente. Pour tout entier N 1, x e (ln k)|u-z| = N k=1 1 k x-|u-z| ζ(x -|u -z|) puisque, pour tout u ∈ D(z, x -1), on a |u -z| < x -1 et donc x -|u -z| > 1 ; d'où la convergence absolue souhaitée. Par suite, pour tout u ∈ D(z, x -1), on a :

6. 2

 2 Unicité du prolongement analytique de ζIl s'agit de montrer que si ϕ est une fonction analytique sur C \ {1}, qui est nulle sur Π 1 , alors elle est identiquement nulle.Soit z ∈ C \ Π 1 ; il existe z 0 ∈ Π 1 tel que le segment [z 0 , z] est inclus dans C \ {1}. Comme ϕ est continue, la réunion de tous les segment [z 0 , v] inclus dans [z 0 , z] sur lesquels ϕ = 0 est un segment [z 0 , u] (c'est le plus grand segment d'origine z 0 inclus dans [z 0 , z] sur lequel ϕ = 0). Puisque Π 1 est ouvert, le segment [z 0 , u] n'est pas réduit à un point. Supposons que u = z. Comme ϕ est développable en série entière au voisinage de u, il existe r > 0 et des nombres complexes c n tels que, pour tout v ∈ D(u, r), on a :ϕ(v) = +∞ n=1 c n (v -u) n et comme ϕ(v) = 0 pour tout v ∈ D(u, r) ∩ [z 0 , u],tous les c n sont nuls ; en effet, dans le cas contraire, si p est le plus petit entier tel que c p = 0, alors, pour tout v ∈ D(u, r), on a ϕ(v) = (v -u) p ϕ 1 (v) où ϕ 1 est continue et vérifie ϕ 1 (u) = c p = 0, ce qui entraine qu'il existe un voisinage de u dans lequel ϕ ne s'annule qu'en u, d'où une contradiction (il s'agit là du principe des zéros isolés). Par suite, ϕ est identiquement nulle sur D(u, r), ce qui contredit la définition de u. D'où u = z et ϕ(z) = 0.

2

  Les polynômes de Bernoulli et les formules d'Euler-Maclaurin 2.1 Les polynômes de BernoulliNotons E l'espace vectoriel sur R des polynômes à coefficients réels. L'application L qui a tout polynôme P de E associe le polynôme L(P ) tel que

	∀x ∈ R,	L(P )(x) =

x+1

x P (t) dt est linéaire et préserve le degré ; il s'ensuit que L est un isomorphisme de E dans lui-même. On définit alors la famille {B n ; n 0} des polynômes de Bernoulli comme étant l'image de la base canonique de E par L -1 ; autrement dit, pour tout entier n 0, on a :

∀x ∈ R,

x+1 x B n (t) dt = x n et ceci caractérise B n . Pour tout n 0, le polynôme B n est de degré n et, si n 1, son intégrale sur [0, 1] est égale à 0. Les trois premiers polynômes de Bernoulli sont :

  Montrons que la fonction ζ est analytique sur Π 1 , c'est-à-dire qu'elle est développable en série entière en tout point z ∈ Π 1 . Pour cela, on va montrer que l'on a (1). Soit donc z

	6.1 Analyticité de la fonction ζ

L'article de Yasushi Matsuoka ne semble pas disponible en ligne, mais est consultable dans plusieurs bibliothèques universitaires.