Version légèrement modifiée d'un article publié dans Radical Philosophy https://www.radicalphilosophy.com/-146 (2007)

Who's Afraid of Gay Parents?

Sylvie Duverger

Symbolic order (dated): see Sexual difference (will never go out of fashion). … Sexual difference, cultural, natural, in a word anthropological. Republican value. Universal. Intellectual buffer, impossible to go beyond, like day and night. Not to be gone beyond in any event, or at least not in French culture (1). Legislation adopted in France in 1999, after some ten years of controversy, ensures that all couples, irrespective of the sex of the partners, can sign a "Pacte civil de solidarité". A PACS is, however, far from being an equivalent to the civil partnerships introduced in the United Kingdom in 2005. Despite amendments in 2006, the PACS does not allow same-sex couples to acquire the rights enjoyed by married couples. More specifically, it does not allow them to adopt, either as a couple or as individuals (where one partner adopts the child and the other acquires the status of co-parent). In the United Kingdom, that right was extended to same-sex couples by the provisions of the Adoption and Children Act of 2002, and same-sex couples have been granted the same right in several other Western countries. Since the end of the 1990s, a growing number of same-sex couples in France have been demanding the right to marry and to obtain parental status. Shortly before he was elected president of the French Republic, Nicolas Sarkozy undertook to introduce a civil union that would grant same-sex couples the right to inherit and all the tax and social rights that had hitherto been the prerogative of married couples, but not the rights to same-sex filiation and adoption. In respect to the latter, he proposed simply that a mandate to educate or an agreement to share parental authority should be granted to a 'trusted third party', 'usually the step-parent':

There is a family model, and it is organized around a father and a mother. It is a social model derived from a biological model. A child is conceived by a man and a woman, and that is how and why the family was created. Knowing that there are same-sex couples who are bringing up children in the real world is one thing … Passing a law which states that a family can consist of a man and a woman, two men or two women, and that this makes no difference, is quite another … One can certainly enhance the status of step-parents, by saying, among other things, that, in the event of the death of the biological parent, the step-parent has specific rights and duties with respect to the child. … We do not need the fiction of adoption to do that. (Têtu, April 2007) These issues are due to be debated again during the parliamentary session that has just opened in France.

A week before the men and women taking part in the Marche des fiertés (formerly Gay Pride) loudly and publicly demanded the right to marry and to adopt, the philosopher Sylviane Agacinski, an apologist for sexual difference and parity in political and cultural terms if ever there was one, published an article in Le Monde in which she summarized her arguments against the extension of marriage and adoption rights to same-sex couples (2). Whilst the 'new French feminism', or at least the feminism promoted by Agacinski, 'affirm[s] sexual duality as the only universal difference within humanity',

'homosexual marriage' would symbolically make individuals of the same sex parental couples, and that would call into question the bilateral filiation of their children (a maternal side and a paternal side). … A homosexual parental couple would abolish the man/woman distinction in favour of a distinction between homosexuals and heterosexuals. (TPF 18)

If same-sex couples were no longer prohibited from marrying and if, therefore, they could acquire the status of a parental couple, it would destroy the 'model that articulates generation, sexual difference and generational difference'. French family law, which restricts marriage, filiation and adoption to 'mixed couples', is supposedly an analogical expression of the zoological order: or the fact that we are born of one male and one female, that we can only be one sex, and that one therefore cannot be both mother and father, only one or the other.

According to Agacinski, preventing same-sex couples from marrying is not a way of guaranteeing a heterosexual hegemony. The institution of marriage does not legalize heterosexuality, it regulates filiation. The point of the prohibition is, rather, to guarantee that every man and woman is inscribed 'within the order of a humanity that is itself sexed and to ensure that he/she accepts that he/she will never be 'the whole human' (PS 33). If same-sex couples were able to marry, the inevitable outcome would be the legitimation of gay parenting, and that would endanger 'the human, social and symbolic order '. Nothing less.

One cannot but be surprised to find that Sylviane Agacinski's arguments support Nicolas Sarkozy's point of view. Perhaps it is less surprising when it is demonstrated that they derive from a differentialist and naturalist feminism and that, paradoxically but inevitably, they converge with the arguments of those Lacanians who support a return to a symbolic patriarchal order that supposedly protects us from, if not psychotic indifferentiation, at least 'the absolute freedom which simply means that everything is equivalent to everything else', to cite Michel Schneider, one of the staunchest defenders of the 'dissymmetry of paternal and maternal functions'. [START_REF] Schneider | La Confusion des sexes[END_REF].

The institutionalization of heterosexuality denied

Let us be clear. Agacinski is not opposed to gay men and women bringing up children, but to the idea that a child can have two parents of the same sex:

It is not desirable to legally attribute to a child a parental couple of the same sex. But I will not confuse this principle with the very different idea that homosexuals ought not to be able to have or adopt children … One can very easily be parent and homosexual … but it is not as homosexual nor, moreover, as heterosexual that one is a father or a mother; it is first of all as man or woman, and thus with a second parent of another sex or alone. (PS xiii, translation altered) She accepts that, within a same-sex couple, the biological parent's partner can perfectly well perform 'the function of a step-father or -mother, with all the concomitant rights and duties' with respect to the child for whom he/she cares, but not that the couple concerned can be recognized as a parental couple. Same-sex couples should not be granted all the rights conferred by marriage, because nature itself does not allow two individuals of the same sex to have children together: 'Humanity is naturally heterosexual … humans, universally sexed, are generally animated by desire for the other and depend on this other to procreate' (PS 83). Refusing same-sex couples the right to marry and to be joint parents does not, however, constitute discrimination against them on the basis of their sexuality because the sexuality of individuals has never been the basis of marriage or parenthood. Their basis is primarily sex, or in other words the anthropological distinction between men and women …. Parenthood has nothing to do with sexuality, as we can see from the case of adoption … The rules of filiation are based upon the bilaterality of the male/female couple … It is never love or desire as such that defines the binary nature of the genitor couple. (EDSS 129) It therefore cannot be argued that 'the very equality of individuals should preclude all discrimination on the basis of sexual orientation' (4) because it is not in so far as they are heterosexual that men and women can get married. They can do so simply because they form a couple made in the image of the genitor couple, the heterosexual or mixed couple (EDSS 129). In other words, they can marry because it is 'likely' that sexual relations between them will be fertile. Agacinski thus urges us to be more perspicacious:

It seems to go unnoticed that the demand for 'gay marriage' or 'gay parenting' can only be formulated on the basis of the construct or fiction of subjects-in-law that have never existed: 'heterosexuals'.

To which one would have to reply that the annointing of mixed couples as the paradigm does not, to say the least, avoid the heterosexism famously denounced by Judith Butler. Of course, Agacinski bluntly rejects Butler's theses. But if heterosexism's primary vocation is to plead the case for biological reproduction, arguing that there is no imposition of the heterosexual norms in the priviledging of the genitor, rather than the heterosexual, couple is surely a surrender to a 'heterosexual phallocentrism' (Derrida) designed to promote a fertile sexuality that obeys the law of the father or conforms to the 'state's desire' (Butler). Should any doubt about this remain, it is removed by the staggering fact that Agacinski has no qualms about laying claim to the heritage of Roman law, despite its undeniable patriarchalism:

In a civilization such as ours, which is the heir to Roman law, marriage has always meant the legal union of one man and one woman, whom he makes the mother of his children: the French word matrimonial retains a trace of the Latin matrimonium (marriage), which is designed to make a woman a mother (mater).

Thus, when gay parenting is at issue, Agacinski appears to forget her own analyses of patriarchalism, analyses based on the work of the anthropologist Françoise Héritier:

The ancestral incertitude that strikes at the parental origin of a child, whereas the maternal origin is clear, constitutes an alienation of masculine fertility. This cannot help but shed light on why men wish to appropriate their offspring by appropriating one or several women. It also explains why they have tried to bestow upon themselves the true power to engender and have, almost everywhere, theorized about the pre-eminent role, even exclusive role, of the father in the transmission of life. (PS xx) 'Fortunately', as Agacinski reminds us, 'we have come a long way since the time of Roman law and the sexual inequality it instituted.' But then, rather than affirming that 'marriage is still based upon the union of the two sexes because they play complementary roles in generation', should we not conclude that it is time to rid ourselves completely of that heritage? Agacinski refuses that conclusion out of fear of the new civilization being outlined by new forms of parenthood. Her position is therefore potentially even more prohibitive than that of the state, which not only accepts the possibility of marriage for mixed couples who are no longer of an age to procreate, who cannot have children or who use contraception.

To assert, as Agacinski does, that heterosexuals are not subjects in law is to deny that homosexuals have been and are the victims of discrimination because they do not conform to the heterosexual norm; at the same time, she argues that the civil law should continue to enforce that norm. To be perfectly consistent, if one wants to deny that family law discriminates between heterosexual and homosexual couples, one would have to agree that homosexual marriage should be legalized on a basis equal to that of its heterosexual counterpart.

Rather than acceptingas many studies have provedthat gay families are neither more nor less pathological than any others, Agacinski prefers to wonder whether 'the establishment of a filiation relating to individuals of the same sex might not play a part in … complicating the child's awareness of sexual alterity and his or her access to his or her own identity' (EDSS 132). Drawing on Freudthough without actually referring to any specific textsshe takes the view that, 'through its contradictions and from its outset', the structure of the parental couple, 'mixed in both the real and the symbolic', 'programmes the need for a child to choose an identity for himself and to choose objects, which are themselves sexed' (PS 88). But if, as she also argues, 'from a psychoanalytic point of view, the homosexual choice … does not call into question either sexual difference or the ultimate anchoring of this difference in generation' (PS 89), her fears for sexed subjectivation are groundless.

