Stéphane Zuber 1 & Marc Fleurbaey 2

Discounting and Intergenerational Ethics

The question of social discounting is central in intertemporal cost-benefit analysis that often shapes economists' recommendations regarding climate policy. The practice of discounting has been the object of heated debates among economists and philosophers, revolving around the issue of intergenerational ethics. In this chapter, we review the different arguments for and against specific values of social discounting. We show that there are actually two different ethical issues at stake: 1) the question of impartiality (or equal treatment of all generations); 2) the question of priority to the worse-off (aversion to inequality in resources, capabilities or welfare). These questions have emerged in the utilitarian approach and can be neatly separated in that case. They also have very different consequences for climate policy. We then argue that the question of social discounting is not confined to the utilitarian framework as it more generally describes the social value of income (or capability or welfare) transfers to future generations. Lastly, we discuss the many limitations of social discounting as a tool for policy analysis.

Introduction

The social discount rate is an important tool in intertemporal cost-benefit analyses used by economists to inform long-term policy choices, for instance mitigation policies aiming at reducing greenhouse gas emissions. The social discount rate is used to convert future monetary costs and benefits in present value. For instance, a rate of 2 % (=0,02) means that a project whose cost is $1,000,000 in 50 years has a present value of $1,000,000 / (1+0,02) 50 ≈ $371,528 today. As illustrated by this simple example, even low discount rates can significantly reduce the value of future costs and benefits. Moreover, if the social discount rate is 3% the present value of the same project would be about $228,107, that is, two thirds of a the present value using the 2% rate. Hence, apparently small changes in the social discount rate have huge implications in terms of how we value future economic impacts.

As acknowledged in the last report of Working Group 3 of the IPCC (Intergovernmental Panel on Climate Change), "the use of a temporal discount rate has a crucial impact on the evaluation of mitigation policies and measures" (Kolstad et al. 2014, p. 211). The Working Group asserts that the appropriate risk-free social discount rate should be between one and three times the anticipated growth rate in real per capita GDP, but also underscores that these values are based on the socalled Ramsey rule and that ultimately there are normative choices to be made. In this chapter, we aim at presenting some of these normative choices. We also argue that one does not need to restrict attention to the Ramsey rule and that the discount rate is a general tool that can be applied in many ethical frameworks.

Heated discussions around specific values of the social discount rate have aroused after the publication of the Stern review's recommending strong action to reduce greenhouse gas emissions [START_REF] Stern | Stern Review: The Economics of Climate Change[END_REF]. A key principle that was discussed was intergenerational equity and the discussion mostly relied on the Ramsey rule. In Section 1 of this chapter, we present the Ramsey rule and explain how it encompasses two different dimensions of equity: impartiality (the equal treatment of all generations) and preference for more equal distributions of resources. We explain why these two aspects can go in opposite directions. We also present the different positions in the debate that followed the Stern review.

The Ramsey rule emerged in the Utilitarian moral framework. In his seminal paper, [START_REF] Ramsey | A mathematical theory of savings[END_REF] explicitly assumed that "enjoyments and sacrifices at different times can be calculated independently and added" (Ramsey, 1928, p. 543) and used a criterion that explicitly adds utilities across periods. An important question then could be whether social discounting is tied to an excessively narrow (e.g., utilitarian, economistic) ethical approach, or whether it can accommodate a variety of relevant principles and values. One thesis defended in this chapter is that, while usual practice of discounting is indeed restricted to an unduly narrow ethical framework, the methodology itself is rather flexible. Still, it is part of a cost-benefit approach, which relies on a consequentialist axiology. Section 2 provides a general ethical framework in which discount rates may be obtained and we argue that this framework can encompass many moral views about who matters, how claims should be balanced, how advantages should be distributed, and how to measure individual advantage.

Despite its salience, the issue of discounting cannot be considered as the only important one in the evaluation of policies that affect future people (for instance, climate policies). We briefly recall why discount rates are theoretically relevant only for the analysis of marginal changes and can therefore not be used to assess abrupt regime shifts or existential risks. They are also not the right tool to use for choice that may affect the size or composition of the future population. Last, the discounting issue should also not obscure other important equity issues like the distribution of cost and benefits within generations. We present these limitations of social discounting in Section 3.

1 Discounting: definition in the utilitarian framework and equity issues As explained in the introduction, the standard approach to determining the social discount rate is based on the Ramsey equation. The perspective adopted to derive this equation is that of a benevolent social planner (or sometimes a representative agent) that seeks to maximize a value function. Such a value function assigns a real number to each possible state of affairs, with better states of affairs being assigned a greater number. The standard approach typically describes states of affairs as streams of consumption for all future generations, with 𝑐 the consumption of generation t (where 𝑡 = 0 denotes the current generation), and the value function is given by 3

𝑉(𝑐) = 𝑒 𝑢(𝑐 ) , ( 1 )
where function u is the utility function (transforming consumption levels into utility numbers) and 𝛿 is the so-called `utility discount rate' or `rate of pure time 3 We do not account for population size in this formula to simplify the exposition. We could more generally write utility as 𝑢(𝑁 , 𝑐 ), with 𝑁 𝑡 the population size in period t. Typically, the total utilitarian approach takes 𝑢(𝑁 , 𝑐 ) = 𝑁 𝑣 𝑐 𝑁 .

preference'. The term 𝑒 can be seen as a decreasing weight put on the utility of future generations.

Equation ( 1) is often further simplified by assuming that 𝑢(𝑐) = 𝑐

(1 -𝜂) ⁄ .

In that case, the 𝜂 parameter measures how rapidly marginal utility decreases when consumption increases (the formula thus accepts that marginal utility is decreasing, a standard assumption in the utilitarian tradition).

The idea of social discounting consists in measuring the value of a small increase in consumption in period t as equal to the value of a small increase in consumption today, discounted by a factor 1/(1 + 𝜌 ) . The  parameter is the social discount rate. From the value function described in Equation ( 1), it can be derived from the Ramsey equation:

𝜌 = 𝛿 + 𝜂𝑔 , ( 2
)
where 𝑔 is the average growth rate of consumption between the current period and period t.

The Ramsey Equation (2) clearly distinguishes two reasons for discounting. One is pure time discounting expressed by parameter 𝛿: we discount future consumption or future damages because we discount the welfare of future people. This first reason has given rise to a debate about intergenerational equity that could better be expressed as a debate about impartiality. The second part of the Ramsey equation combines the elasticity parameter 𝜂 and the growth rate of consumption. This is "discounting for growth": given that future generations are richer when 𝑔 is positive, their consumption has less priority. The rate of decrease in the value of future consumption is proportional to growth, and the proportionality parameter 𝜂 represents the strength of the redistributive motive. A higher 𝜂 represents a higher willingness to reduce consumption inequalities. We now separately discuss these two rationales for discounting.

1.1 Discounting and intergenerational impartiality: "normative" and "positive" approaches

The main controversy in the economics of climate change was probably the Nordhaus-Stern debate that arose after the publication of the Stern review [START_REF] Stern | Stern Review: The Economics of Climate Change[END_REF]. The Stern review promoted a strong action to reduce greenhouse gas emissions, while Nordhaus promoted a gradual response [START_REF] Nordhaus | A Question of Balance: Weighing the Options on Global Warming Policies[END_REF]. As explained by Nordhaus himself [START_REF] Nordhaus | A review of the Stern Review on the Economics of Climate Change[END_REF], a big part of the difference lies in using different values of the social discount rate: the Stern review is based on a 1.4% discount rate, while Nordhaus preferred a 5.5% discount rate. Part of the difference is due to using a different value of the utility discount rate: Stern argued in favor of 𝛿 = 0.1% while Nordhaus chose 𝛿 = 1.5%.

The line of argument used by Stern to justify his low value of the utility discount rate is based on the principle of impartiality. This was hardly a new line of argument: it can be traced back to Sidgwick who argued that "[...] the time at which a man exists cannot affect the value of his happiness from a universal point of view" (Sidgwick, 1907, p. 414). Ramsey himself used the idea of impartiality to justify a zero-utility discount rate. In the end, Stern does not use a zero-utility discount rate but a very low rate to account for a small probability that future generations may not exist. 4The impartiality argument is the main argument in favor of a zero or near-zero utility discount rate. 5Despite this strong impartiality argument, several authors have insisted that several reasons may explain a positive pure time discount rate. A first line of argument in favor of a positive utility discount rate was provided by [START_REF] Koopmans | Stationary ordinal utility and impatience[END_REF] who produced an influential axiomatization of discounted utilitarianism based on the Pareto principle combined with rationality and parsimony principles of time consistency and invariance of social evaluation. From this initial contribution stemmed a very rich (but technical) literature showing the incompatibility between the Pareto and impartiality principles when one considers an infinite sequence of successive generations [START_REF] Diamond | The evaluation of infinite utility streams[END_REF][START_REF] Basu | Aggregating infinite utility streams with intergenerational equity: The impossibility of beingParetian[END_REF][START_REF] Zame | Can intergenerational equity be operationalized?[END_REF][START_REF] Lauwers | Ordering infinite utility streams comes at the cost of a non-Ramsey set[END_REF].

One possible conclusion from this strand of literature would be that we cannot maintain the impartiality requirement in the intergenerational context because the Pareto principle is deemed to have more normative appeal. There are however several objections. A first objection is that it does not seem plausible that there will be infinitely many future generations, at least if we only consider generations of humans on Earth (there may exist other living species in other galaxies or worlds, but one may dispute whether they are morally relevant for us). A key problem then is that we do not know how many future generations there will be: we cannot plausibly specify a number of years until humans disappear. A well-known remedy has been proposed: to introduce an extinction (or existential) risk [START_REF] Dasgupta | Economic theory and exhaustible resources[END_REF][START_REF] Stern | Stern Review: The Economics of Climate Change[END_REF]. In the utilitarian case, this risk provides a foundation for a utility discount rate 𝛿 equal to the hazard rate of extinction as explained in footnote 2 (for a more general treatment, beyond the utilitarian case, see Fleurbaey and Zuber, 2015b).

Another objection is that violations of impartiality occur only because we want to have a value function that is able to rank and compare all situations. We may consider a less demanding moral theory based on incomplete criteria (that may not always say which situation is morally better). Several incomplete versions of utilitarianism -without utility discounting -have been proposed, in particular the overtaking criterion [START_REF] Weizsäcker | Existence of optimal programs of accumulation for an infinite time horizon[END_REF] and the Gale criterion (Gale, 1967). Last, let us underline that Koopmans ' (1960) argument in favor of a positive value of 𝛿 does not provide any guidance about the exact value of this parameter. In principle, it could be as low as one wishes so that, for any practical purpose, we may just neglect this part of the Ramsey equation.

