Pedro S F Mendes

André F C Gregório

Antoine Daudin

Christophe Bouchy

João M Silva

M Filipa Ribeiro
email: filipa.ribeiro@tecnico.ulisboa.pt

Elucidation of the zeolite role on the hydrogenating activity of Pt-catalysts

Keywords: platinum, zeolite, bifunctional catalyst, nanoscale intimacy, toluene hydrogenation

Toluene hydrogenation was studied over model catalysts with a fixed hydrogenating function (Pt/Al 2 O 3 ) mechanically mixed with either HUSY or HBEA zeolites. Such mechanical mixtures showed improved platinum turnover frequencies compared to single Pt/Al 2 O 3 . For a same Pt content, similar activities were obtained for both zeolites catalysts indicating that the contribution of acid sites is fairly independent of the zeolite properties.

Comparing to Pt-impregnated zeolite catalysts, the lower activities per site confirmed the detrimental effect of distance between Pt clusters and acid sites. Therefore, toluene hydrogenation can be used as a powerful tool to evaluate Pt-acid site intimacy in bifunctional catalysts.

Introduction

Bifunctional catalysts, namely Pt/zeolite, play a key role in modern conversion processes in industry [START_REF] Bouchy | Fischer-Tropsch Waxes Upgrading via Hydrocracking and Selective Hydroisomerization[END_REF][START_REF] Hanaoka | Jet fuel synthesis from Fischer-Tropsch product under mild hydrocracking conditions using Pt-loaded catalysts[END_REF][START_REF] Iliopoulou | Producing high quality biofuels: Pt-based hydroisomerization catalysts evaluated using BtL-naphtha surrogates[END_REF] and most of the aspects that affect their performances have been studied along the last decades [START_REF] Weitkamp | Catalytic Hydrocracking-Mechanisms and Versatility of the Process[END_REF]. For hydrocracking catalysts, special attention hasve been dedicated to the metal-acid balance [START_REF] Alvarez | Hydroisomerization and hydrocracking of alkanes -Influence of the balance between acid and hydrogenating functions on the transformation of n-decane on PtHY catalysts[END_REF] and the degree of intimacy between both functions [START_REF] Batalha | n-Hexadecane hydroisomerization over Pt-HBEA catalysts. Quantification and effect of the intimacy between metal and protonic sites[END_REF] which, nevertheless, are criteria difficult to quantify.

The difficulty of understanding fundamental phenomena occurring in complex catalytic systems can be overcome by the use of model reactions. Namely, the hydrogenation of onering aromatic compounds has been used to evaluate the accessibility and location of the metal sites [START_REF] Bond | Hydrogenation of the Aromatic Ring[END_REF][START_REF] Chupin | Influence of the metal and of the support on the activity and stability of bifunctional catalysts for toluene hydrogenation[END_REF]. On the other hand, in the presence of acid sites (i.e. in a bifunctional catalyst) the activity can also be influenced by a support effect [START_REF] Lin | Toluene Hydrogenation Over Supported Platinum Catalysts[END_REF]. In most of the cases, the increased hydrogenation activity obtained over acidic supports has been attributed to the additional sites available to reactant adsorption that can be hydrogenated by spilled-over hydrogen (H sp ) [START_REF] Lin | Toluene Hydrogenation Over Supported Platinum Catalysts[END_REF][START_REF] Lin | Hydrogenation of Aromatic Hydrocarbons over Supported Pt Catalysts .III. Reaction Models for Metal Surfaces and Acidic Sites on Oxide Supports[END_REF]. The modest sum of studies published on zeolite-based bifunctional catalysts lack systematization of clear properties-activity relationships, due to either disregard for Pt dispersion [START_REF] Wang | Acid function of Al-MCM-41 supported platinum catalysts in hydrogenation of benzene, toluene and o-xylene[END_REF][START_REF] Wang | The effect of metal-acid balance in Pt-loading dealuminated Y zeolite catalysts on the hydrogenation of benzene[END_REF][START_REF] Wang | Influence of the Bronsted and Lewis acid sites in platinum/solid acid catalysts on the hydrogenation of benzene and toluene[END_REF] or by holding the focus on other questions as deactivation mechanisms [START_REF] Chupin | Influence of the metal and of the support on the activity and stability of bifunctional catalysts for toluene hydrogenation[END_REF]. To fully use the potential of this model reaction on the advanced characterization of bifunctional catalysts, a background understanding of the support characteristics that affect hydrogenating activity is required.

In our previous work [START_REF] Mendes | Hydrogenating activity of Pt/zeolite catalysts focusing acid support and metal dispersion influence[END_REF], the toluene hydrogenation was studied over Pt/zeolite catalysts based on HUSY and HBEA zeolites which are widespread in industrial catalysts. The TOF per Pt site was observed to differ by a 6-fold depending on both metal content and the support. Acknowledging the role of zeolite acidity, catalysts were evaluated by their activity per total adsorbing (metal + acid) sites which was observed to increase rapidly with n Pt /(n Pt +n A ). A unique relationsihip between these two parameters was obtained for two different series of Pt/HUSY catalysts showing that the contribution of the acid sites could be well accounted by the total acidity. Comparing HBEA and HUSY catalysts, it was suggested that the proximity between Pt clusters and acid sites significantly impacted the hydrogenating activity of the catalysts. However, it precluded the direct evaluation of the zeolite intrinsic properties on activity.