Agacinski's contradictions are in fact new variations on Lacan's ambiguous pronouncements on the relationship between sex difference and sexual positions, between the 'real lack' (the fact of being 'subject to the of sexed reproduction') and lack-in-being ('the fact that the subject depends on the signifier and that the signifier is first of all in the field of the Other') [START_REF] Lacan | The Four Fundamental Concepts of Psychoanalysis[END_REF]. The argument that gay parenting obliterates the difference between the sexes, making it impossible for the child to be inscribed in 'the symbolic, social and human order' also echoes the warnings that certain Lacanians have felt obliged to address to French public opinion about the future development of the children of parents of the same sex.

Lacanianism, patriarchalism, naturalism

Children's rights: speak on their behalf. As all the experts know, children need a father and a mother (gay parenting and adoption by single parents are delicate issues) [START_REF] Fabre | Dictionnaire des idées reçues sur le mariage gay' in Liberté, égalité, sexulités[END_REF].

If, according to Agacinski, individuals of the same sex cannot be legally recognized as co-parents, this is because 'the human is necessarily masculine or feminine … Neither man nor woman constitutes the "whole human"' (PS 35). But we are obliged, further, to accept that our origins lie in a conflictual 'sexual bilaterality (a father and a mother)' (EDSS 131) and that there is necessarily some differend or misunderstanding between individuals who are different: 'Men and women depend on each other for the satisfaction of their desires as well as for procreation … the game does not derive solely from a natural difference, but from conflicts of interest and mutual dependence' (PS 23-4); 'Without this originary discord, human relations would be deprived of their first enigma, their first doubt about the identity of the other and the problem forever posed by the coexistence with him or her. Uncertainty and misunderstanding are the motivating forces in relations between the sexes' (PS 23).

At the same time, Agacinski argues, we should recognize that we are of oneand only onesex and can therefore fulfil only one of two interdependent reproductive functions. If we do not, our very psychosexual identity is at stake and we fall victim to hubris in denying our finitude. The mixed-sex couple bears exemplary witness to the fact that, as men and women, we are dependent on one another with respect to an undeniable difference whose enigma supposedly 'puts the paradigm of alterity at the heart of the human condition' [START_REF] Feher | Quelques réflexions sur Politique des sexes[END_REF].

It is as though families that conform to the heterosexual matrix were the necessaryand sufficientcondition for the recognition of human finitude, and for our awareness of the fact that our individuality, at once distinct, other and separate, whilst also being bound up with and dependent upon others, is inscribed in the sequence of generations: that it is, in a word, human. It is also as though the only possibility offered by same-sex couples were that of non-differentiation; as though they denied the alleged difference between the sexes, and could never recognize their finitude-castration; as though neither party in a same-sex couple could fulfil the function which supposedly devolves upon the father and plays a necessary role in the child's becoming-subject, one's entry into the order of signification and socialization. All this, despite the obvious fact that today the roles of father and mother are constantly being exchanged within mixed couples. Such considerations, among others, take us to the 'heart of all homophobic phantasies … in the incestuous world of the organic sexual magma that homosexuals supposedly never escape' [START_REF] Théry | Différences des sexes, homosexualité et filiation[END_REF]. It might be objected that Agacinski recognizes that 'far from being ignorant of the difference between the sexes, homosexuality refers to the categories of same and other in precisely the same way as heterosexuality' and reminds us that 'the other is desirable only if he, like me, is sexed, is always an other's other and therefore an other to oneself' (EDSS 128). But she does not, to say the least, draw the obvious conclusion from all this. For Agacinski's position is naturalist. It suggests that the two reproductive sex functions are so different that alterity itself ultimately boils down to the difference between the sexes:

The mixity of humanity, relative to the division of sex roles in generation, is not only a given of physical anthropology but, further, from our point of view, it also structures cultural duality and value by generating singularity and diversity. In effect, it is through the union of the two parents' genes that each individual draws one's singularity. (PS 109)

Here Agacinski fails to take into accountalthough she recognizes it elsewherethe fluctuations in maternal and paternal roles, and in gender. The occasional assertion that gender is a construct or, for example, that 'one must not confuse reference to the natural reality of the sexes with submission to a natural order' (PS 21) is not enough to escape naturalism. We also have to ask ourselves to what extent our conception of 'sex' might be a construct, and query the value of the hypothesis that the social or symbolic order is based upon the biological order. This naturalism explains why her comments on gay parenting are consistent with those of the Lacanians whose phallogocentrism is not in doubt. As Michel Schneider argues, the move to recognize gay parents is an 'assault on the symbolic order' (Le Figaro, 18 May 2002), just like political parity, the feminization of job descriptions and titles (CS 88-94), allowing children to take their mother's family name (CS 80-87) and, of course, the failure to respect roles that are supposedly prescribed sub specie aeternitatis: 'It is because fathers are no longer fathers and mothers are no longer mothers that sons find it so hard to be men and that daughters find it so hard to be women' (CS 16).

Agacinski fears that dual parentage without mixity invites 'the danger that we will construct the institutional fiction of homosexual fertility' (EDSS 130), as if same-sex couples are incapable of explaining to their children that it was not the stork that brought them. This also meets with the approval of Jean-Pierre Winter, who takes the view that in their structuring attempts to imagine how they came to be born, the children of same-sex parents will be faced with an impossible task (Le Monde, 30 June 2007). Agacinski, like Schneider and Winter, seems here to forget the plasticity of the human psyche. Acknowledging that the well-being of an individual is proportional to his or her mental plasticity and creativity, she then forgets that conformity to the heterosexual family form provides no special guarantees in that respectfar from it. This is a necessary amnesia for those who reject 'swaying in and out of one's sex and one's roles', preferring the hypostatized structure of a sexual difference in which there is no differance. On the grounds, allegedly, that they are safeguarding alterity… Agacinski obviously does not completely overlook the constructedness of sex and gender -'the division of the sexes is always simultaneously natural and cultural, real and symbolic, biological and psychic' (PS 12) -or that there is 'a certain vagueness about "the genders" -in a psychological or cultural sense' (PS xvi). But she is very careful not to investigate the basis of the biological definition of sexual difference, which is, in her view the 'starting point' for both femininity and masculinity (PS xvi). She does, of course, recognize here and there that nature and the natural never speak anything other than the language of power:

The simple anatomical or physiological description of male and female characteristics has always been dominated by hierarchical models borrowed from other fields (technical or political, for example). (PS ix) Natural difference, in its essential connection with birth, says nothing to us about the way in which relations between men and women find themselves concretely regulated … Although very real, natural differences never directly engender social or cultural norms. This norm is always moral, political and aesthetic. (PS 22) But, having made these perfectly valid points, she seems immediately to ignore their implications. In Agacinski's view the cultural/symbolic/psychic is no more than a translation of the natural/real/biological. Agacinski tries to avoid the differentialism that boasts of knowing what femininity would or should be were it not devalued by phallogocentrism, but still speaks of 'femininity' and 'masculinity' without questioning or clarifying their meaning, leaving open the possibility of invasion by all kinds of stereotypes.

The differences that the patriarchal system tries to establish between men and women (symbolic/cultural versus corporeal/natural, rational versus emotional, economic and political versus domestic), on a claimed basis in nature, cannot resist their deconstruction by gender theory and the ongoing redistribution or roles and tasks that result from feminist demands. But it is, perhaps, the public recognition of the existence of homosexual couples and same-sex parenting that deals the final blow to the institutionalization of rigid differences between men and women. This is why neodifferentialist thinking must refute the universalist demands of gays and lesbians. And Agacinski comes close to admitting as much:

Homosexuality does not constitute an erasure of sexual difference; it is a different way of living it …. But in cultural terms, that difference is probably more fluid than it was in the past. Men, and especially women, are less constrained in stereotypical roles and much more able to play on the ambiguities of the masculine and feminine in their behaviours, their roles and even their sexual roles. That is why I had to rediscover what the hard core or basis of sexual difference is. Its basis is generation, or the difference between paternal and maternal roles. It takes a masculine element and a feminine element to create life … Procreation and the father/mother differential are decisive. (QAF 22-3)

The fear that the difference between the sexes might be eroded dates from the late nineteenth century, emerging just as patriarchal authority began to go into decline. But if phallogocentrist psychoanalysts will inevitably try to block the deconstruction of this difference, should not feminists see it as cause for celebration? Does Sylviane Agacinski really represent a 'new French feminism'?

  Clarisse Fabre and Eric Fassin, 'Dictionnaire des idées reçues sur le mariage gay', Liberté, égalité, sexualités, Belfond, Paris, 10/18, 2004, pp. 312, 308. 2.. Sylviane Agacinski, 'L'Homoparentalité en question', Le Monde, 21 June 2007. All further unattributed quotations refer to this article. These arguments are developed in 'Questions autour de la filiation' [QAF], Ex Aequo 20, July-August 1998; Parity of the Sexes [PS], trans. Lisa Walsh, Columbia University Press, New York, 2001, pp. xii-xvii, 90-119, 127-33; 'The Turning Point of Feminism: Against the Effacement of Women' [TPF], trans. Mary Schwartz, in R. Celstin, E. DalMolin and I. de Courtivron, eds, Beyond French Feminism: Debates on Women, Politics and Culture in France, 1981-2001, Palgrave Macmillan, New York, 2003;


Béatrice Boulu-Reshef
email: beatrice.boulu-reshef@univ-paris1.fr

Samuel H Brott
email: sbrott@thinkbrg.com

Adam Zylbersztejn
email: zylbersztejn@gate.cnrs.fr

Does Uncertainty Deter Provision of Public Goods?