In another vein, [START_REF] Arrow | Discounting, morality and gaming[END_REF] argued that the present bias introduced by utility discounting is not only a mathematical necessity, related to the infinite horizon framework, but is also ethically justified, on the grounds that it reflects a permissible agent-relative preference for ourselves and our own projects. A similar form of agent relative morality was defended by [START_REF] Dasgupta | Birth and Death[END_REF] who proposed a form of generation-relative utilitarianism. One key intuition developed by Arrow and Dasgupta in favor of a large enough utility discount rate is that it is not morally acceptable to demand excessively high savings rates of any one generation: simple growth models with a value of 𝛿 close to zero typically imply a very large savings rate (see [START_REF] Mirrlees | Optimum growth when technology is changing[END_REF]. This drawback of undiscounted utilitarianism was already mentioned by John Rawls who declared that "the utilitarian doctrine may direct us to demand heavy sacrifices of the poorer generations for the sake of greater advantages for the later ones that are far better off" (Rawls, 1971, p. 253). He went on to say that "these consequences can be to some degree corrected by discounting the welfare of those living in the future" (Rawls, 1971, p. 262). However, Rawls never argued in favor of discounted utilitarianism but simply wanted to point out a flaw of utilitarianism in the intergenerational context. The concern that utilitarianism may demand too large sacrifices from the current generation may also not be real: what matters for optimal savings is the whole consumption discount rate 𝜌 , not simply the utility discount rate: reasonable levels of investments can be obtained in the undiscounted utilitarian framework if one chooses large enough values of 𝜂 (see [START_REF] Asheim | The malleability of undiscounted utilitarianism as a criterion of intergenerational justice[END_REF].

While the two previous lines of arguments against the zero-utility discount rate may be related to normative considerations, many economists have preferred to offer reasons that are not directly stated as ethical reasons. Some scholars have labelled approaches relying on these reasons as "descriptive" approaches (see [START_REF] Arrow | Intertemporal equity, discounting, and economic efficiency[END_REF] or "positivist" approaches [START_REF] Posner | Climate Change Justice[END_REF]. These approaches mainly use a revealed preference argument. 6 Most people do in fact discount their future utility, as revealed for instance in market interest rates. Given that collective actions should be selected on the basis of aggregating individual preferences, a utility discount rate should reflect people's present bias. In particular, [START_REF] Nordhaus | A review of the Stern Review on the Economics of Climate Change[END_REF] famously declared Stern's approach as undemocratic and depicted it as a situation where a utilitarian elite (epitomizing "the dying embers of the British empire") makes decisions based on its own rather than the population's belief. Several objections can be made to the revealed preference argument. First, even if markets do aggregate preferences in some way (provided markets are wellfunctioning) they do so in a very specific way that may not be democratic. Indeed, the aggregation depends only the preferences of those people who are active on the market and on their initial wealth, so that poorer people preferences are typically not represented. Furthermore, future people's interests and preferences are not represented (at least not directly: they may be partially represented only to the extent that current people care about them). Hence, even if the descriptive approaches do not explicitly take an ethical stance on how advantages should be distributed across generations, they do implicitly rely on ethical assumptions. These assumptions are broadly that only current generations, and among them mostly the wealthier people or at least those who are active on markets, may have a say on how to allocate goods between periods, even in the long term.

The many objections to arguments in favor of a strictly positive utility discount rate explain why the authors of the 5 th Assessment report of the IPCC mention a "relative consensus in favor of 𝛿 = 0" (Kolstad et al. 2014, p. 230).

Discounting and aversion to intergenerational inequalities

The second part of the Ramsey equation ( 2) has to do with the fact that future generations may be richer. It is the product of the growth rate of consumption, which is clearly an empirical quantity (albeit a very uncertain one), with the elasticity parameter 𝜂. In the economic literature and in most presentations of the Ramsey rule, three main interpretations of this parameter have been offered (see for instance [START_REF] Greaves | Discounting for public policy: A survey[END_REF]. It may represent:

• Individuals' relative risk aversion;

• Individuals' inverse elasticity of intertemporal substitution;

• Aversion to inequality.

The choice of one of these interpretations is consequential. As highlighted in [START_REF] Atkinson | Siblings, not triplets: Social preferences for risk, inequality and time in discounting climate change[END_REF], empirical estimates of these three quantities are usually very different, which may explain the very wide range of value found in the literature (from 1 to 3 or 4 according to Kolstad et al. 2014, p. 230). Although the economic literature mentions these three interpretations, it mainly presents them as three empirical strategies to calibrate 𝜂 rather than appealing to normative reasons to choose one of them. We would like to argue that they correspond to specific interpretations of the utilitarian formula (1), including non-utilitarian ones.

The two first interpretations (in terms of individuals' risk or temporal preferences) correspond to a specific view on utility, namely utility as preference satisfaction. Although this is the standard interpretation of utility in economic theory, this is not uncontroversial for utilitarians that may prefer a hedonistic interpretation, or definitions appealing to people' judgement on their own life. But even if one accepts the preference satisfaction definition of utility, it appears that one has to choose what preferences are relevant: risk preferences or instantaneous temporal preferences.

The use of risk preferences to measure utility has a long history in economics. Harsanyi (1953Harsanyi ( , 1953) ) famously provided two frameworks to justify a representation of utility by risk preferences. Harsanyi's (1953) impartial observer theorem assumes that, behind a veil of ignorance, individuals have extended preferences on prospects of outcomes and identities, so that they can compare welfare across outcomes and preferences. Harsanyi assumed that all deliberators behind the veil of ignorance will have the same extended preferences that are revealed by choices under uncertainty. [START_REF] Harsanyi | Cardinal welfare, individualistic ethics and interpersonal comparison of utilities[END_REF] aggregation theorem, which involves the Pareto principle applied to risky situations, characterizes utilitarianism as a sum of Von Neumann-Morgenstern individual utilities. 7 Although they have attracted less attention, time preferences can give a foundation for individual utility by arguments similar to those in [START_REF] Harsanyi | Cardinal welfare, individualistic ethics and interpersonal comparison of utilities[END_REF]. Zuber (2011, Prop. 1) showed that when individual preference are time separable, social aggregation satisfying the Pareto principle must a sum of these utilities.

Interestingly, both time and risk preferences have been used to construct welfare-like measures for the case of health. Indeed, QALYs (Quality Adjusted Life Years) that are widely used health measures are often calibrated using two methods [START_REF] Drummond | Methods for the Economics Evaluation of Health Care Programmes[END_REF]: the standard gamble methodology (using risk preferences) and the time trade-off methodology (using time preferences).

The other prominent interpretation of the elasticity parameter 𝜂 is that it controls social attitudes towards inequality (it is the coefficient of relative inequality aversion in the terminology used in economics). The foundation here may not be in terms of decreasing marginal utility of consumption, as in the standard utilitarian approach, but in terms of social priority for consumption of poorer people. Most empirical approaches to estimate this inequality aversion parameter are based on individuals' attitudes towards redistribution in statedpreference experiments (see, e.g., [START_REF] Atkinson | Siblings, not triplets: Social preferences for risk, inequality and time in discounting climate change[END_REF] or on actual redistribution policies (see, e.g., [START_REF] Tol | International inequity aversion and the social cost of carbon[END_REF]. A method providing intuition about the inequality aversion parameter is the leaky bucket thought experiment [START_REF] Okun | Equality and Efficiency: The Big Trade-Off[END_REF], where one has to declare how much one is willing to lose in a transfer of money from richer to poorer people. But some authors argue that in the intergenerational context, we should rather ground our moral intuitions in experiments concerning the saving rate [START_REF] Dasgupta | Discounting Climate Change[END_REF].

In all cases, the parameter will control the marginal social value of consumption for any individual, depending on her initial consumption level. A larger value of 𝜂 implies a greater priority for poorer people or poorer generations. Thus, a value function exhibiting "greater concerns for intergenerational inequality" (in the specific sense of inequality aversion with respect to consumption) typically exhibits a larger social discount rate.

We then end up with two similarly opposite effects of ethical principles of justice on the social discount rate. The principle of impartiality, discussed in 1.1, would require a very low level of 𝛿 (actually 𝛿 = 0 if we do not account for a risk of extinction) and thus a low level of the discount rate. A principle of equality (or aversion to inequality), as applied to consumption levels in the discussion above, would require a high level of 𝜂 and thus a high level of the discount rate, at least in the standard case where the consumption growth rate 𝑔 is positive. 8 Of course, there is no direct logical connection between the principles of impartiality and 8 See Equation (2). Of course, if 𝑔 𝑡 < 0, i.e., when future generations are poorer than current generations, more aversion to consumption inequality will decrease the social discount rate, possibly yielding a negative discount rate. This will also happen in a framework with uncertainty about future growth, provided that it is sufficiently likely that situations where 𝑔 𝑡 < 0 may happen (see [START_REF] Fleurbaey | Climate policies deserve a negative discount rate[END_REF].

equality, but one may expect that foundations of the social discount rate that are based on the idea of intergenerational justice will promote both a low value of 𝛿 and a high value of 𝜂.9 All in all, one can conclude that there is no clear reason why intergenerational justice in general would promote high or low values of the social discount rate.

Discounting beyond utilitarianism

Discounting is not restricted to Utilitarianism. However, it still entails making strong ethical assumptions. Indeed, any notion of discounting is derived from the existence of a value function, whose general form can be as follows:

𝑉 = 𝐹(𝑐 , 𝑐 , … , 𝑐 , … ), ( 3
)
where 𝑐 is a vector of all "goods" that matter in period t and that can be discounted to compare with the value of good(s) in period 0.

The existence of a value function means that we are trying to assess and compare situations in terms of their goodness or, more accurately, their betterness. This is in contrast with ethical frameworks pertaining to the theory of the right or other approaches involving notions of harm, virtues or duties. Also, it means that we are primarily concerned with outcomes or consequences of actions. Therefore approaches for which discounting can be a relevant tool belongs to a broad class of maximizing consequentialist theories. Note however that some people who don't think that consequentialism is the only relevant ethical view may still think that consequences are part of the ethical considerations we should rely on. For instance, [START_REF] Broome | Climate Matters: Ethics in a Warming World[END_REF] argues public morality can focus on the pursuit of goodness while individuals can focus on avoiding actions that harm other (future) people, including by compensating potential harms through carbon offsetting. This division of labor can be debated but cost benefit analysis and thus discounting are important tool for coordinating the pursuit of the common good and thus pertain to axiological public morality. More generally, discounting may be relevant in an ethical theory that is not purely consequentialist: it is relevant only in so far as consequences are relevant.