Keywords: Cooperation, uncertainty, voluntary contribution mechanism, public good, expected return

We study a finitely repeated public goods game (based on the voluntary contribution mechanism) played under complete uncertainty about the marginal benefit of the public good relative to the private consumption (commonly known as the marginal per capita return): neither one's marginal per capita return nor other players' marginal per capita returns are known at the time of decisionmaking. We show that contributions are equivalent when the rate of return is predetermined and when it is uncertain, and display a similar decay over time. Combined with the previous experimental findings, our results suggest that the cooperation in public goods games is sensitive to the source of uncertainty about marginal per capita return.

Introduction

The voluntary contribution mechanism (hereafter, VCM) is a simple and elegant way to capture the social dilemmaa trade-off between individual and group interestunderlying the production of public goods. Over the years, investigating the determinants of public good provision via voluntary contributions experiments has become a classic topic in economics and other social sciences [START_REF] Marwell | Experiments on the Provision of Public Goods. I. Resources, Interest, Group Size, and the Free-Rider Problem[END_REF][START_REF] Kim | The Free Rider Problem: Experimental Evidence[END_REF][START_REF] Ledyard | Public goods: a survey of experimental research[END_REF].

Nonetheless, certain features of standard VCM experiments might call their external validity into question. For instance, the marginal per capita return on each unit of wealth contributed to the public good (hereafter MPCR), i.e. the marginal benefit of the public good relative to the private consumption, is common knowledge in numerous laboratory experiments. 1 However, this absence of uncertainty may not be a general feature of real world public goods. This might be the case because one's needs cannot be defined with certainty (like the demand for health care), or because the value of a public good only becomes known ex post (like the value of education in a newly established public school or the value of a newly opened public park). In addition, the value of a public good for the other individuals in a group may not be known to those who are in a position to contribute to a public good and these values may vary over time.

Whether and how uncertain MPCR may affect the provision of public goods remains an open question. Existing experimental methodology is based on partial uncertainty (related either to personal MPCR or others' MPCRs) and yields mixed results. For instance, the design of [START_REF] Fisher | Heterogenous Demand for Public Goods: Behavior in the Voluntary Contributions Mechanism[END_REF] combines uncertainty about the distribution of MPCR given to other players with certainty about one's own MPCR. In [START_REF] Fischbacher | Heterogeneous Reactions to Heterogeneity in Returns from Public Goods[END_REF], in turn, the distribution of MPCRs 1 Note that the MPCR has to be lower than the return from consumption (i.e. from keeping the unit for oneself) to generate the social dilemma. The details of our experimental VCM game can be found in Section 3. in a group is common knowledge, while one's own MPCR remains uncertain. The contribution levels are found to be neutral to the first type of partial uncertainty, and diminished slightly due to the second type of partial uncertainty.

In this paper, we complement this line of literature by investigating the effect of uncertainty in a novel way and combine both types of partial uncertainty in what we call a complete uncertainty environment. In this setting, all group members either have a predetermined MPCR that is known ex ante (Certainty treatment), or face a uniformly distributed and mean-preserving MPCR that is drawn ex post and iid across rounds and individuals (Uncertainty treatment).

The main results are the following. First, we confirm the classic finding that contributions tend to diminish over time in the Certainty treatment. Second, we show that uncertain returns to cooperation do not negatively affect provision to the public good, and that our subjects well comprehend the random nature of MPCR.

Related literature

A seminal study by [START_REF] Fisher | Heterogenous Demand for Public Goods: Behavior in the Voluntary Contributions Mechanism[END_REF] is the first attempt to introduce uncertainty about MPCR in what we call a standard VCM game where the decision environment is repeated over multiple rounds with multiple players matched in a partner setting and facing a quasi-continuous action set.

Each player knows his MPCR which may equal either 0.30 and 0.75. In addition, in the homogenous treatments all group members have the same MPCR (which is common knowledge), whereas in the heterogeneous treatments MPCRs involve uncertainty: they might vary within a group and their exact distribution is unknown to players. Fisher et al. find that while players react to the changes in their own MPCR, their contributing behavior is neutral to the presence of uncertainty about others' MPCRs. In a recent study, [START_REF] Fischbacher | Heterogeneous Reactions to Heterogeneity in Returns from Public Goods[END_REF] swap the sources of Documents de travail du Centre d'Economie de la Sorbonne -2017. [START_REF] Marwell | Experiments on the Provision of Public Goods. I. Resources, Interest, Group Size, and the Free-Rider Problem[END_REF] uncertainty and show that uncertainty about personal MPCR slightly (yet significantly) decreases contributions when MPCR is heterogeneous within a group and has a predetermined distribution that is common knowledge.

In other related experiments, [START_REF] Dickinson | The voluntary contributions mechanism with uncertain group payoffs[END_REF] and [START_REF] Gangadharan | Experimental analysis of risk and uncertainty in provisioning private and public goods[END_REF] use the standard VCM framework to study uncertainty about receiving the return from the public good.

They find that this type of uncertain environment also decreases contributions to the public good.

In addition, [START_REF] Stoddard | Probabilistic Production of a Public Good[END_REF] suggests that this effect may be sensitive to the way uncertainty comes about: moving from certain to uncertain benefits from public goods is found to decrease contributions.

Finally, our endeavor also echoes [START_REF] Levati | Voluntary Contributions with Imperfect Information: An experimental Study[END_REF]. Their control treatment consists of a quasicontinuous-action-space two-person prisoner's dilemma game with a 0.75 MPCR. In their "imperfect information" treatment, it is common knowledge that the MPCR is identical for both players and may equal either 0.4 or 1.1 (with the same probability). Hence, the expected marginal benefits from cooperation are the same as in the control treatment which makes this game a social dilemma in expectation. However, none of the possible scenarios is a social dilemma per se: the MPCR is either as low so as to make any cooperation unbeneficial, or as high so as to make contribution to the public good a dominant strategy even for selfish players. Herein, we refine the framework of [START_REF] Levati | Voluntary Contributions with Imperfect Information: An experimental Study[END_REF] by maintaining the social dilemma environment under all information structures, and extend it to the standard VCM framework applied in the previous studies reviewed above.

Our method

Documents de travail du Centre d'Economie de la Sorbonne -2017. [START_REF] Marwell | Experiments on the Provision of Public Goods. I. Resources, Interest, Group Size, and the Free-Rider Problem[END_REF] We study the effect of uncertainty on the return to cooperation in a conventional voluntary contribution game played for 10 rounds. Each participant is first randomly assigned to a group of four players that remains unchanged throughout the experiment. At the beginning of each round, every player holds an endowment of E=10 tokens and decides on his personal contribution (ci) to the public good. The individual return per round is defined as πi = α1(Eci) + α2C where C is the total group contribution, α1 is the marginal benefit from keeping a token to oneself, while α2 (with α2<α1) is the MPCR from a contributed token.

Standard predictions for finitely repeated VCM are straightforward. Since the personal return to each unit of wealth contributed to the public good is lower than the benefit from keeping it to oneself, the Nash equilibrium for all individuals is to contribute nothing. However, individual interest stands at odds with social welfare: Pareto-improvement may be achieved if all players contribute their entire endowment. This prediction also holds for a repeated game.

Within this standard framework, we implement two different treatments. In the Certainty treatment, we have α1 = 0.1 and α2 = 0.0525. In the Uncertainty treatment, players must contribute before learning about their MPCR. It is common knowledge that each player's MPCR in a given round of the game is an independent draw from a uniform distribution U[0.03, 0.075]. This calibration guarantees that the latter treatment is mean-preserving with respect to the previous one. 2 The experiment was carried out at the Veconlab at the University of Virginia with students who signed-up to be part of the participant pool. The experiment was programmed and conducted with the experiment software z-Tree [START_REF] Fischbacher | z-Tree: Zurich Toolbox for Ready-made Economic Experiments[END_REF]. We ran 8 sessions (4 per treatment) involving 2

The empirical average MPCR observed in our experiment equals 0.0529 --fairly close to the expected value of 0.0525. Documents de travail du Centre d'Economie de la Sorbonne -2017.04 a total of 96 participants. Each session consisted of 12 participants divided into three fixed group of four players. Each session proceeded as follows. Paper instruction sheets were distributed and read aloud. 3 It was common knowledge that the experiment consisted of 10 rounds. The MPCR was presented in terms of the token-USD exchange rate. Each token kept to oneself was worth $0.10 in both treatments. Each token contributed to the public good in the Certainty treatment brought $0.0525 to every group member. In the Uncertainty treatment, the value of a contributed token could vary uniformly between $0.03 and $0.075. Next, subjects were asked to answer practice questions to ensure full comprehension of these rules. Finally, the first round of the experimental game started. At the end of each round, each participant learned about his/her personal realization of the MPCR (but not other players' realizations), the total group contribution, as well as his/her payoff (in tokens) in that round. At the end of the experiment, individual gains were totaled over 10 rounds. Payments in USD were made in cash to each participant. Sessions lasted about one hour with an average payoff of $20.40, including a show-up fee of $6.00.

Results

Our empirical results are as follows. First, we confirm the classic behavioral phenomenon observed in the vast majority of experiments using a repeated VCM game: a systematic decay of contributions over time in the Certainty treatment. More importantly, we also show that this behavioral pattern also holds (with a similar magnitude) in our Uncertainty treatment. Second, we investigate the effect of experienced randomness on behavior in an uncertain world. We robustly reject the hypothesis that past MPCRs affect players' contributions: neither the most recent realization of random MPCR, nor the accumulated individual experience with its past realizations can predict players' behavior. This suggests that our subjects have a rational perception of randomness in the experiment. Together, these two findings point to the conclusion that players' behavior is neutral to the presence of uncertainty about MPCR. Our results are consistent with what could be expected from a pool of risk neutral subjects in the sense that uncertainty about MPCR has no impact on behavior. This stands in line with an observation by [START_REF] Harrison | Risk Aversion in the Laboratory[END_REF] that the degree of risk aversion among experimental subjects is usually modest.

Result 1. Aggregate mean contributions in both treatments are not statistically different. Furthermore, both treatments reveal similar aggregate dynamics: a significant decay of contributions over time. 1 suggest that the difference between the mean contributions in both treatments is not statistically significant (H0: b1=0, p=0.770). 4Furthermore, Figure 1 suggests that the average contribution in both treatments decreases over time. This observation is confirmed by Model 2 reported in Table 1. The null hypothesis that Round has no effect on the average contribution is rejected in both the Certainty treatment (H0: b2=0, p<0.001) and the Uncertainty treatment (H0: b2+b3=0, p=0.034). Furthermore, the difference between the effects of the Round variable in both treatments is not significantly different from zero (H0: b3=0, p=0.173).

Support. The estimates of Model 1 reported in Table

INSERT TABLE 1 HERE INSERT FIGURE 1 HERE

Result 2. Individual contributions do not depend on players' past experience with stochastic MPCRs. 2 uses the data from our Uncertainty treatment to investigate the relationship between individual contributions and the realizations of MPCRs experienced in the past. The latter is measured in two distinct ways: as MPCR observed in the previous round and the cumulative average of all MPCRs observed since the first round. 5 They both suggest that there is no significant effect of the historical information about MPCR on present behavior: the p-values from the tests of nullity of coefficients b1 and b2 range from 0.718 to 0.953 across the three models, and their joint nullity in Model 3 cannot be rejected (p=0.930).

Support. Models presented in Table

INSERT TABLE 2 HERE

Conclusion

Understanding the effect of uncertainty on the provision of public goods is an important empirical challenge in public economics. This paper uses experimental evidence to determine the effect of complete uncertainty about marginal returns to public goods on contributions in a standard setting of a finitely repeated voluntary contribution mechanism. We report no significant differences in individuals' contributions between a treatment where this rate of return is fixed and commonly known, and a treatment where it is determined individually and randomly in every round using a mean-preserving uniform distribution, and becomes known ex post.

5

As suggested by a referee, the past experience with random MCPRs can be also related to gains of losses relative to the theoretical expected value. We have also estimated an alternative specification of the model in which the independent variables are: a dummy set to 1 if the MPCR in t-1 is below the expected value of 0.0525, and an accumulated proportion of such realizations between rounds 1 and t-1. This new model fully confirms our finding that behavior is neutral to the experience with random MPCRs. Documents de travail du Centre d'Economie de la Sorbonne -2017.04

Figure 1 .

 1 Figure 1. The evolution of aggregate mean contributions across treatments

  du Centre d'Economie de la Sorbonne -2017.04

Table 1 .

 1 Contributions across treatments and rounds: OLS regression analysisNote. OLS estimates in both models are based on observations coming from 10 rounds of both treatments. Uncertainty is a dummy variable set to 1 for Uncertainty treatment (and 0 otherwise), Round represents the consecutive rounds of the game. Standard errors are clustered at the group level (12 groups per treatment, 24 in total) and corrected using a leave-one-out jackknife procedure; their estimates are provided in parentheses underneath the coefficient estimates.

		Model 1	Model 2
	Constant (b0)	3.860***	5.568***
		(0.667)	(0.912)
	Uncertainty (b1)	0.279	-0.493
		(0.945)	(1.184)
	Round (b2)		-0.310***
			(0.066)
	Round X Uncertainty (b3)		0.140
			(0.100)
	N	960	960
	R 2	0.002	0.043
	Prob>F	0.770	<0.001

Documents de travail du Centre d'Economie de la Sorbonne -2017.04

Documents de travail du Centre d'Economie de la Sorbonne -2017.04

Instructions for both treatments are provided in the online supplementary material. Documents de travail du Centre d'Economie de la Sorbonne -2017.04

Left-censored tobit estimation of the models from Tables 1 and 2 yields similar results as our OLS estimation.Documents de travail du Centre d'Economie de la Sorbonne -2017.04

Acknowledgments

We would like to thank Charles A. Holt, Nicolas Jacquemet and Jean-Marc Tallon for their help and suggestions. We also thank two anonymous referees for their useful comments and suggestions. Samuel Brott thanks the financial support of the Small Research and Travel Grant and the Marshall Jevons Fund of the University of Virginia. Part of this research was performed within the framework of the LABEX CORTEX (ANR-11-LABX-0042) of Université de Lyon, within the program Investissements d'Avenir (ANR-11-IDEX-007) operated by the French National Research Agency (ANR).

Accumulated evidence from different uncertainty settings suggests that cooperation in public goods games is sensitive to the source of uncertainty. A null result from [START_REF] Fisher | Heterogenous Demand for Public Goods: Behavior in the Voluntary Contributions Mechanism[END_REF] arises in an environment with uncertainty about the distribution of MPCR amongst others, but without uncertainty about one's own MPCR.


Kamal Goudjil

Computer Visualization of the Riemann Zeta Function

Keywords: Riemann Zeta Function, Number Theory, 3D polar contour maps

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The Riemann zeta function ζ(s) has been explored for over a century and it is still investigated and continues to be the subject of numerous studies. The Riemann zeta function was constructed by Bernhard Riemann and published in his 1859 famous paper entitled "On the Number of Primes Less Than a Given Magnitude." [START_REF] Riemann | Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse[END_REF] The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. The notation s, σ, and t has been adopted in the study of the zeta function following Riemann's notation [START_REF] Edwards | Riemann's Zeta Function[END_REF] .

𝜁(𝑠) = � 1 𝑛 𝑠 ∞ 𝑛=1
where s = σ + it, σ and t ϵR However, it is Leonard Euler who first introduced and studied the zeta function in 1740 without using the complex variable which was not known at that time. Euler studied the zeta function as a function of a real variable (i.e., with "σ only"). Riemann extended Euler's definition of the zeta function to the complex space by introducing the complex variable s = σ + it. Euler further provided a relationship between the zeta function and the prime numbers. An elegant derivation of this relationship known as the Euler Product is provided by William Dunham in "Euler, The Master of Us All." [START_REF] Dunham | Euler, The Master of Us All[END_REF]

𝜁(𝑠)

= � �1 - 1 𝑝 𝑠 � -1 𝑝 = � � 1 1 -𝑝 -𝑠 � 𝑝
Although this relationship was proven by Euler in the case where s is a real variable, this relationship holds true in the complex space, where s is a complex variable.

The Riemann zeta function has a number of zeros. There are zeros located at σ = -2, -4, -6, …and t = 0, and are known as the trivial zeros. The Riemann zeta function also has the so-called non-trivial zeros that are located within the critical strip 0≤σ≤1. The Riemann hypothesis states that all non-trivial zeros of the zeta function are located along the line σ=1/2. A number of zeros have been computed and found to lie on the line σ=1/2. Andrew Odlyzko provides an extensive list of non-trivial zeros of the zeta function [START_REF] Odlyzko | Tables of zeros of the Riemann zeta function[END_REF] . For example, the first three zeros are located at, approximately, t=14.13, 21.02, 25.01. However, at the present time, no proof is available to show that all non-trivial zeros lie on the line σ=1/2. Proving that all non-trivial zeros lie on the line σ=1/2 is part of Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems. It is also one of the Clay Mathematics Institute's Millennium Prize Problems. This paper is not intended to develop any new mathematical proof but simply provide various graphical visualizations of the zeta function to show its various facets depending on the selected real variables or dimensions. The zeta function being a complex function of a complex variable is necessarily a function in a four-dimensional space. However, since we cannot visualize a four-dimensional space, the zeta function can be "projected" in the three-dimensional space. In the following paragraphs, instead of the traditional notation s = σ + it, the notation s = 𝑅𝑅𝑅 + 𝑖 • 𝐼𝐼𝑅 is used. The zeta function 𝜁(𝑠) can then be expressed as follows:

𝑅𝑅𝑅 + 𝑖 • 𝐼𝐼𝑅 = 𝜁(𝑅𝑅𝑅 + 𝑖 • 𝐼𝐼𝑅)
In this case, the 4 variables or dimensions are ReS, ImS, ReZ and ImZ, where ReS and ImS are, respectively, the real part and the imaginary part of the variable s, and ReZ and ImZ are, respectively, the real part and the imaginary part of the zeta function.