Equation ( 3) is a very broad and general definition of the value function. It may include several "goods" or ethical dimensions that may range from human-centered individualistic and materialistic considerations (the amount and distribution of personal consumption goods in a population in a given period) to more holistic and non-speciesist (we may include dimensions like the quality of social relations, the level of biodiversity and protection of other species in the list of goods). The key assumption made by Equation ( 3) is that these different goods are measurable as well as comparable: we can derive overall good from them (and thus implicitly tradeoff the different dimensions).

With the general value function described in Equation (3), we cannot obtain a precise description of the main elements of the social discount rate. We will thus focus on a more specific value function encompassing many individualistic consequentialist ethical theories.

A general formula for individualistic consequentialist ethics

The more specific class of value functions we will focus on is described by the following formula:

𝑉 = 𝑊 𝑤(𝑖, 𝑐 ) ∈ , 𝑤(𝑖, 𝑐 ) ∈ , … , 𝑤(𝑖, 𝑐 ) ∈ , … , (4)
where each 𝑐 is a vector of "goods" available to person i and each person i belongs to a specific generation (𝑁 is the set of all individuals leaving in generation t; typically generation 0 is the current generation, but we may imagine that some past generation could be the "first" generation). 10 Function w is an individual advantage function that depends on the identity of each individual (therefore w depends on i) and on the consumption vector.

Formula (4) provides a very flexible framework to encompass many ethical theories. Different theories can be described as taking a stance on three different issues: the scope of justice (the population 𝑁 of individuals included in each generation); the currency of justice (the individual advantage functions w and the goods included in the vectors 𝑐 of personal goods); the shape of justice (the "aggregator function" W that combines and weighs the advantages of the different individuals).

The scope of justice: The question is to decide which entities are the legitimate recipients of burdens and benefits. Formally, in Equation (4) the question is to decide who is included in population 𝑁 in each period. We may even ask whether population 𝑁 should appear for 𝑡 > 0. As suggested in "descriptive" approaches to 10 There is an ambiguity in the economic literature with the notion of generation: usually economic models consider only time periods but name such periods a "generation". Most ethical theories would use indices of individual lifetime well-being to weigh the claims of different people, while in most applications only indices of momentary well-being appear. See Greaves (2017, p. 396) for a discussion of this matter. social discounting, some may argue that only current generations can have legitimate claims and that future generations claim are taken into account only insofar as some people in current generation care about them. However, most approaches to climate ethics would include both current and future generations.

Within a period, one may then wonder whether we should include all persons irrespective of the country they belong to. [START_REF] Schelling | Intergenerational discounting[END_REF] famously suggested distance in space might justify different treatment of individuals and that we may care less about people in the far away countries. Again, the impartiality principle seems to prevent us for making a difference between individuals on the basis of where and when they live. But the literature about global justice and cosmopolitanism has produced (controversial) arguments to justify some bias against aliens in defining social priorities (Rawls 1999;[START_REF] Nagel | The problem of global justice[END_REF].

Then we could also argue that the scope of justice includes not only humans but also nonhuman animals or even other nonhuman species. For instance, climate change is an important stressor for biodiversity so that it may overwhelm species that are slow to move or adapt. Of course, the instrumental value for humans of the environment may be included in the vector of goods 𝑐 that is available to a person. But this completely ignores any possible intrinsic value. The literature on animal ethics and climate change is developing quickly (see Hsiung and Sunstein 2007, McShane forthcoming, Sebo forthcoming) and emphasizes the need to broaden the scope of justice. One of the key difficulties remains to identify principles of crossspecies comparisons: how can we trade off human against non-human interests? Even within a hedonistic utilitarian approach, the question is not easy to settle: it is not at all obvious how to compare emotions across differently structured brains, or neural systems more generally.

In principle formula (4) could cover any scope discussed above. As highlighted by the problem of nonhuman species, it is however not simple in practice to extend the scope of justice as far as one would like and most applications restrict attention to the human population in all countries and all present and future periods.

The currency of justice: The question is to define what should be distributed and how the situation of the different people composing the population should be assessed and compared. In the philosophical and economic literature on social justice, many answers have been provided. Of course, one classical answer that we have already discussed before is that the function w in Equation ( 4) is an individual advantage function, as used in Utilitarianism. But even then, as emphasized before, there are several approaches to such an individual advantage function, including preference-satisfaction, extended-preferences, hedonistic or other mental-state approaches. One can give the generic label of welfarism to approaches relying on such individual advantage functions.

Several alternatives to welfarism have been proposed at least since the seminal book by [START_REF] Rawls | A Theory of Justice[END_REF]. Rawls proposed to replace welfare metrics with indices of primary goods: in that case, the vector 𝑐 would be a vector of primary goods and 𝑤(𝑖, 𝑐 ) the corresponding index. [START_REF] Sen | Commodities and Capabilities[END_REF] proposed the concept of capabilities reflecting the freedom or ability to achieve valuable functionings (i.e. "beings and doings"). In that case, the vector 𝑐 should be not thought of as a vector of commodities but as access to functionings and the function 𝑤 becomes an index of capabilities as developed in the economic literature (see Alkire 2016 for a recent review). [START_REF] Roemer | Theories of Distributive Justice[END_REF] proposed to develop indices of opportunities that should be equalized to achieve social justice. Opportunities are distributions of outcomes or advantages that people may choose or achieve through effort. [START_REF] Fleurbaey | A Theory of Fairness and Social Welfare[END_REF] and [START_REF] Fleurbaey | Beyond GDP. Measuring Welfare and Assessing Sustainability[END_REF] recently revived theories of equivalent-income to combine multiple dimensions of human achievement and welfare into a single index formally similar to function 𝑤(𝑖, 𝑐 ). There are many other possible views on the appropriate currency, including objective goods approaches (see [START_REF] Adler | The Oxford Handbook of Well-Being and Public Policy[END_REF] for presentations and comparisons of many different approaches).

Formally, any of these approaches could give rise to a specific application of the social discounting methodologies. There are however practical restrictions depending on the specific case. First, only goods that are included in the vector 𝑐 can be discounted and converted into some corresponding present value. Some of the approaches we have discussed may include non-material goods (or even no material goods at all), which raises the question of how such non-material goods should be measured. Some of the approaches also consider opportunities described as menus or distributions of outcomes or achievements. What we should be discounting, then, are changes in those distributions, which is not mathematically as straightforward as discounting changes in simple quantities. But distributions can be seen as risky outcomes and a whole methodology has been developed to discount risky outcomes. 11The shape of justice: The question concerns the criteria to use to determine how to weigh the benefits accruing to different people. Beside the additive formula of Utilitarianism exhibited in Equation ( 2), where (weighted) welfare numbers are simple added, many other forms for the "aggregator function" W have been proposed and studied. A prominent alternative defended by [START_REF] Parfit | Equality or priority? Ratio[END_REF] and [START_REF] Broome | Weighing Lives[END_REF] is an additively separable formula that yields Prioritarianism. A general formula would be:

𝑉 = ∑ ∑ 𝜙 𝑤(𝑖, 𝑐 ) ∈ , (5)
With Prioritarianism, welfare numbers are transformed using a concave function 𝜙 that grants more priority to welfare that accrues to worse-off people compared to that accruing to better-off people. Such a formula has been applied to social discounting and climate policy (Fleurbaey andZuber 2015b, Adler et al. 2017).

Another prominent option is Egalitarianism either in its strict form proposed by [START_REF] Temkin | Inequality[END_REF] or in the modified form of maximin or leximin as suggested by [START_REF] Rawls | A Theory of Justice[END_REF]. They have usually been considered too extreme (including by Rawls himself, as recalled earlier) for application to intergenerational equity. Strict egalitarianism faces the levelling-down objection that we may prefer to reduce the welfare of everyone in society to promote equality and therefore has not been used by economists who seek efficient allocations. The maximin approach has counter-intuitive implications in the case of climate change policy: given that it only focuses on the worst-off, it completely discount impacts or outcome changes of all other individuals and thus, in all existing models, of all future generations. Given that the current worst-off generation pays for the cost of climate policy, we do not want to make any sacrifice for the sake of the future.

Sufficientarianism is the doctrine that the notion of sufficiency, understood as having a decent (or good enough) life, should be the key consideration for distributive justice. A version of sufficientarianism holds that as many people as possible should enjoy conditions of life that place them above a sufficiency threshold [START_REF] Frankfurt | Equality as a Moral Ideal[END_REF][START_REF] Frankfurt | The Moral Irrelevance of Equality[END_REF]. Another version holds that we should give greater priority to helping worst-off persons up to the point at which these persons attain a good enough quality of life, but otherwise we should only maximize total welfare [START_REF] Crisp | Equality, Priority, and Compassion[END_REF].

The economic literature has also provided several social criteria to aggregate individual welfare or advantage with the idea to promote a notion of sustainability. [START_REF] Chichilnisky | An axiomatic approach to sustainable development[END_REF] proposed sustainable social preferences that combine a discounted sum of utilities and a long-run value. Asheim, Mitra and Tuggoden (2012) introduced a sustainable discounted utilitarian criterion similar to discounted utilitarianism in the sustainable case where future generations are betteroff than the current generation, but which is similar to Maximin case for unsustainable paths. [START_REF] Zuber | Justifying social discounting: The rankdiscounted utilitarian approach[END_REF] have introduced a rank-dependent model that implies a relative priority (in contrast to the absolute priority of prioritarian criteria) to worst-off people. The model shares some similarities with discounted utilitarianism except that the pure-time discount rate is conceived as a social weight prioritizing the interests of least-advantaged people.

The W function is therefore very flexible, so that the discounting technique can be applied to many views regarding how interests should be balanced. Most consequentialist approaches would fit into the model proposed by Equation (4).

The main elements of the discounting formula

In Fleurbaey and Zuber (2015a), we developed a general methodology to compute a social discount rate for the general value function described in Equation ( 4). The social discount rate then generally represents the rate of change in the value of a specific (or composite) good in the future period t compared to a reference (composite) good in the current period.12 This rate will depend on two key elements: the rate of change in marginal advantage derived from the consumption of the good; the rate of change in the social priority of individual advantage. The "marginal advantage derived from the consumption of the good" is similar to the concept of marginal utility of consumption. The only difference is that the individual advantage function 𝑤(𝑖, 𝑐 ) is not necessarily a utility function that measures changes in pleasure or happiness, but may measure changes in opportunities, capacities or other concepts.

What we call the "social priority of individual advantage" is a generalized notion of priority that does not necessarily correspond to the concept developed in the prioritarian theory. 13 It simply measures the social or ethical marginal value of advantage for a specific person. This makes it possible to compare and balance the distributive claims or needs of different people: is such or such increase in the advantage of the current generation more or less valuable than such or such increase in the advantage of a generation living in one thousand years?

In a utilitarian formula, social priority of utility would be the same for all individuals in all generations. In a discounted utilitarian formula, this social priority would be 𝑒 for a person of generation t, and thus decreasing through time. In a prioritarian formula, this social priority would be lower the better-off a person is. In a sufficientarian formula, the social priority would be greater for badly-off individuals up to some level of advantage and then the same for all. In an egalitarian maximin formula, all individuals would have zero priority except the worst-off persons, implying an extreme discounting formula. 14The simple decomposition in terms of marginal advantage and social priority of individual advantage can actually become more complex in practice for three reasons: 1) there are multiple goods; 2) there are several people in a generation; 3) there may be some risk or uncertainty about future outcomes.

Contrary to the simple case of formula (1) and the associated Ramsey equation ( 2), the value function (4) allows the advantage measure for individuals to depend on several goods (or resources or attributes). These different goods will generally have different discount rates, reflecting future changes in their relative value or price. One way out of this complication is to construct an "aggregate" good that can be discounted with a single discount rate. But, as explained in [START_REF] Greaves | Discounting for public policy: A survey[END_REF], we must then be careful not to forget the issue of changing relative prices: if some good (for instance environmental quality, the level of biodiversity, etc.) becomes less abundant in the future compared to another good (for instance material consumption), its relative price will increase making it more valuable. And more to the point: not only the relative market price may change, but the relative value for the ethical assessment may change (where these relative value is computed as a shadow price using the value function in Equation ( 4), which may not be reflected in market prices). Retaining good-specific discount rates (see for instance Gollier 2010), on the other hand, is more transparent and makes it possible to highlight the importance of certain goods when assessing policies affecting several generations: for instance, the "ecological discount rate" for future damages on the environment may become negative if we think that environmental quality will decrease in the future [START_REF] Gollier | Ecological discounting[END_REF].

Future generations are composed of several individuals that may be affected by the decisions we take today. For ethical assessment, we do not want to assume that there exists a representative agent of these different individuals as is generally assumed when using the Ramsey equation. This raises the issue of the inequality within future generations. One standard methodology to take inequality into account relies on equity weights when we compute the future costs and benefits of a policy (see Anthoff, Hepbrun and Tol 2009 for an application to climate policy). In that case, we compute an "average" social discount rate for the future (that looks at the average level of a good enjoyed by the future generation) and transform the measure of impacts in the future to account for inequalities. An alternative consist in incorporating directly (intra-generational) inequality considerations in the measure of the social discount rate as suggested by [START_REF] Gollier | Discounting, inequality and economic convergence[END_REF] and Fleurbaey and Zuber (2015a). The idea is that a less unequal future generation can be considered as better-off so that we may want to put less weight in increases in their consumption of goods and resources. In that case, the social discount rate can be viewed as an aggregation of individual or personalized discount rates (for person-to-person transfers) and it also makes it possible to include the consideration of inequalities in the distribution of consequences (Fleurbaey and Zuber 2015a).

Risk is also a pervasive phenomenon in many problems involving intergenerational ethics and in particular for climate justice. There is a lot of uncertainty about the level of resources available to future uncertainty and also epistemic uncertainty about the models we can use to foresee the future consequences of our actions. Risk has attracted a lot of attention in the economic literature. Several attempts have been made to adjust the discounting formula in that case as well as providing alternative decision model that may disentangle attitudes towards risk and attitudes towards intergenerational distribution in the value function (see [START_REF] Greaves | Discounting for public policy: A survey[END_REF][START_REF] Fleurbaey | The social cost of carbon: Valuing inequality, risk, and population for climate policy[END_REF] for extensive surveys of this issue). Formally, the question of risk is very similar to the question of inequality. But there is one key difference: in the case of risk, the discounting formula may include an additional term that reflects the correlation between individual advantage and the aggregate advantage of people in all generations (Fleurbaey and Zuber 2015b). This involves a notion of correlated or aggregate risk, which is a risk on the overall value function.

Conclusion: Beyond social discounting

In this chapter, we have argued that social discounting is a flexible tool for policy evaluation for problems spanning several generations. Social discounting does not necessarily entail a violation of the principle of impartiality among generations: on the contrary, we have argued that ethically defensible versions of social discounting typically satisfy this principle. Social discounting may also involve different ethical positions regarding how resources, well-being or advantages should be distributed across people in different generations. We have actually argued that the social discounting methodology can be developed for a wide range of ethical views. The main restriction is that it remains within the scope of consequentialist axiology, and thus is not the appropriate tool to deal with considerations of right, harm, virtues or duties. But even people who do not view consequentialism as the only relevant moral theory may accept that consequentialist considerations are part of the overall moral picture: in that sense, the social discounting methodology may still be useful for them.

However, even within the scope of cost-benefit analysis, the social discounting approach (and the associated methodology of net present values) should not be considered an all-purpose tool that can serve for all evaluations and issues. We would like to conclude this chapter with some caveats.

First, we must emphasize a well-known point that is sometimes overlooked. As stressed for instance by the Stern review (Stern 2006 § 2A.2), the social discount rate is useful to evaluate small transfers of consumption across individuals living at different times. It is not adapted to large-scale changes. For instance, there is evidence that climate change may affect future growth and therefore that a climate policy may alter the underlying consumption path (Dell et al. 2012, Moore and[START_REF] Moore | Temperature impacts on economic growth warrant stringent mitigation policy[END_REF]. Similarly, climate change may worsen inequalities or hinter the development of some regions in the world, and such changes may alter our willingness to implement policies that reduce more greenhouse gas emissions in the near term (Hallegate et al. 2016, Budolfson et al. Similarly, policy may change the size or the composition of the future population. For instance, climate change and climate policy could influence patterns of fertility and mortality thereby changing who will exist in the future. Thus, our value function should incorporate population sizes, thereby raising issues of population ethics [START_REF] Broome | Climate Matters: Ethics in a Warming World[END_REF], Kolstad et al. 2014, p. 211). Population ethics is known to raise difficult puzzles and no single approach has emerged that is consistent with all attractive intuitions [START_REF] Parfit | Reasons and Persons[END_REF]Blackorby et al. 2005). A broad divide is between theories that value population size even at the expense of average well-being (like Total Utilitarianism) and theories that regard average well-being as the most important aspect even if it implies reducing population size (like Average Utilitarianism). Population ethics can significantly modify our view on policy especially in cases when we are not sure about the future population trajectory [START_REF] Scovronick | Impact of population growth and population ethics on climate change mitigation policy[END_REF][START_REF] Méjean | Intergenerational equity under catastrophic climate change[END_REF].

Population ethics is particularly important in the case of catastrophic or existential risks that may drastically reduce future population size (or even lead to human extinction). Climate policy in general has focused on future impacts of climate on consumption or on the goods (in a broad sense) available to future generations. But climate change may not only alter future resources, it may alter the risk that future generations (not only human, but also for other species) do not exist. In economic cost-benefit analysis, the technique used to evaluate changes in probabilities of a risk on the existence (of an individual) consists in computing the 'value of a statistical life'. The methodology can be extended in the case of risks on the existence of future generations (the idea was suggested, but used in a very different way by [START_REF] Weitzman | On modeling and interpreting the economics of catastrophic climate change[END_REF]. In that case, the social discount rate is not any more the key parameter to value future consequences and alternative methods must be developed [START_REF] Bommier | Models-as-usual for unusual risks? On the value of catastrophic climate change[END_REF][START_REF] Méjean | Intergenerational equity under catastrophic climate change[END_REF].

For all three cases mentioned above (effects of the policy on growth, on population size or on large-scale risk), the computation of net present values using some social discount rate cannot provide the right guidance if the underlying ethical theory can be represented by a value function like the one exhibited in Formulas (1) or (4). For these cases (and other similar cases) policy evaluation has to rely directly on the underlying value function. This reminds us that the social discounting methodology is only an approximation of our ethical assessment. And that the soundness or attractiveness of a discounting formula only derives from the soundness and attractiveness of the underlying ethical theory. Discounting is good only insofar as it relies on sound ethical principles. 15

Paris School of Economics -CNRS, France. Address: Centre d'Economie de la Sorbonne, 106-112 boulevard de l'Hôpital, 75013 Paris, France. E-mail: Stephane.Zuber@univ-paris1.fr.

Woodrow Wilson School and University Center for Human Values, Princeton University, USA. E-mail: mfleurba@princeton.edu.

With this interpretation, the utility discount factor 𝑒 -𝛿𝑡 is the probability that the future generation of period t exists.

Greaves (2017, p. 405) mentions another argument based on an extended Pareto principle. We believe that this argument, which involves a choice made an individual between existing today with some consumption level or living tomorrow with this same level, is not real realistic unless we consider a veil of ignorance context that actually serves to promote the impartiality principle.

[START_REF] Posner | Climate Change Justice[END_REF] have a different "positivist" line of argument to choose the social discount rate (without reference to the Ramsey equation): an opportunity cost argument. We should never choose projects with returns lower than the market interest rate because otherwise we would incur a loss -we could have invested in a market fund and get a larger benefit in the future. But the identity between the social discount rate and the market rate is justified only in the very specific case where we are already at the economic optimum (so that no more investments are required). It also assumes that the maturity of the market investment is the same as that of the proposed policy, which is unlikely for very long term policies like climate policy. We have discussed this line of arguments in more details in[START_REF] Fleurbaey | Climate policies deserve a negative discount rate[END_REF].

[START_REF] Sen | Welfare inequality and Rawlsian axiomatics[END_REF] andWeymark (1991) argued that, while Harsanyi's theorem establishes that social welfare is a sum of VNM utilities, it does not follow that it is the sum of individuals' welfare levels. But this is beside the point of the theorem, whose potent message is that social evaluation must rely on a (weighted) sum of VNM utilities. See[START_REF] Fleurbaey | The utilitarian relevance of Harsanyi's theorem[END_REF] for further discussion.

And indeed, for instance,[START_REF] Dasgupta | Discounting Climate Change[END_REF] criticized the Stern report for using principles of intergenerational justice to justify a low value of 𝛿 without considering such principles when setting the value of 𝜂.

We cannot extensively cover the case of risky outcomes in that chapter. The question has been discussed in depth in other papers, for instanceFleurbaey and Zuber (2015a),[START_REF] Greaves | Discounting for public policy: A survey[END_REF] and[START_REF] Fleurbaey | The social cost of carbon: Valuing inequality, risk, and population for climate policy[END_REF].