Fig. 1 shows a contour map of the real and imaginary parts of the zeta function as a function of the real and imaginary parts of the s variable. The contour map is plotted for ReS in the range [-13, 0.9] and for ImS in the range [-5, 27]. The real part ReZ at value 0 is shown as contours with solid thicker lines. The imaginary part ImZ at value 0 is shown as contours with solid thinner lines. The position of the three first non-trivial zeros can be seen at the intersection of a thicker line corresponding to ReZ = 0 and a thinner line corresponding to ImZ = 0, at ReS = 0.5 and ImS = 14.13, 21.02, and 25.01. This contour plot is similar to a contour plot provided by J. Arias-de-Reyna in "X-Ray of Riemann's Zeta-Function" [START_REF] Arias-De-Reyna | X-Ray of Riemann's Zeta-Function[END_REF] . The non-trivial zeros can be seen at ReZ = 0 and ImZ = 0, corresponding to the cyan areas for ReZ and green areas for ImZ. A superposition of these two maps provides the location of the three first zeros at ImS equal to about 14.13, 21.02, and 25.01.

A more precise location of the zeros can be seen when performing a cross-cut through the above contour maps at exactly ReS = 0.5. An interesting three-dimensional graphical representation of the zeta function is a 3D representation in which the x and y axes are, respectively, the real part ReZ and the imaginary part ImZ of the zeta function, and the z axis is the real part ReS or the imaginary part ImS of the complex variable s. Fig. 5 shows such a 3D plot of the zeta function where the x-axis corresponds to ReZ, the y-axis corresponds to ImZ, and the z-axis corresponds to either ReS or ImS. This 3D plot is obtained for a fixed value ReS = 0.5 (in blue) and for ImS in the range [0, 27] (in red). The position of the zeros of the zeta function is at ReZ = 0 and ImZ = 0. The zeros are located at the intersection point of the blue curve with itself, and at ImS=14.13, ImS = 21.02, and ImS = 25.01, for the red curve. For example, the blue line intersects three times with itself at ReZ=0 and ImZ=0, which correspond to the three non-trivial zeros in the interval ImS = [0, 27]. Similar plots representing the relationship between ReZ, ImZ and ReS and ImS can be found in various publications. However, it is worthwhile to provide a representation of the magnitude and phase of the zeta function 𝜁(𝑠) as a function of ReS and ImS, but most importantly as a function of the magnitude of the complex variable s and the phase of the complex variable s. Indeed, the zeta function 𝜁(𝑠) can also be expressed in the polar form as follows:

|𝑧| • 𝑅 𝑖•𝜑 𝑧 = 𝜁�|𝑠| • 𝑅 𝑖•𝜑 𝑠 �
In this case, the 4 variables or dimensions are |𝑠|, 𝜑 𝑠 , |𝑧|, 𝜑 𝑧 , where |𝑠| is the magnitude of the complex variable s, 𝜑 𝑠 is the phase of the complex variable s, |𝑧| is the magnitude of the zeta function 𝜁, and 𝜑 𝑧 is the phase of the of the zeta function 𝜁. These 4 variables are expressed as follows:

|𝑠| = �𝑅𝑅𝑅 2 + 𝐼𝐼𝑅 2 𝜑 𝑠 = arctan � 𝐼𝐼𝑅 𝑅𝑅𝑅 � = atan2(𝐼𝐼𝑅, 𝑅𝑅𝑅) |𝑧| = �𝑅𝑅𝑅 2 + 𝐼𝐼𝑅 2 𝜑 𝑧 = arctan � 𝐼𝐼𝑅 𝑅𝑅𝑅 � = atan2(𝐼𝐼𝑅, 𝑅𝑅𝑅)
In the following paragraphs, the following notation will be used instead of the above notation.

|𝑠|, 𝜑 𝑠 , |𝑧|, 𝜑 𝑧  MagS, PhS, MagZ, PhZ

It may be worthwhile to show the behavior of the phase of the zeta function and/or the phase of variable s. In computing the phase PhS or PhZ, "atan2" is used instead or "atan" to take into account the sign of ImZ, ReZ, ImS and ReS. The phase is computed in the interval [-π, π] for both the complex variable s and the zeta function. Phase plots can provide novel ideas [START_REF] Wegert | Phase Plots of Complex Functions: a Journey in Illustration[END_REF] . Fig. 7 below is a contour map of the phase PhZ of the zeta function versus the real part ReS of the complex variable s (the x-axis) and the imaginary part ImS of the complex variable s (the y-axis). The third dimension which is represented by color corresponds to the value of the phase angle PhZ. 7, for certain values of ReS, ImS, the phase PhZ jumps from negative to positive. In fact, the phase shifts along the "ribs" of the zeta function (the term "ribs" being borrowed from reference [START_REF] Arias-De-Reyna | X-Ray of Riemann's Zeta-Function[END_REF] ). Along the critical line, every time the zeta function changes sign, a discontinuous jump by π in the phase is introduced [START_REF] Bhaduri | Phase of the Riemann Zeta Function and the Inverted Harmonic Oscillator[END_REF] . As can be seen in Fig. 7, the phase contour is symmetric relative to the zero axis ReS. The negative values of the phase of the zeta function on negative ImS side are the "mirror image" of the positive values of the phase of the zeta function on positive ImS side. 9, similar to Fig. 7, for certain values of ReS, ImS, the phase PhZ jumps from negative values to positive values (blue to red in the contour map). Fig. 9 can be seen as a projection of a sphere wherein the radial distance (radius) of the sphere represents the magnitude of s (MagS), the azimuth angle represents the phase angle of s (PhS), and the inclination or elevation angle represents the phase of the zeta function (PhZ). Fig. 10 above is also a polar contour map of the phase PhZ of the zeta function as a function of the magnitude MagS of the complex variable s and the phase PhS of the complex variable s. However, in this plot, the phase PhS and the magnitude MagS are computed using values of ReS in the interval [0.0, 0.9] within the critical strip, and ImS in the interval [-26, 26]. The zeros can be seen at PhS around 90° with a magnitude of about 14, 21and 25, and with a phase of the zeta function (PhZ) shifting from negative values (blue) to positive values (red). 12 above is a polar contour map of the phase of the complex variable s, where the radius represents the magnitude of the zeta function, i.e, MagZ, and the azimuthal angle represents the phase angle PhZ of the zeta function. The various levels of color represent the value of the phase angle PhS of the complex variable s. This plot is similar to the plot in Fig. 11 but obtained when using values of ReS in the narrower interval [0, 1] (with a step of 0.01) and ImS in the narrower interval [12, 28] (with a step of 0.01). For a phase of s (PhS) around 87° to 88° (yellowgreen color), a forward shape is generated in the zeta function with a phase angle (PhZ) -45° to 45°. A lobe can be seen for MagZ between 0 and 1.5, a semi-ring can be seen for MagZ of about 2.5, and a further outer semi-ring can be seen for MagZ of about 3.5. It appears that for the smallest value for phase PhS (in blue) the phase PhZ of the zeta function alternates from positive to negative at consecutive zeros. For the first zero (at ImS = 14.13), the phase PhZ is located between 0° and 45° in the first quadrant (Fig. 13a). For the second zero (at ImS = 21.02), the phase PhZ is located between 0° and -45° (315°) in the fourth quadrant. Thereafter, the phase alternates from positive to negative values at consecutive zeros.

In Figs. 14a-d, the ImS range is further narrowed around the ImS value corresponding to the zeros of the zeta function. The ReS interval is not modified and is maintained as [0, 1]. These plots show a reversal of direction in the obtained "snail" shape at consecutive zeros, from clockwise to counter-clockwise and from counter-clockwise to clockwise, etc. Figs. 15a and 15b are two plots obtained for ReS in the interval [0,1] for a zero at two ImS intervals centered around two consecutive larger ImS values that generate non-trivial zeros, respectively, 10000.92 and 10002.28. Similarly, the "snail-like" form reverses direction for consecutive zeros. Hence, shape orientation in the 3D polar contour map projections can be used to provide a location of consecutive zeros. The next logical step would then be to look into the variation of the phase of the zeta function as a function of the phase of the complex variable s, around some non-trivial zeros. Figs. 16a-f show plots of the phase of the zeta function the phase of the variable s around consecutive zeros. The phase of the zeta function reverses its shifting direction for consecutive zeros. This feature can be used to locate consecutive zeros of the zeta function.

Fig. 1 :

 1 Fig. 1: Contour map of real part ReZ and imaginary part ImZ of the zeta function versus ReS and ImS. Thicker lines correspond to contour lines at ReZ=0, and thinner lines correspond to contour lines at ImZ = 0.

Fig. 2a and

  Fig. 2a and 2b show contour maps of the real and imaginary parts of the zeta function as a function of the real and imaginary parts of the s variable. The contour map is plotted around the line ½, for ReS in the range [0.4, 0.6], and for ImS in the range [0, 27]. The values of the real part ReZ are shown in various colors with the color cyan corresponding to the zero values and the red color corresponding to the higher value 2.5, as shown on the color-coded strip. The values of the imaginary part ImZ are shown as various colors with the color blue/purple corresponding to the negative value -1.6, the zero values corresponding to the green color, and the red color corresponding to the higher value 1.6, as shown on the color-coded strip.

Fig. 2a :

 2a Fig. 2a: Contour map of real part ReZ of the Fig. 2b: Contour map of imaginary part ImZ of the zeta function as a function of ReS and ImS. zeta function as a function of ReS and ImS.