The term "specific good" means that we consider a particular good in the vector 𝑐 𝑖 for a specific individual 𝑖 ∈ 𝑁 . The term "composite good" means that we consider an aggregate or equivalent quantity to represent the level of all goods either for an individual or the society at large in a given period. Typically, economic models focus on average consumption in a population.

See Parfit (1997) for the introduction of this notion of priority and[START_REF] Broome | Equality versus priority: a useful distinction[END_REF] for a defense of the concept.

For instance, the maximin case with one person per generation and one good, the discount rate would be infinitely large if the current generation is the worst-off generation.


est destinée au dépôt

Evaluation of the spider (Phlogiellus genus) phlotoxin 1 and synthetic variants as antinociceptive drug candidates

P33-EVALUATION OF THE SPIDER (PHLOGIELLUS GENUS) PHLOTOXIN 1 AND SYNTHETIC VARIANTS AS ANTINOCICEPTIVE DRUG CANDIDATES

Evelyne BENOIT ; Tânia c. GONçALVES ; Pierre LESPORT ; Sarah KUYLLE ; Enrico STURA ; Justyna CIOLEK; Gilles MOURIER; Denis SERVENT; Emmanuel BOURINET ; Nicolas GILLES ; Over the two last decades, venom toxins have been explored as alternatives to opioids to treat chronic debilitating pain. Actually, approximatively 20 potential analgesic toxins, mainly from spider venoms, are known to inhibit with high affinity the NaV1.7 subtype of voltage-gated sodium (NaV) channels, the most promising genetically validated antinociceptive target identified so far. The present study aimed to consolidate the development of phlotoxin 1 (PhlTx1), a 34-amino acid and 3-disulfide bridge peptide of a Phlogiellus genus spider, as an antinociceptive agent by improving its affinity and selectivity for the human (h) NaV1.7 subtype. The synthetic homologue of PhlTx1 was generated and, as the natural peptide, equilibrated between two active forms on reverse-phase liquid chromatography and exhibited potent analgesic effects in a mouse model of inflammatory pain. The effects of PhlTx1 and 8 successfully synthetized alanine-substituted variants were studied (by automated whole-cell patchclamp electrophysiology) on cell lines stably overexpressing hNaV subtypes, as well as two cardiac targets, the hCaV1.2 and hKV11.1 subtypes of voltage-gated calcium (CaV) and potassium (KV) channels, respectively. PhlTx1 and D7A-PhlTx1 were shown to inhibit hNaV1.1-1.3 and 1.5-1.7 subtypes at hundred nanomolar concentrations, while their affinities for hNaV1.4 and 1.8, hCaV1.2 and hKV11.1 subtypes were over micromolar concentrations. Despite similar analgesic effects in the mouse model of inflammatory pain and selectivity profiles, the affinity of D7A-PhlTx1 for the NaV1.7 subtype was at least 5 times higher than that of the wild-type peptide. Computational modelling was performed to deduce the 3D-structure of PhlTx1 and to suggest the amino acids involved in the efficiency of the molecule. In conclusion, the present structure-activity relationship study of PhlTx1 results in a low improved affinity of the molecule for the NaV1.7 subtype but without any marked change in the molecule selectivity against the other studied ion channel subtypes. Further experiments are therefore necessary before considering the development of PhlTx1 or synthetic variants as antinociceptive drug candidates.

  Evelyne Benoit, Tânia C. Gonçalves, Pierre Lesport, Sarah Kuylle, Enrico Stura, Justyna Ciolek, Gilles Mourier, Denis Servent, Emmanuel Bourinet, Nicolas Gilles To cite this version: Evelyne Benoit, Tânia C. Gonçalves, Pierre Lesport, Sarah Kuylle, Enrico Stura, et al.. Evaluation of the spider (Phlogiellus genus) phlotoxin 1 and synthetic variants as antinociceptive drug candidates. 30th Ion Channel Meeting, Sep 2019, Sète, France. hal-04467245


L Stefanuto

L Chomienne

Bossard

& Montagne

In search of the signature of perceptual-motor expertise in 4x100 m relay

The 4x100 m relay is a very demanding sport as performance depends on the ability of athletes to run at the highest possible speed of course, but also and above all to pass the baton at a high speed in a constrained space. The coordination between runners is one of the main determinants of performance and requires highly developed anticipation skills in athletes. Indeed, success presupposes for the receiver to pick-up information early on their running partner to be able to initiate their race at the right time. In this context, the go-mark's placement on the ground is supposed to help receivers to initiate their run at the right time (i.e., when the partner passes over of the go-mark). The ambition of the present study is twofold. The first one is to better understand the perceptual-motor mechanisms allowing athletes to successfully perform this highly constrained synchronization task. The second one is to analyze the way expertise shapes the implemented mechanisms. The participants were equipped with a virtual reality headset to be immersed in a virtual stadium at the receiver's position. Three groups were formed according to their level of expertise: beginner, intermediate (regional level) and expert (national/international level). They were instructed to initiate their race when the avatar of their partner, who could arrive in a straight line or in a bend, passed over the go-mark. They performed 3 blocks of 10 trials (5 trials per condition). Their perceptual-motor strategies have been characterized according to several levels of analysis: the timing error, the temporal sequence of remarkable events (athlete's preparatory movement initiation, end of taking information on the avatar, initiation of the race...) and the information sampling strategies employed. The results revealed that the strategies implemented depended on the level of expertise: the timing error was lower for experts and performance was more stable. Interestingly, the time sequence of the remarkable events was more efficient for the experts, reflecting the importance of the movement preceding the initiation of the race in the process of anticipation. In the same way, the analysis of the eye tracking data also revealed specific strategies for taking information according to the level of expertise, with experts spending more time scrutinizing the most information-rich areas. Taken together, these results provide a first insight regarding the effect of expertise on athletes' anticipatory behavior during the initiation of the race while approaching the partner. Even if new studies are necessary to provide more information about the involved mechanisms, our study already underlines the close links between perception and action in the anticipation process of experts and the extreme sophistication of the implemented mechanisms.

Acknowledgements

This work is part of the REVEA project supported by a government grant managed by the French National Research Agency (ANR) as part of the "France 2030" program (20-STHP-0004).

Meeting abstract presented at ECSS 2023


Olivier Ramar
email: olivier.ramare@univ-amu.fr

G K Viswanadham
email: viswanadh@iiserbpr.ac.in

Olivier Ramaré

EXPONENTIAL SUMS OVER PRIMES WITH MULTIPLICATIVE COEFFICIENTS

Keywords: 2010 Mathematics Subject Classification. 11L07, 11N13 (11L20) Exponential sums, Bilinear decomposition, Fourier coefficients of cusp forms

We consider exponential sums of the form X<p≤2X f (p)(log p)e(pα) ,

where the sum runs over the prime numbers p ∈ (X, 2X] and f is a multiplicative function satisfying certain growth conditions. As a consequence of our result, we consider the normalized Fourier coefficients (a g (n)) of any eulerian GL(n)cuspform g that satisfies the Ramanujan conjecture as well as an estimate of the form max α∈R | n≤X a g (n)e(nα)| ≤ X η for some η < 1. For such a form, we get that X<p≤2X a g (p)(log p)e(pα)

where α is a real number such that α -a q X -1+ 1-η 120 for some q ≤ X (1-η)/15 . Under stronger restrictions and the same conditions on α and a/q, we also prove that X< ≤2X a g ( )µ( )e(pα) X/ √ q .

Introduction

We are concerned here in getting estimates for the trigonometric polynomial p≤N τ (p)e(pα) where p is a prime number and τ the Ramanujan function. We consider more precisely phases α that are close to a rational a/q with a small denominator q. It can be seen in the general light of Sarnak's conjecture [START_REF] Sarnak | Three lectures on the Mobius function randomness and dynamics[END_REF] as showing that τ (p) do not correlate with the additive characters we consider, continuing the work [START_REF] Fouvry | Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms[END_REF] of É. Fouvry and S. Ganguly.

Our method fits in a more general framework and continues our previous query [START_REF] Ramaré | Modular ternary additive problems with irregular or prime numbers[END_REF] where we obtained 'optimal' bounds for p≤N e(pα) log p. We add a multiplicative function f as a coefficient to consider p≤N f (p)e(pα) log p and want to show that this sum is small provided reasonable hypotheses on the values of f over the integers are met. We present these assumptions just below, but we want to stress out that the main difficulty with respect to our previous work is that we do not assume |f (p)| ≤ 1.

Let us present the three assumptions we make on our multiplicative function f . (H 1 ) There exists an integer k ≥ 1 such that

|f (n)| ≤ τ k (n) for all n ≥ 1 .
Here and in what follows τ k denotes the kth divisor function defined by τ k (n) = d 1 •••d k =n 1. We fix such an integer k once and for all. (H 2 ) On denoting by f -1 the Dirichlet inverse of f , we assume that for each > 0,

|f -1 (n)| n .
(H 3 ) There exists an η ( 1 2 ≤ η < 1) such that n≤X f (n)e(nα) X η uniformly in α ∈ R. These hypotheses have for instance been investigated in the context of automorphic forms and we describe this situation some more below. The class C(k, η) of functions satisfying (H 1 ), (H 2 ) and (H 3 ) could be extended somewhat, but we keep it for simplicity. Notice that the convolution product of two multiplicative functions, one from C(k 1 , η 1 ) and the other from C(k 2 , η 2 ), belongs to

C k 1 + k 2 , 1-η 1 η 2 -(η 1 -η 2 ) 2 2-η 1 -η 2
(this is a simple consequence of the Dirichlet hyperbola formula). Broadly speaking, the question we address is to infer properties on the primes from properties on the integers as in sieve theory. Here is our first result, where ϕ denotes the Euler's function.

Corollary 1.1. Assume (H 1 ), (H 2 ) and (H 3 ). Let X ≥ 1 be a real number. Let q be a positive integer such that q ≤ X (1-η)/15 . Then for any real number α with |α -a q | X -1+ 1-η 120 we have

X< ≤2X f (p)(log p)e(pα) η √ qX ϕ(q) .
The constant implied in the η -symbol depends only on the constants implied in hypotheses (H 1 ), (H 2 ) and (H 3 ). This is a direct corollary of Theorem 1.4 below. Our proof goes by building in Section 3 (more precisely (19)) a family of bilinear representations of the characteristic function of the primes, a task for which we rely heavily on the prior work of Y. Motohashi in [START_REF] Motohashi | Sieve Methods and Prime Number Theory[END_REF] (see also [START_REF] Motohashi | Primes in arithmetic progressions[END_REF]). The averaging effect on this family will save the last log X.