Fig. 3

 3 is a plot of ReS and ImZ versus ImS at ReS = 0.5. ReZ is plotted as a blue line and ImZ is plotted as a red line. The intersection of these two lines at ReZ (blue) equal to 0 and ImZ (red) equal to 0 indicates the position of first three non-trivial zeros which are highlighted by green triangles in the interval ImS=[0,27], at ImS = 14.13, ImS = 21.02, and ImS = 25.01, respectively.

Fig. 3 :

 3 Fig. 3: Plot of ReZ and ImZ versus ImS in the the interval [0,27]. The position of the first three non-trivial zeros is indicated by 3 green triangles.The position of the first three non-trivial zero can also be seen when plotting the magnitude of the zeta function, i.e., MagZ = sqrt(ReZ 2 +ImZ 2 ) versus ImS in the interval [0, 27], as shown in Fig.4. The zeros correspond to MagZ = 0.

Fig. 4 :

 4 Fig. 4: Plot of Magnitude of the zeta function as a function of ImS.

Fig. 5 :

 5 Fig. 5: 3D plot of the zeta function where the x-axis corresponds to ReZ, the y-axis corresponds to ImZ and the z-axis corresponds to ReS (blue) or ImS (red). The above 3D representation can be projected onto the 2D plane to plot the argand diagram of the zeta function where the real part of the zeta function ReZ is plotted along the x-axis and the imaginary part of the zeta function ImZ is plotted along the y-axis, for ReS = 0.5 and ImS in the interval [0, 27], as shown in Fig. 6. The argand diagram shows closed loops of the zeta function where the intersection points, at ReZ = 0 and ImZ = 0, correspond to the zeros of the zeta function.

Fig. 6 :

 6 Fig. 6: Argand diagram of the zeta function .

  The color-coded strip shows the corresponding values of the phase angle PhZ in the interval [-π,π] or [-180°, 180°]. The phase contour map is plotted for values of ReS in the interval [-27, 27] and values of ImS in the interval [-27, 27]. As shown in Fig.

Fig. 7 :

 7 Fig. 7: Contour map of the phase PhZ of the zeta function versus ReS and ImS.

Fig. 8 :

 8 Fig. 8: Plot of phase PhZ of the zeta function versus the phase PhS of complex variable s.

Fig. 8

 8 Fig. 8 shows the variation of phase of the zeta function as a function of the phase of the complex variable s, for ReS = 0.5 and ImS in the interval [12, 26]. At the zeros, the phase of the zeta function shifts from a negative value to a positive value by π, as shown by the vertical lines in Fig. 8.

Fig. 9

 9 Fig.9below is a polar contour map of the phase PhZ of the zeta function as a function of the magnitude MagS of the complex variable s and the phase PhS of the complex variable s. The

Fig. 9 :

 9 Fig. 9: Polar contour map of the phase PhZ of the zeta function as a function of the magnitude MagS of the complex variable s and the phase PhS of the complex variable s.

Fig. 10 :

 10 Fig. 10: Polar contour map of the phase PhZ of the zeta function as a function of the magnitude MagS of the complex variable s and the phase PhS of the complex variable s.

Fig. 11 :

 11 Fig.11: Polar contour map of the phase of the complex variable s, the radius representing the magnitude of the zeta function, the azimuthal angle representing the phase angle PhZ of the zeta function, and the color representing the phase angle PhS of the complex variable s.

Fig. 11

 11 Fig. 11 above is a polar contour map of the phase of variable s as a function of the magnitude MagZ of the zeta function and the phase PhZ of the zeta function, where the radius represents the magnitude of the zeta function, i.e, MagZ, and the azimuthal angle represents the phase angle PhZ of the zeta function. The various levels of color represent the value of the phase angle PhS of the complex variable s. The phase angle PhS varies from -90° (blue color) to +90° (orange color). This plot is obtained using values of ReS in the interval [-5, 5] and ImS in the interval [-28, 28]. The zeros are obviously located at the origin where the magnitude of the zeta function is equal to zero.

Fig. 12 :

 12 Fig.12: Polar contour map of the phase of the complex variable s, the radius representing the magnitude of the zeta function, the azimuthal angle representing the phase angle PhZ of the zeta function, and the color representing the phase angle of the complex variable s.

Fig.

  Fig. 12 above is a polar contour map of the phase of the complex variable s, where the radius represents the magnitude of the zeta function, i.e, MagZ, and the azimuthal angle represents the phase angle PhZ of the zeta function. The various levels of color represent the value of the

Fig. 13a :

 13a Fig. 13a: ReS=[0,1] and ImS=[14.0, 14.5] Fig. 13b: ReS=[0,1] and ImS=[20.5, 21.5]

Fig. 14a :

 14a Fig. 14a: ReS=[0,1] and ImS=[14.13, 14.14] Fig. 14b: ReS=[0,1] and ImS=[21.01, 21.03]

Fig. 15a :

 15a Fig. 15a: ReS=[0,1] and ImS=[10000.91, 10000.93] Fig. 15b: ReS=[0,1] and ImS=[10002.27, 10002.29]

Fig. 16a :

 16a Fig. 16a: ReS=[0,1] and ImS=[14.13, 14.14] Fig. 16b: ReS=[0,1] and ImS=[21.01, 21.03]

DISCLAIMER: THE CONTENT OF THIS PAPER IS AN INDEPENDENT WORK BY THE AUTHOR AND IS NOT "A WORK PERFORMED FOR HIRE" NOR FUNDED BY ANY ENTITY. THE CONTENT OF THE PRESENT PAPER DOES NOT REPRESENT IN ANY WAY, SHAPE, OR FORM THE VIEWS OF FORMER OR PRESENT EMPLOYERS OF THE AUTHOR.

In conclusion, there are various ways to "slice" or project the four-dimensional zeta function. For example, graphical visualization on a polar contour map provides another perspective of analyzing the zeta function. It may be worthwhile to explore this type of polar projection to further analyze the zeta function at various real and imaginary values of the complex variable s (for example, at higher values). The above described projections can also be applied to any complex function in the four-dimensional space. All the above plots were obtained using a Python code written by the author.


Olivier Labeau

Philippe Tamarat

Brahim Lounis

Temperature Dependence of Single CdSe/ZnS Quantum Dots Luminescence Lifetime

We study the temperature dependence of the luminescence decay of single CdSe/ZnS quantum dots between 2 and 140 K. For the first time, we observe a biexponential decay which was completely hidden in ensemble measurements. We find that the long time component strongly depends on temperature. This demonstrates that the band edge luminescence arises from two thermally mixed fine structure states, the dark ground state and the lowest bright one. To interpret our results, we derive the analytical expressions for the decay using a three level model. Fitting the experimental data leads directly to the lifetime of the states as well as their energy splitting.

Over the past decade, interest in low-dimensional, mesoscopic systems, such as quantum dots (QDs), has grown dramatically. These semiconductor QDs bridge the gap between single molecules and the bulk solid state, thereby offering the opportunity to study the evolution of bulk properties. Additionally, their size-dependent optical properties make them ideal candidates for tunable absorbers and emitters in applications such as nanoscale electronics [1][2][3] , laser technology 4 and biological fluorescent labeling 5,6 . Chemically synthesized CdSe QDs are promising for many of these applications due to their bright luminescence, which is size tunable across the visible spectrum 7,8 .

The electronic properties of nanocrystals with dimensions smaller than the bulk exciton Bohr radius are dominated by quantum confinement effects. The band edge exciton (1S 3/2 1S e ), which is eightfold degenerate in spherically symmetric dots 9,10 , is split into five levels by the crystal shape asymmetry, the intrinsic crystal field and the enhanced electron-hole exchange interaction. The exciton ground state is located at few meV below the lowest-energy optically active (J = ± 1 L ) exciton state, and has a total spin projection on the crystal hexagonal axis J = ± 2. While optical transitions to and from this state are normally forbidden within the electric dipole approximation, radiative recombination from this state can occur with the emission or absorption of phonons. At low temperatures, these processes are much slower than those of direct optical recombination. This leads to the formation of a long-lived exciton state.

Low temperature fluorescence line narrowing and photoluminescence spectra, measured on ensembles of CdSe QDs, support the picture of dark exciton formation via excitation of higher energy fine structure states followed by rapid thermalization to the exciton ground state 9 . The photoluminescence comes primarily from the recombination of the dark state. It is red shifted with respect to a resonant excitation frequency and has long radiative lifetimes (~1 µs at 10 K) relative to the bulk exciton recombination time (~1ns). Mixing of the bright and dark excitons in an external magnetic field allows a direct optical recombination of the dark exciton and shortens the luminescence decay time of CdSe QDs 10,11 . The radiative pathway of this nominally forbidden transition is not fully understood.

Time correlated single photon counting technique (TCSPC) performed on ensembles

of QDs always led to multiexponential luminescence decays [9][10][11][12] because of the sample inhomogeneities (e.g. in size, surface and shape). This prevented any quantitative temperature dependence investigation and hid any multiple decay behavior of the system. Since its inception, single nano-objects methods have improved our understanding of fluorescent systems by elucidating the photophysics of isolated species in the absence of ensemble averaging 13 . The study of QDs has already benefited from these methods, including the discovery of spontaneous spectral shifts 14 , ultranarrow transitions 15 , and fluorescence intermittence 16 .

At room temperature, lifetimes of the excitonic ground manifold were first deduced from the luminescence intensity correlation function of single CdSe QDs in photon antibunching measurements 17 . The lifetimes ranged between 12 and 28 ns with a mean value around 18 ns.