Examples of the situation we consider is given by non-trivial Dirichlet characters; more interesting situations are given by cuspforms. For instance, we can consider a normalized Hecke eigen cuspform g of weight m on SL 2 (Z) and its sequence {a g (n)} of normalized Fourier coefficients (this means for instance that rather than considering the Ramanujan τ function, we prefer to investigate τ (n)n -11/2 ). Then it is known that a g is a multiplicative function that belongs to C(2, 1/2). Indeed, Deligne's bound says that a g satisfies (H 1 ) with k = 2. It can also be seen, looking at the Euler product of L-function associated to g, that the function a g satisfies (H 2 ). And it is proved by M. Jutila [START_REF] Jutila | On exponential sums involving the Ramanujan function[END_REF] that (H 3 ) is satisfied by a g with η = 1 2 , refining the work of J.R. Wilton in [18, Lemma 3]. Assumption (H 1 ) in this context would be a consequence of the Ramanujan Conjecture, which is known to hold in several cases. A similar setting can be developped for GL(n)-cuspforms; Assumption (H 1 ) would be a consequence of the proper Ramanujan Conjecture, Assumption (H 2 ) would be given by the rationality of the Euler-factor of the corresponding Dirichlet series. We refer to Chapter 9 by J.W. Cogdell from the book [START_REF] Bump | An introduction to the Langlands program[END_REF] for the theory of GL(n)-L-functions and in particular the eulerianity of Fourier expansions. These two 'size' hypotheses are somewhat stronger than the corresponding ones used for defining the Selberg class of [START_REF] Selberg | Old and new conjectures and results about a class of Dirichlet series[END_REF]. Assumption (H 3 ) has been investigated by S.D.

Miller in [START_REF] Stephen | Cancellation in additively twisted sums on GL(n)[END_REF] for GL(3)-forms and is believed to hold in general. The reader will find a discussion of this in the paper [START_REF] Lü | Exponential sums with Fourier coefficients of automorphic forms[END_REF] by Guangshi Lü.

Hence by Corollary 1.1 we have the following.

Corollary 1.2. Let g be an eulerian GL(n)-cuspform that satisfies the Ramanujan conjecture and (H 3 ) for some η < 1. Let q be a positive integer such that q ≤ X (1-η)/15 . Then for any real number α with |α -a q | X -1+ 1-η 120 we have X< ≤2X

a g (p)(log p)e(pα)

√ qX ϕ(q) .
Let us now be more precise. We introduce L(f, s) (for (s) > 1) the L-function associated to f , i.e.

L(f, s) = n≥1 f (n) n s = p 1 + f (p) p s + f (p 2 ) p 2s + • • • . Let L (f,s) L(f,s) = -n Λ f (n)n -s
be the logarithmic derivative of L(f, s), see [START_REF] Levin | Application of some integral equations to problems of number theory[END_REF]. Then clearly Λ f has support only on prime powers. In fact we have: Lemma 1.3. Let p be a prime number. For any integer m ≥ 1 we have

Λ f (p n ) = k+ =n ≥1 f -1 (p k )f (p ) log(p ) .
From (H 1 ) and (H 2 ) together with the above lemma and the well-known

τ k (n) n , we get (1) |Λ f (p n )| p n .
In view of this bound we have

(2)

X<p m ≤2X m≥2 |Λ f (p m )| X 1 2 +
for each > 0. Here is the main theorem of this paper when expressed in terms of Λ f . Theorem 1.4. Let X ≥ 1 be a real number and let f be a multiplicative function satisfying (H 1 ), (H 2 ) and (H 3 ). Let q be any positive integer such that q ≤ X (1-η)/15 . Then for any real number α with |α -a q | X -1+ 1-η 120 we have

X< ≤2X Λ f ( )e ( α) η √ qX ϕ(q) .
Since Λ f (p) = f (p) log p for all primes p, Corollary 1.1 can be easily obtained on using (2). We did not try to optimize the exponents that appear but only aimed at producing a clean proof.

The method of proving Theorem 1.4 is flexible enough to obtain similar results for the sum of f -1 over square-free integers. We however need to control n≤M |f (n)| 2 . Concerning holomorphic modular form, this (and more) follows from the work [START_REF] Rankin | Contributions to the theory of Ramanujan's function τ (n) and similar arithmetical functions. I. The zeros of the function ∞ n=1 τ (n)/n s on the line Rs = 13/2. II. The order of the Fourier coefficients of integral modular forms[END_REF] of Rankin. See also [START_REF] Fomenko | Mean value theorems for automorphic L-functions[END_REF] by O.M. Fomenko and [5] by H. Lao for symetric powers L-functions.

Theorem 1.5. Let X ≥ 1 be a real number and let f be a multiplicative function satisfying (H 1 ), (H 2 ) and (H 3 ). Let q be any positive integer such that q ≤ X (1-η)/15 . Then for any real number α with |α -a q | X -1+ 1-η 120 we have

X< ≤2X µ( )f ( )e ( α) η X √ q W f (X) (3)
where

(4) W f (X) = max M ≤X M <m≤2M |f (m)| 2 /M .
Let us note here that, if considering convolutions in Theorem 1.4 is rather pointless as the corresponding value at the primes is simply the sum of the one of each factor. The situation changes drastically here. As already mentioned, Hypothesis (H 3 ) in this contexct has been investigated by Guangshi Lü in [START_REF] Lü | Exponential sums with Fourier coefficients of automorphic forms[END_REF]. Since the Möbius function satisfies our hypotheses under the Generalized Riemann Hypothesis, we see that the above theorem is conjecturally optimal at least when q = 1 and α = 0. The coefficient W f (X) is introduced to accomodate possible powers of log X. Notice that ( 5)

M <m≤2M |f (m)| ≤ 2 W f (X)M whenever M ≤ X.
Acknowledgements. This paper started in 2018 when the second author was suported by the Indo-French Institute of Mathematics, which we thank warmly for its support, on this occasion as well as for numerous other meetings where, among other things, this work has been pursued.

Organisation of the proof. The proof starts with Equation ( 27) which proposes a decomposition of the relevant trigonometric sum in three parts. The first part, L

r (α), is studied in Section 4, while the second one is studied in Section 5. These are technical but rather straightforward as no particular precision is required. The third and last part, which we call the bilinear sum is handled in Section 6. The sum S b in ( 6) is the central quantity and contains the averages over r (i.e. the averages over the family bilinear decompositions). We first need to reduce the evaluation to an L 2 -problem, then we localize the variables and finally we separate them properly (they are linked with a mild X < mn ≤ 2X). At the same time, we take care of the offset between α and a/q. This introduces the error terms E 1 (δ, r), E 2 (δ, r) and δE 3 (r). It is then a matter of bookkeeping to reduce the expression obtained to the hybrid large sieve inequality we have recalled earlier in Lemma 2.7. The proof of Theorem 1.5 is mutatis mutandis to the proof of Theorem 1.4. We give a sketch of it in Section 7.

Preliminaries

We state the following lemma of Motohashi, see [10, page 25].

Lemma 2.1. Let d be a square-free integer. Then for any positive integer n, we have

f (dn) = µ(d) u|n, u|d ∞ f n u f -1 (du) .
Here u | d ∞ means that u divides some power of d.

Lemma 2.2. Let and d be positive integers. Then for each > 0 we have

n≥1 n|[ ,d] ∞ f -1 ([ , d]n) n 1 2 (d ) .
Let θ > 0 (we shall finally choose θ = 1/k 2 ) and z > 1 be fixed. For any positive integers k and d, define Λ

(k ) d by Λ (k ) d = 1 k ! (θ log z) -k k j=0 (-1) k -j k j λ (j,k ) d , where λ (j,k ) d =    µ(d) log z 1+jθ d k if d < z 1+jθ ;
0 otherwise. We have the following lemma. (1)

Λ (k ) d = µ(d) if d < z.
(2) Let c > 0 be some parameter. We have

n≥1 τ k (n) d|n Λ (k ) d 2 n -ω 1 , provided ω ≥ 1 + c log z .
The weights Λ (k ) d further satisfy the following bound.

Lemma 2.4. Let θ > 0. Then

|Λ (k ) d | ≤ (2k ) k k ! .
Proof. Since the quantity on the right hand side is greater than 1, we can assume that d ≥ z. Then we have

|Λ (k ) d | ≤ 1 k !(θ log z) k k j=0 k j (log z jθ ) k .
Cancelling (log z θ ) k , using j ≤ k and k j=0 k j = 2 k gives the desired upper bound.

The function f being given to verify assumptions (H 1 ), (H 2 ) and (H 3 ). They imply the two parameters k and η that we keep fixed throughout the proof. We use the above weights with k = k 2 and some θ > 0 that we keep as a parameter until the end of the proof where we choose θ = 1 k 2 . For any r ≥ 1, let M r (s) be defined by

(6) M r (s) = d≤z 1+k 2 θ Λ (k 2 ) d |r µ r [ , d] s µ([ , d]) n≥1 f -1 [ ,d] (n) n s ,
where

f -1 [ ,d] (n) = f -1 ([ , d]n) when n | [ , d] ∞ ; 0 otherwise Lemma 2.5. Let (s) ≥ 1 2 . Then for each > 0 we have M r (s) r 1+ z (1+k 2 θ)( 1 2 + ) .
Proof. We have for (s)

≥ 1 2 , |M r (s)| ≤ d≤z 1+k 2 θ |Λ (k 2 ) d | |r [ , d] 1 2 n≥1 |f -1 [ ,d] (n)| n 1 2 k d≤z 1+k 2 θ 1 d 1 2 |r n≥1 n|[ ,d] ∞ f -1 ([ , d]n) n 1 2 . We have by Lemma 2.2 n≥1 n|[ ,d] ∞ f -1 ([ , d]n) n 1 2 ( d) .
From this we get

|M r (s)| d≤z 1+k 2 θ 1 d 1 2 - |r 1+ . Hence M r (s) ,k r 1+2 z (1+k 2 θ)( 1 2 + ) .
From (H 3 ) we obtain the following estimate by partial summation.

(

) n≤X f (n)(log n)e(nα) X η log X . 7
Now we state one of the crucial lemmas.

Lemma 2.6 (Lemma 41, [START_REF] Ramaré | Modular ternary additive problems with irregular or prime numbers[END_REF]). Let δ ∈ (0, 1/2), β and X ≥ 1 be three real parameters. There exists a C 1 -function H such that for any sequence (a ) of complex numbers, we have

X< ≤2X a e(β ) = ∆ -∆ ≥1 a iu H(u)X iu du + O * X< ≤(1+δ)X, or (2-δ)X< ≤2X |a | + 2δ ≥1 |a | , where ∆ = 100 δ -1 +(βX) 2 δ . We have furthermore |H(u)| ≤ 25 73 (1 + |β|X)/(1 + |u|) and ∞ -∞ |H(u)| 2 du = (log 2) 2 (2 -2δ)/(4π) 2 .
For any positive integers r and n, the Ramanujan sum c r (n) is defined by

(8) c r (n) = a(mod) * r e an r ,
where the sum runs over all the coprime residue classes modulo r. It is known that c r (n) can also be expressed as

c r (n) = |n, |r µ r .
By definition of Ramanujan sum it is clear that |c r (n)| ≤ ϕ(r). Now we state the following version of Large sieve inequality:

Lemma 2.7 (Theorem 10, [START_REF] Ramaré | Modular ternary additive problems with irregular or prime numbers[END_REF]). Let q be some fixed modulus and N 0 be some real number. Let (u n ) n be a sequence of complex numbers that is such that

n (|u n | + n|u n | 2 ) < ∞.
Then we have, for any T ≥ 0, r≤R/q, (q,r)=1

1 ϕ(r) a mod q T -T n u n c r (n + N 0 )n it e(na/q) 2 dt ≤ 7 n |u n | 2 (n + R 2 max(T, 10)).
Let us recall the classical definition

(9) G q (D) = d≤D, (d,q)=1 µ 2 (d) ϕ(d) , G(D) = G 1 (D).
We quote from [START_REF] Van Lint | On primes in arithmetic progressions[END_REF]:

(10) G(D) ≤ q ϕ(q) G q (D) ≤ G(qD).
We quote from [11, Lemma 3.5] (see also [START_REF] Riesel | On sums of primes[END_REF])

(11) G(D) ≤ log D + 1.4709, (D ≥ 1)
and, concerning a lower bound, (

) log D + 1.06 ≤ G(D), (D ≥ 6). 12
For any given integer r ≥ 1, we define

(13) ν r (n) = f (n)c r (n)   d|n Λ (k 2 ) d   .
We have the following estimate.

Lemma 2.8. Let B ≥ 1 be a real number. Then

r≤ R q µ 2 (r) ϕ(r) n≤B |ν r (n)| 2 n B c log z R q 2 for each c > 0.
Proof. By definition of ν r (n), we have

r≤ R q µ 2 (r) ϕ(r) n≤B |ν r (n)| 2 n = r≤ R q µ 2 (r) ϕ(r) n≤B f 2 (n)|c 2 r (n)| d|n Λ (k 2 ) d 2 n .
Let c > 0 be a real number. Then

r≤ R q µ 2 (r) ϕ(r) n≤B |ν r (n)| 2 n ≤ r≤ R q µ 2 (r)ϕ(r)B c log z n≤B f 2 (n) d|n Λ (k 2 ) d 2 n 1+ c log z . By Lemma 2.3 the last sum is bounded since f 2 (n) ≤ τ 2 k (n) ≤ τ k 2 (n), and hence r≤ R q µ 2 (r) ϕ(r) n≤B |ν r (n)| 2 n ≤ B c log z R q 2 .
The following lemma will be used in Section 6. Lemma 2.9. Let M be a sufficiently large real number and q be a positive integer. Then [START_REF] Riesel | On sums of primes[END_REF] b(mod)q m≡b(q) m∼M

Λ f (m) m iu 2 M 2 ϕ(q) .
Proof. We have

b mod q m≡b(q) m∼M Λ f (m) m iu 2 = b(mod) * q m≡b(q) m∼M Λ f (m) m iu 2 + b mod q (b,q)>1 m≡b(q) m∼M Λ f (m) m iu 2 . ( 15
)
We have b(mod) * q m≡b(q) m∼M

Λ f (m) m iu 2 b(mod) * q p≡b(q) p∼M Λ f (p) p iu 2 + b(mod) * q p t ≡b(q) p t ∼M t≥2 Λ f (m) m iu 2 .
For the first sum we use |Λ f (p)| ≤ k log p after applying Cauchy-Schwarz inequality.

The sum inside the modulus in the second sum is trivially M 1+ in view of (2) . Hence we have * q m≡b(q) m∼M

Λ f (m) m iu 2 M ϕ(q) log M b(mod) * q p≡b(q) p∼M log 2 p + M 1+ ϕ(q) .
Hence by prime number theorem we have b(mod) * q m≡b(q) m∼M

Λ f (m) m iu 2 M 2 ϕ(q)
Similarly we can show that the second sum on the right hand side of ( 15) is

M 2 ϕ(q) .

Bilinear decomposition of Λ f and µf

For any square-free integer r ≥ 1, let

V r (s) = n≥2 ν r (n) n s ,
where ν r (n) be as in [START_REF] Rankin | Contributions to the theory of Ramanujan's function τ (n) and similar arithmetical functions. I. The zeros of the function ∞ n=1 τ (n)/n s on the line Rs = 13/2. II. The order of the Fourier coefficients of integral modular forms[END_REF]. By Lemma 2.3 we can see that ν r (n) = 0 if n ≤ z.

We have

1 + V r (s) = 1 + n≥2 f (n)c r (n) n s   d|n Λ (k 2 ) d   = d≤z 1+k 2 θ Λ (k 2 ) d |r µ r [ , d] s n≥1 f ([ , d]n) n s .
By Lemma 2.1 the above identity becomes

1 + V r (s) = d≤z 1+k 2 θ Λ (k 2 ) d |r µ r [ , d] s µ([ , d]) n≥1 f * f -1 [ ,d] (n) n s =L(f, s) m≥1 h r (m) m s . (16) Let M r (s) = m≥1 hr(m) m s . Write (17) 1 = -V r (s) + (1 + V r (s)) .
Multiplying both sides by -L (f,s) L(f,s) we get

(18) - L (f, s) L(f, s) = L (f, s) L(f, s) V r (s) -L (f, s)M r (s) .
This gives the following decomposition for Λ f (n):

(19) Λ f (n) = -(Λ f * ν r )(n) + (f log * h r )(n) . Define L (f, s) = n≥1 µ(n)f (n) n s = p 1 - f (p) p s .
We can write this L as

(20) L (f, s) = 1 L(f, s) p 1 + h≥2 f (p h ) -f (p h-1 )f (p) p hs . Denoting f (p h ) -f (p h-1 )f (p) by f 2 (p h ), we get that (21) L (f, s) = 1 L(f, s) p 1 + h≥2 f 2 (p h ) p hs := 1 L(f, s) n≥1 g(n) n s .
Multiplying both sides of [START_REF] Van Lint | On primes in arithmetic progressions[END_REF] with L (f, s) gives the following decomposition for µ( )f ( ):

(22) µ( )f ( ) = -(ν r * µf )( ) + (h r * g)( ) .
Now we consider the sum

(23) S(α) = ∼X Λ f ( )e ( α) ,
where ∼ X means X < ≤ 2X. Using (19) we get

S (α) = - ∼X (Λ f * ν r )( )e ( α) + ∼X (f log * h r )( )e ( α) = mn∼X f (n) log(n)h r (m)e (mnα) - mn∼X m≤M 0 Λ f (m)ν r (n)e (mnα) - mn∼X m>M 0 Λ f (m)ν r (n)e (mnα)
for some positive real number M 0 which will be chosen later.

Let

(24)

L (1) r (α) = mn∼X f (n) log(n)h r (m)e (mnα) (25) L (2) r (α) = mn∼X m≤M 0 Λ f (m)ν r (n)e (mnα)
and call these sums as first linear sum and second linear sum respectively. With these notations we have ( 26)

S (α) = L (1) r (α) -L (2) r (α) - mn∼X m>M 0 Λ f (m)ν r (n)e (mnα) .
We call the last sum the bilinear sum. Let R be a sufficiently large real number which we will choose later. Multiplying both sides of (26) by µ 2 (r) ϕ(r) and summing over 1 ≤ r ≤ R q with (r, q) = 1 gives us (27)

G q R q S (α) = r≤ R q (r,q)=1 µ 2 (r) ϕ(r) (L (1) r (α)-L (2) r (α))- r≤ R q (r,q)=1 µ 2 (r) ϕ(r) mn∼X m>M 0 Λ f (m)ν r (n)e (mnα) .

Estimating the first linear sum L

(1) r (α) We have

L (1) r (α) = mn∼X f (n) log(n)h r (m)e (mnα) = m≤2X h r (m) n∼ X m f (n) log(n)e (mnα) .
Use ( 7) and Lemma 2.5 to get

L (1) r (α) X η log X m≤2X |h r (m)| m 1 2 X η r 1+ z (1+k 2 θ)( 1 2 + ) log(X) . ( 28
L (2) r (α) = m≤M 0 mn∼X Λ f (m)   d|n Λ (k 2 ) d   f (n)c r (n)e (mnα) . Since Λ (k 2 ) d = 0 if d > z 1+k 2 θ
, the equation above can be written as

L (2) r (α) = m≤M 0 Λ f (m) d≤z 1+k 2 θ Λ (k 2 ) d n∼ X m d|n f (n)c r (n)e (mnα)
.

By the properties of Ramanujan sum c r (n) it can be written as

L (2) r (α) = m≤M 0 Λ f (m) d≤z 1+k 2 θ Λ (k 2 ) d n∼ X m d|n f (n)e (mnα)
|r, |n µ r

= m≤M 0 Λ f (m) d≤z 1+k 2 θ Λ (k 2 ) d |r µ r n∼ X m d|n, |n f (n)e(mnα)
.