Recently, TCSPC experiments performed on single QDs showed a multiexponential luminescence decay which was attributed to changes of the single QD lifetime during the accumulation time 18 . Correlation between the blinking effect and a fluctuating dynamic process was hypothesized.

In this letter, we study for the first time the temperature dependence of single QD luminescence lifetimes in the range 2-140 K. This dependence provides a new insight into the exciton recombination mechanisms.

Samples of ZnS-coated CdSe QDs (1.9 nm radius, 565 nm peak emission) are prepared by spin coating clean glass coverslips with a 10 -9 M solution of QDs in toluene, followed by 2 % by weight PMMA/toluene solution. The excitation beam is the 514 nm line of a mode-locked Ar + laser (120 ps pulse width). The repetition rate of the laser beam is reduced with a pulse-picker in order to study the long decays. Single QDs are imaged with a home-built laser scanning confocal microscope. It is based on a pair of galvano-driven mirrors, a telecentric lens system, and a 0.95 NA objective which is inserted in a helium cryostat together with the sample. The emitted photons are filtered from the scattered excitation light by a band pass filter (70 nm FWHM) and sent to a single-photon-counting avalanche photodiode. The luminescence decay is measured using classical TCSPC with a time resolution limited to few hundreds of pico-seconds. single QDs decreases dramatically below this temperature (more than a factor of ten between 140K and 5K). These observations suggest that the emission occurs from two thermally populated states, a weakly emitting long-lived ground state, and a strongly emitting upper one.

The temperature dependence of the decay curves is modeled on the basis of a three level system (see figure 1): a zero exciton ground level G , and two excitonic states

( ) 1
0 ) 0 ( = + A F p p
). The luminescence signal ( )

t S is proportional to ( ) F F F A A A p p Γ + Γ η η
, where A η and F η are the radiative yields of states A and F respectively. Considering the approximation

F A Γ >> Γ >> 0 γ , we find: ( ) ( )       -       + - Γ +       - + Γ + Γ = S B B A B F F B A t N N p N N S τ η τ η η exp 2 1 0 t exp 2 1 t A A L A , (1)
where S τ and L τ are respectively a short and a long characteristic time given by: ( )

B S N 2 1 0 1 + = - γ τ and       ∆       Γ - Γ - Γ + Γ = - T k E B F A F A L 2 tanh 2 2 1 τ . ( 2
)
In the high temperatures regime, where

E T k B ∆ >> ( 1 >> B N
), the amplitude of the fast component vanishes for ( ) ( )

2 1 0 0 = = F A p p
. This means that the two lowest excitonic levels are equally populated from the high energy continuum states which are initially prepared by the laser pulse 19 . In this case, the signal shows a single exponential decay and writes:

( )

      Γ - Γ = > 2 exp 2 t S A T t A A η
. The luminescence stems from the bright exciton state and exhibits a decay time which is twice the lifetime of the state A . However, in the low temperatures regime (

E T k B ∆ << and 1 << B N
), the signal shows a biexponential decay and is given by

( ) ( ) ( ) t t S A A F T 0 F F exp 2 - exp t γ η η - Γ + Γ Γ = < .
The ratio between the time integrated signals of the fast and slow components is then 9,20 , this ratio is less than a few percents, in agreement with our experimental observations. The stability of our experimental setup allows us to follow the behavior of a single QD on a large temperature range. The inset of figure 3 The expression (2) of the long component decay rate 1 -L τ reproduces satisfactorily the temperature dependence of the average rates calculated from the histograms (see figure 3 (a)).

0 2 γ η η F A A Γ . For typical values A Γ ~ 0.1 ns -1 , 1 F A η η and 0 γ >1 ns -1
The solid curve is plotted for average values of the rates A Γ = 0.13 ns -1 and F Γ = 2 µs -1 and an energy splitting E ∆ = 3.5meV (typical for 1.9 nm QD radius).

When we study smaller size QDs (1.5 nm radius, peak emission 538 nm), we find the same overall luminescence decay behavior. The decay rates are comparable to those of the previous QDs. As expected, we obtain higher activation temperatures (see right curve of figure 3(b))

corresponding to larger splitting energies (3.6 meV < E AF <7 meV for 6 QDs). This is due to the enhancement of the electron-hole exchange interaction when the QDs size decreases 9 . The large dispersion in the splitting energies is attributed to the size and shape distribution of the QDs.

However, most of the measurements performed on QDs with a larger size (5.6 nm radius, peak emission 640 nm) show a multiexponential luminescence decay on a large time scale at low temperatures. For these large QDs, the size is comparable to the bulk exciton Bohr radius. Therefore, the strong confinement regime is not reached and the presented model is no longer valid. Some QDs (~20%) show a short time multiexponential decay, followed by a long time monoexponential one. The left curve of figure 3(b) displays a typical evolution of 1 -L τ with temperature for such QDs. The fit gives a splitting energy of ~ 2meV. These QDs may be small-radius particles of the size distribution edge, or may present deviant shapes.

In summary, we studied for the first time the temperature behavior of the luminescence decay of single QDs. We showed that the luminescence arises from the two thermally mixed lowest states of the band edge exciton fine structure. Contrarily to room temperature measurements 17,18 which only lead to an effective lifetime resulting from the thermalization of all the fine structure levels of the exciton state, the temperature dependence of the luminescence decay gives access to the bright and the dark exciton lifetimes, as well as their energy splitting. In contradiction with what was reported for self assembled CdSe/ZnSe QDs 21 , our results show that the bright exciton radiative decay rate A Γ is much smaller than the spin flip

F A → rate 0 γ .
For the dark state, a dipole radiative transition is forbidden and the emission can occur through interaction with phonons. A long lifetime 1 -Γ F of the dark state is consistent with a longitudinal optical (LO) phonon assisted recombination 19,20 . However, the zero LO phonon line observed in the photoluminescence suggests an alternate recombination pathway through coupling to acoustic phonons 22 . Other recombination mechanisms may involve spin flip transitions induced by paramagnetic defects in the lattice or a high order interaction with phonons. Surface states may also have a strong influence on the relaxation mechanisms of this weakly emitting dark state. This would explain the large relative distribution of L τ observed at very low temperatures.

Further experiments will be the investigation of the effect of an applied magnetic field on the luminescence decay, as well as the measurement of the optical coherence lifetime of the dark exciton state. the "red" QD, radius 5.6 nm; squares for the "yellow" QD, radius 1.9 nm; triangles for the "green" QD, radius 1.5 nm). The experimental points are well fitted with the expression of 1 -L τ deduced from the thermal mixing model. The fits give Γ F = 5.4 µs -1 , Γ A = 57 µs -1 , E AF = 2.1 meV for the "red" QD; Γ F = 1.4 µs -1 , Γ A = 92 µs -1 , E AF = 3.8 meV for the "yellow" QD; Γ F = 2.0 µs -1 , Γ A = 82 µs -1 , E AF = 5.0 meV for the "green" QD.

Figure 1 (

 1 Figure 1(a) presents the luminescence decay of a bulk solution of QDs recorded at 16K. Because of the inhomogeneous distribution of QDs lifetimes, it shows a multiexponential behavior distributed on a large time scale. In figure 1(b), we present a typical decay curve of a single QD recorded at the same temperature. It clearly shows a biexponential behavior which is completely hidden in the ensemble measurement of figure 1(a). It has a very fast decay occurring within our time resolution scale (a few hundreds of ps) and a much slower one with a lifetime L τ in the 100 ns range. The time integrated signal

F

  F and A representing respectively the dark (J = ± 2) and the bright (J = ± 1 L ) excitons, with recombination rates F Γ and A Γ . The energy splitting AF E between these two states is in the few meV range, depending on the size and shape of the QDs 10 . Emission and absorption of acoustic phonons from a mode whose energy E ∆ matches AF E induces a thermalization of the excitonic states. The F A → and A → rates are respectively ( ) -Einstein phonon number at the temperature T. 0 γ is the zero temperature relaxation rate of the F A → transition.We use rate equations to derive the time evolution of the two populations F p and A p of the excitonic states, after a short pulsed laser excitation at t = 0 which leads to initial

  (b) shows the long time component of the luminescence decay recorded on the same QD at different temperatures (between 2.7 and 28 K). The shortening of the lifetime with temperature is well reproduced by the expression (2) (see middle curve of figure3(b)). The fit leads to A Γ , F Γ and E ∆ , the three relevant parameters involved in the QD luminescence. The energy E ∆ of the acoustic phonon mode responsible for the thermalization matches the energy splitting AF E between the lowest exciton states. We performed this measurement on 15 single QDs and found 1.3 meV < E AF < 3.8 meV.

Figure captions:Figure 1 :

 1 Figure captions:

Figure 2 : 3 K

 23 Figure 2: Histograms of the long component decay time τ L built from about a hundred single QDs at various temperatures (between 2.3 and 140 K). The mean τ L values are 865 ns at 2.3 K, 209 ns at 5.7 K, 85 ns at 16 K, 51 ns at 77 K and 19 ns at 140 K.

Figure 3 : 1 - L τ deduced from the histograms of figure 2 .

 312 Figure 3: (a) Evolution with temperature of the mean decay rate 1 -L τ deduced from the histograms of figure 2. The continuous curve is theoretical rates of equation (2), plotted with

  Figure 1

Figure 3

 3 Figure 3

Acknowledgement:

We thank Michel Orrit for helpful discussions. This research was funded by CNRS (Individual Nano-Object Program), Région Aquitaine and the French Ministry for Education and Research (MENRT).


High resolution resonant photoluminescence excitation of CdSe/ZnS nanocrystals at low temperatures

We present a new technique to perform high resolution resonant photoluminescence excitation of CdSe/ZnS nanocrystals. The method takes advantage of the long photoluminescence decay times (~ 1 µs) observed in this system at liquid helium temperatures. Resonant photoluminescence excitation can be performed using a tunable pulsed excitation and a time-gated detection. Spectral hole burning investigations on an ensemble of CdSe/ZnS nanocrystals lead to homogeneous linewidths of ~100 µeV for the band edge exciton state.

Semiconductor nanocrystals (NCs) featuring atomic-like electronic energy level structures can be used in a wide variety of applications, such as nanoscale electronics 1 and biological labelling 2 . In particular, colloidal CdSe NCs are promising for many of these applications due to their bright luminescence which is tunable across the visible spectrum [START_REF] Klimov | Semiconductor and Metal Nanocrystals: Synthesis, Electronic and Optical Properties[END_REF] .

Photoluminescence from single CdSe NCs exhibit a strong photon antibunching [START_REF] Lounis | [END_REF] , making these particles good candidates for the realization of triggered single photon sources 5 .

For certain applications 6 in quantum information such as linear optics quantum computing 7 , the single photons should display a Fourier-limited spectrum which is determined by the lifetime of the excited state. The case of pure radiative decay of an ideal two-level system leads to perfectly indistinguishable photons. However, dephasing and spectral diffusion, which manifest as fast and slow fluctuations of the transition frequency, introduce distinguishability between photon wavepackets. It is therefore of crucial interest to investigate the homogeneous linewidth, and thus the coherence decay time, of the transition involving the band edge exciton. Spectral hole burning is a powerful method to investigate the homogeneous linewidth of organic molecules in the condensed phase at liquid helium temperatures 8 . Since the molecules emit in a wide spectral range of vibrational lines (~ 10-350 meV) which are redshifted with respect to the zero phonon line (ZPL), the separation of the fluorescence photons from the excitation light can easily be performed using appropriate optical filters. Direct implementation of this technique on CdSe NCs is not possible because of a very small red shift of the luminescence spectrum with respect to the absorption transition, as shown by fluorescence line narrowing experiments 9 . This red shift is only a few meV (depending on the particle size) for the sharp ZPL. The luminescence spectrum of the NCs also displays a longitudinal optical (LO) phonon replica of the ZPL, with a ~25 meV red shift 10 which makes very difficult the spectral filtering of the luminescence from the scattered excitation light.

In this letter, we present a new experimental method to perform high resolution spectroscopy of the CdSe NCs band edge exciton. This method is a spectral hole-burning technique based on "time filtering" of the luminescence photons from the excitation light. It exploits the long photoluminescence decay times (~ 1 µs) observed in this system at liquid helium temperatures.

In order to measure the homogeneous width of the 1S e 1S 3/2 band edge exciton transition, we first irradiate an ensemble of CdSe NCs with a fixed frequency cw beam delivered by a single longitudinal mode dye laser. The spectral hole burnt in the red wing of the lowest absorption band is then probed by recording a photoluminescence excitation (PLE) spectrum using an optimized combination of pulsed excitation and time-gated detection. The pulses are prepared from the same cw laser with two acousto-optical modulators in double pass configuration and the hole burning spectra are recorded by scanning the laser frequency around the burning wavelength. In practice, we prepare rather long pulses of 20 ns duration which offer a good spectral resolution of 35 neV and efficient excitation of the NCs. Timeshifted detection of the emitted photons is performed during the inter-pulse period (see Fig. We studied ZnS coated CdSe NCs (1.9 nm radius, 565 nm peak emission) dispersed in a thin polymer (PMMA) film. The sample is prepared by depositing on a silica plate a small drop of a solution of NCs (10 -4 M) in toluene containing 2% by weight PMMA. It is then placed in a helium flow cryostat at the focal region of a high numerical aperture parabolic mirror, used in a confocal configuration to excite and collect the luminescence from the NCs. Fig. 1.b shows a luminescence excitation spectrum recorded at 3 K after a 30 minutes irradiation at 571.63 nm laser wavelength. The burning intensity was set to 500 W.cm -2 , below the saturation intensity to ensure that the photon absorption rate of the NCs is well below their average luminescence decay rate. The reading intensity was fifty times weaker (peak intensity 10 W.cm -2 ). The spectrum shows a hole at the burning laser frequency with a relative depth of about 10 %. Similar results were also observed at other burning frequencies within the inhomogeneously broadened 1S e 1S 3/2 absorption band. The spectral hole is well fitted by a lorentzian shape with a total width at half maximum of 200 µeV, corresponding to a homogeneous width of 100 µeV (half of the measured value 8 ). Reducing the burning intensity by a factor of ten led to spectral holes with the same width, confirming the absence of power broadening in the presented spectra. The spectral holes disappear several hours after the irradiation, showing that the mechanisms involved in hole burning are due to photoinduced spectral jumps.

In order to explain the observed hole width one has to take into account the fine structure of the band edge exciton which consists of five sublevels. Three of them are optically active (bright). The two others including the ground state are optically passive (dark). Since the spectral hole is burnt in the red wing of the lowest absorption band,, the resonant laser light creates preferentially the excitons in the lowest energy active state L 1 ± , from which fast relaxation to the dark ground state 2 ± occurs at low temperatures. While luminescence from the ground state is normally forbidden within the electric dipole approximation, radiative recombination is made possible through coupling to phonons, spin flip transitions induced by paramagnetic defects, or mixing of bright and dark excitons induced by surface states 12 . The measured homogeneous width should give access to the bright exciton optical coherence lifetime. Dephasing processes due to phonons vanish at liquid helium temperatures, but a contribution of spectral diffusion to this width cannot be excluded 13 . Therefore from our measurement we can conclude that the coherence lifetime is longer than 13 ps, which sets a lower limit for spin relaxation within the band-edge exciton manifold.

We also investigated the effect of the hole burning in the emission spectrum of NCs.

The photo-induced spectral dispersion of the emitting states gives photoluminescence spectra broader than the excitation spectra. Nevertheless, interesting informations can be extracted by comparing luminescence spectra before and after a burning irradiation. Fig. 2 shows the differential emission spectrum (difference between the luminescence spectrum taken after and the one taken before irradiation) of the sample at 3 K after the irradiation described in Fig. 1.

The luminescence is excited at the fixed wavelength 514 nm. Spectral holes in the luminescence spectra appear with a red shift compared to the burning frequency. The observation of a hole which is 6 meV red-shifted with respect to the burning frequency supports the picture that excitation creates the exciton in the

L 1 ±
state and luminescence arises from the "dark" exciton ground state 2 ± . The holes (ii) and (iii) display a progression energy of 25 meV, attributed to the longitudinal optical (LO) phonon mode 10 .

A characteristic feature of the differential emission spectrum is the increase of the luminescence intensity in the blue side of the burning frequency. This anti-hole, accompanied with phonon replica, can be interpreted as follows. Let us assume that the photo-induced spectral jumps responsible for hole burning are spectrally symmetric (blue shifts and red shifts of the NC absorption lines occur with equal probabilities). While the red-shifted NCs can still interact with the burning laser via absorption in the upper fine structure sub-levels ( U 1 ± , U 0 ), the blue-shifted NCs are out of resonance and cannot be re-excited by the burning beam. The latter will therefore lead to the observed increase of the blue shifted luminescence. , followed by fast relaxation to the dark ground state 2 ± at low temperatures. Luminescence stems from this ground state with a lifetime of a few hundred ns.Main figure: Difference between luminescence spectra recorded after and before hole burning, at 3 K. The main spectral hole (i) is red-shifted compared to the burning energy, evidencing burning in the "bright" state and emission from the "dark" state. Red-shifted replicas (ii) and (iii) of the main spectral hole (i) are attributed to a LO phonon mode at 25 meV.

  Labeau, Philippe Tamarat, and Brahim Lounis* Centre de Physique Moléculaire Optique et Hertzienne, CNRS et Université Bordeaux I, 351 Cours de la Libération, 33405 Talence Cedex, France * corresponding author. E-mail: b.lounis@cpmoh.u-bordeaux1.fr

  1.a) by a triggered single photon counting avalanche photodiode synchronized with the excitation pulses. A compromise between a high excitation rate of the NCs and a long detection time window has to be found for an optimum detection rate. Indeed, for experiments performed at 3 K where the average NCs lifetime 11 is 300 ns, the collection of the luminescence is optimum at a repetition rate of 1 MHz. Nearly half of the emitted photons are collected in the 800 ns detection time window.

  Figures

  FIG.2

  photoluminescence excitation of NCs with a high spectral resolution, based on the long emission decay of CdSe/ZnS NCs at low temperature. Although measured at low saturation intensity, the 100 µeV homogeneous linewidth found for the first excitonic state cannot exclude a contribution from spectral diffusion. This open question can be adressed by extending the presented method to perform time-gated PLE of individual NCs.

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13 H. Htoon, P. J. Cox, and V. I. Klimov, Phys Rev Lett 93 (18), 187402 (2004); P. Palinginis, S. Tavenner, M. Lonergan et al., Physical Review B 67, 201307 (2003).

Acknowledgement:

This research was funded by CNRS, Région Aquitaine and the French Ministry for Education and Research (MENRT).