By Lemma 2.1 we get

L (2) r (α) = m≤M 0 Λ f (m) d≤z 1+k 2 θ Λ (k 2 ) d |r µ r n∼ X m[d, ] f (n[ , d])e (mn[d, ]α) = m≤M 0 Λ f (m) d≤z 1+k 2 θ Λ (k 2 ) d |r µ r n∼ X m[d, ] e (mn[d, ]α) µ([d, ]) u|n u|[ ,d] ∞ f n u f -1 ([ , d]u) .
An interchange of summation yields

L (2) r (α) = m≤M 0 Λ f (m) d≤z 1+k 2 θ Λ (k 2 ) d |r µ r µ([ , d]) u|[ ,d] ∞ f -1 ([ , d]u) n∼ X mu[d, ] f (n)e (mnu[d, ]α) .
By our assumption we have

|L (2) r (α)| X η m≤M 0 |Λ f (m)| m η d≤z 1+k 2 θ |Λ (k 2 ) d | |r u|[ ,d] ∞ |f -1 ([ , d]u)| ([ , d]u) η X η m≤M 0 |Λ f (m)| m η d≤z 1+k 2 θ |Λ (k 2 ) d | d η |r u|[ ,d] ∞ |f -1 ([ , d]u)| u η .
We use Lemma 2.2 to estimate the last sum on the right hand side. This gives

|L (2) r (α)| X η m≤M 0 |Λ f (m)| m η d≤z 1+k 2 θ |Λ (k 2 ) d | d η- |r 1+ . Since η ≥ 1 2 , we get (29) |L (2) r (α)| X η r 1+2 z (1+k 2 θ) m≤M 0 |Λ f (m)| m 1 2   d≤z 1+k 2 θ |Λ (k 2 ) d | d 1 2   .
We use the bound (1) to estimate the first sum in the brackets and Lemma 2.4 for the second sum to get

|L (2) r (α)| X η r 1+2 z (1+k 2 θ)( 1 2 + ) M 1 2 + 0 . (30) 6. Estimating the bilinear sum Let (31) S r (α, M, N ) = mn∼X m∼M,n∼N Λ f (m)ν r (n)e (mnα) .
Let δ > 0 be a sufficiently small real number which we will choose later. Let a be such that (a, q) = 1. We apply Lemma 2.6 with β = α -a q and

ϕ = mn= m∼M,n∼N Λ f (m)ν r (n)e mna q to get S r (α, M, N ) = ∆ -∆ m∼M n∼N Λ f (m) m iu ν r (n) n iu e mna q H(u)X iu du (32) + (E 1 (δ, r) + E 2 (δ, r) + 2δE 3 (r)) ,
where

E 1 (δ, r) = X<mn≤2 δ X m∼M,n∼N |Λ f (m)ν r (n)|, (33) E 2 (δ, r) = 2X 2 δ <mn≤2X m∼M,n∼N |Λ f (m)ν r (n)|, (34)
E 3 (r) = m∼M,n∼N |Λ f (m)ν r (n)|, (35) ∆ =100 δ -1 + (βX) 2 δ . (36)
We have the following lemma concerning the error term in (32). Lemma 6.1.

E 1 (δ, r) + E 2 (δ, r) + 2δE 3 (r) δM n∼N |ν r (n)| .
Proof. We have

E 1 (δ, r) = n∼N |ν r (n)| m∼M X n <m≤ 2 δ X n |Λ f (m)| n∼N |ν r (n)| X n ≤m≤ X n + 7δM 5 |Λ f (m)| .
The second inequality follows since the interval [max(M, X n ), min(2M,

2 δ X n )] is con- tained in [ X n , X n + 7δM 5 ].
E 3 (r) = n∼N |ν r (n)| m∼M |Λ f (m)| M n∼N |ν r (n)|
again by prime number theorem.

Consider the sum

S b = r≤ R q (r,q)=1 µ 2 (r) ϕ(r) mn∼X m>M 0 Λ f (m)ν r (n)e (mnα)
.

By using Cauchy Schwarz inequality we get (37)

S b ≤ G q R q      r≤ R q (r,q)=1 µ 2 (r) ϕ(r) mn∼X m>M 0 Λ f (m)ν r (n)e (mnα) 2      1 2
.

Consider the sum inside the brackets and call it S 1 , i.e.

S 1 = r≤ R q (r,q)=1 µ 2 (r) ϕ(r) mn∼X m>M 0 Λ f (m)ν r (n)e (mnα) 2
.

We now examine the last sum and localize the variables m and n. Notice that n > z. So we start at N = z, go until 2z, etc until 2 t z ≤ 2X/M 0 < 2 t+1 z, i.e. 0 ≤ t ≤ log(2X/(M 0 z))/ log 2. Concerning M , we have N < n ≤ N ≤ 2N , and thus 1 2 (X/N ) ≤ X/n < m ≤ 2X/N . So for each N , we have two values of M , namely M 1 = 1 2 (X/N ) and M 2 = X/N .

After localizing the variables and applying Cauchy-Schwarz, we reach

S 1 log 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) M,N mn∼X m∼M,n∼N Λ f (m)ν r (n)e (mnα) 2 .
Using Equation (32) for the sum inside the modulus we get

S 1 log 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) M,N ∆ -∆ m∼M n∼N Λ f (m) m iu ν r (n) n iu e mna q H(u)X iu du 2 + log 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) M,N (E 1 (δ, r) + E 2 (δ, r) + 2δE 3 (r)) 2 .
By Lemma 6.1 the second term is

log 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) M,N δM n∼N |ν r (n)| 2 .
Hence we have (38)

S 1 log 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) M,N ∆ -∆ b(q) m≡b(q) m∼M Λ f (m) m iu n∼N ν r (n) n iu e abn q H(u)X iu du 2 + log 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) M,N δM n∼N |ν r (n)| 2 .
We consider the integral I( a q , M, N ) in the above equation and after an application of Cauchy -Schwarz inequality, we obtain

I a q , M, N ∆ -∆    b(q) m≡b(q) m∼M Λ f (m) m iu 2    1 2   b(q) n∼N ν r (n) n iu e bn q 2   1 2 |H(u)|du .
Since the sum in the first bracket inside the integral is M 2 ϕ(q) by Lemma 2.9, we get

I a q , M, N M 2 ϕ(q) 1 2 ∆ -∆   b(q) n∼N ν r (n) n iu e bn q 2   1 2 |H(u)|du .
We use the Cauchy-Schwarz inequality again to get

I a q , M, N M 2 ϕ(q) 1 2 ∆ -∆ |H(u)| 2 du 1 2   ∆ -∆ b(q) n∼N ν r (n) n iu e bn q 2 du   1 2
.

Using the fact that R |H(u)| 2 = R | H(u)| 2 = (2 -2δ) (log 2) 2 (4π) 2 leads to I a q , M, N M 2 ϕ(q) 1 2   ∆ -∆ b(q) n∼N ν r (n) n iu e bn q 2 du   1 2
.

Put this estimate in (38) to get

S 1 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) M,N M 2 ϕ(q) ∆ -∆ b(q) n∼N ν r (n) n iu e bn q 2 du + log 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) M,N δM n∼N |ν r (n)| 2 .
An application of large sieve inequality (Lemma 2.7) gives

S 1 log 2X M 0 z M,N M 2 ϕ(q)   n∼N f 2 (n) d|n Λ (k 2 ) d 2 (n + R 2 4π∆ log 2 )   (39) + log 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) M,N δM n∼N |ν r (n)| 2 .
We assume that R 2 δ -2 z and |α -a q | (zR -2 δ)

1 2 X so that R 2 ∆ z. Hence S 1 log 2X M 0 z M,N M 2 N 2 ϕ(q)   n∼N f 2 (n) n d|n Λ (k 2 ) d 2   + log 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) M,N δ 2 M 2 N n∼N |ν r (n)| 2 .
Let c > 0 be a small real number. Since M N X we get

S 1 X 2 ϕ(q) log 2X M 0 z 2X z c log z   n≤ 2X z f 2 (n) n 1+ c log z d|n Λ (k 2 ) d 2   + δ 2 X 2 log 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) n≤ 2X z |ν r (n)| 2 n . Since |f 2 (n)| ≤ τ 2 k (n) and τ 2 k (n) ≤ τ k 2 (n), Lemma 2.3 gives us S 1 X 2 ϕ(q) log 2X M 0 z 2X z c log z + δ 2 X 2 log 2X M 0 z r≤ R q (r,q)=1 µ 2 (r) ϕ(r) n≤ 2X z |ν r (n)| 2 n .
By Lemma 2.8 we have

(40) S 1 X 2 ϕ(q) log 2X M 0 z 2X z c log z + δ 2 X 2 log 2X M 0 z 2X z c log z R q 2 .
On keeping this bound in (37), we obtain

(41) S b ≤ G q R q log 2X M 0 z X 2 ϕ(q) 2X z c log z + δ 2 X 2 2X z c log z R q 2 1 2
. Putting (28), (30), ( 41) and (27) together gives (with θ = 1/k 2 )

G q R q S (α) X η z (1+1)( 1 2 + ) G q R q R q 1+2 (log X + M 1 2 + 0 ) + X ϕ(q) G q R q log 2X M 0 z 2X z c log z 1 + δ 2 ϕ(q) R q 2 1 2
.

We choose δ -1 = R = z (1-η) to get

S (α) X η+ 1-η 2 (1+ (2+ 1 2 +1)) 1 q + X ϕ(q) log X G q R q
.

For sufficiently small positive , the power of X in the first sum will be smaller than 1 and for the second sum we use [START_REF] Motohashi | Sieve Methods and Prime Number Theory[END_REF] and ( 12) (with q ≤ R 14/15 = X (1-η)/15 ) together to get

S (α) √ qX ϕ(q) .
This completes the proof of Theorem 1.4.

7.

Proof of Theorem 1.5

Since the proof of Theorem 1.5 is similar to that of Theorem 1.4, we only give a sketch of the proof. We use |g(n)| log 2 n, g is supported only on square-full integers and the number of such integers upto X is X 1/2 while estimating the sum L

r . We can get L (1) r (α) X 1 2 r 1+ z (1+k 2 θ)( 1 2 + ) log 2 X .

The sum L

|L (2) r (α)| X η r 1+2 z (1+k 2 θ)( 1 2 + ) M 1 2 + 0 . ( 43
)
We can estimate the third sum on the right hand side as in Section 6 with some modifications which we will write down here. The corresponding bound for the bound in Lemma 6.1 will be E 1 (δ, r) + E 2 (δ, r) + 2δE 3 (r) W f (X)δM n∼N |ν r (n)| , when using [START_REF] Lao | Mean square estimates for coefficients of symmetric power L-functions[END_REF] . Instead of Lemma 2.9 one has to use (44) b(mod)q m≡b(q) m∼M f (m)

m iu 2 W f (X)M 2 q .
With these changes and choosing parameters exactly as in Section 6, we get the mentioned result.

Lemma 2 . 3 (

 23 Motohashi, Theorem 4,[START_REF] Motohashi | Sieve Methods and Prime Number Theory[END_REF]). The weights Λ (k ) d satisfy the following:

) 5 .

 5 Estimating the second linear sum L

1 4

 1 to ensure that R 2 δ -2 z and M 0 = z = X 2 7

  By (22), we have the following decomposition:(42) ∼X µ( )f ( )( )e( α) = L (1) r (α) -L (2) r (α) -mn∼X m>M 0 µ(m)f (m)ν r (n)e(mnα) )f (m)ν r (n)e(mnα).

( 2 )

 2 r (α) can be estimated exactly as in Section 5, where one has to use the bound follows from the bound |f (m)| m .

  The contribution from primes for the second sum of the above equation is δM since |Λ f (p)| ≤ k log p and by prime number theorem. It can be easily seen that the contribution from the higher prime powers is negligible.

	Hence we have
	E 1 (δ, r)	δM .
	Similarly we can show that
	E 2 (δ, r)	δM .
	The result follows since