In this work, we complete our comprehensive study on properties-activity relationships by addressing the role of the zeolite on the hydrogenating activity. Therefore, toluene hydrogenation was studied over powder model catalysts with a fixed hydrogenating function (Pt/Al 2 O 3 ) mixed with either HUSY or HBEA zeolites and the results were compared with those of HUSY and HBEA zeolites directly impregnated with Pt.

Experimental

Commercial zeolites HUSY (CBV720) and NH 4 -BEA (CP814E) were supplied by Zeolyst with a global Si/Al atomic ratio of 16.7 and 13.6 respectively. NH 4 -BEA zeolite was calcined to obtain the protonic form using a protocol published elsewhere [START_REF] Marques | Semiquantitative estimation by IR of framework, extraframework and defect Al species of HBEA zeolites[END_REF] Nitrogen adsorption measurements were carried out at 77 K. Total acidity was calculated from IR spectra of thermo desorbed samples at 423 K, after pyridine adsorption. Accessible

Pt was determined by hydrogen titration of chemisorbed oxygen on Pt/Al 2 O 3 . CO chemisorption experiments were performed at room temperature, until saturation. All the details were published in our previous paper [START_REF] Mendes | Hydrogenating activity of Pt/zeolite catalysts focusing acid support and metal dispersion influence[END_REF]. Particle size was determined by laser granulometry using the Fraunhofer approximation with a Mastersizer 3000 (from Malvern Instruments) equipped with dry particle dispersion.

The toluene hydrogenation was performed at 383 K under atmospheric pressure with a hydrogen to toluene molar ratio of 45 over pre-reduced catalysts. Initial conversion for each catalyst was kept lower than 15% by changing the toluene space velocity. Catalytic initial activity was calculated as the toluene consumption (mmol) per catalyst gram per hour at the earliest reaction time possible (t=1 min). All the details were published in our previous paper [START_REF] Mendes | Hydrogenating activity of Pt/zeolite catalysts focusing acid support and metal dispersion influence[END_REF].

3 Results and discussion

Physico-chemical properties of solids

The structural and textural properties of the three solids are depicted in Table 2. Pt/Al 2 O 3 is fully mesoporous whereas zeolites are both micro and mesoporous. HBEA zeolite has a higher volume of mesoporous volume than HUSY, but lower microporous volume. Results in Table 2 show that HBEA zeolite exhibits two times the acid sites of HUSY (572 vs. 289 μmol/g) and six times the total acid sites of Pt/Al 2 O 3 . Both zeolites report similar concentration of BAS (200 -220 µmol/g) and so the main difference is the 4-fold higher LAS concentration of HBEA (350 vs. 90 μmol/g).

Concerning adsorption of CO followed by FTIR, IR bands in the region 2150 -2000 cm - 1 result from CO stretching mode when linearly adsorbed onto Pt 0 surface atoms [START_REF] Hadjiivanov | Characterization of oxide surfaces and zeolites by carbon monoxide as an IR probe molecule[END_REF] which for Pt/Al 2 O 3 is a single band centred at 2094 cm -1 . For Pt supported in the same zeolites used in this study, the contribution of Pt 0 is centred at 2097 cm -1 [START_REF] Mendes | Hydrogenating activity of Pt/zeolite catalysts focusing acid support and metal dispersion influence[END_REF] and so, in our case of study, Pt clusters have similar electronic density. The mean particle size of Pt clusters on Al 2 O 3 estimated through metal dispersion is around 2.2 nm which fits within the same range of particle size for Pt dispersed on HUSY and HBEA zeolites estimated before: 1.8-3.0 nm [START_REF] Mendes | Hydrogenating activity of Pt/zeolite catalysts focusing acid support and metal dispersion influence[END_REF].

Toluene hydrogenation

Catalysts were obtained by intimate mixture of HUSY or HBEA zeolite with Pt/Al 2 O 3 in different proportions (see Table 1). The composition of the mixtures were was chosen to cover a range of Pt/zeolite weight ratio from 0.001 to 0.007 2 as used in a previous work [START_REF] Mendes | Hydrogenating activity of Pt/zeolite catalysts focusing acid support and metal dispersion influence[END_REF]. Catalytic results are summarized on Table 3. Focusing the discussion on the catalytic initial activities (at t=1 min), the turnover frequencies of platinum (TOF Pt ) for the all set of catalysts vary from 800 to 1700 h -1 .

Zeolite-based catalysts have significantly higher TOF Pt than Pt/Al 2 O 3 individual catalyst and so improved activity per metal site. In literature, the enhancement of one-ring aromatics hydrogenation activity has been related to the presence of acid sites through the hydrogenation of reactant molecules that are adsorbed on these additional acidic sites by spilled over hydrogen (H sp ) [START_REF] Lin | Toluene Hydrogenation Over Supported Platinum Catalysts[END_REF][START_REF] Lin | Hydrogenation of Aromatic Hydrocarbons over Supported Pt Catalysts .III. Reaction Models for Metal Surfaces and Acidic Sites on Oxide Supports[END_REF]. In our previous work on Pt/zeolite catalysts [START_REF] Mendes | Hydrogenating activity of Pt/zeolite catalysts focusing acid support and metal dispersion influence[END_REF], TOF Pt varied by 6-fold depending on both metal content and zeolite structure. To account the contribution of the acid sites, it was suggested to consider the activity per total number of adsorbing sites. This activity was shown to increase rapidly with the ratio of hydrogenating to adsorbing sites, n Pt /(n Pt +n A ).

Concerning the deactivation rate, all catalysts exhibit important deactivation which has been reported in literature as being due to the plugging of micropores by "coke" molecules derived from toluene [START_REF] Chupin | Influence of the metal and of the support on the activity and stability of bifunctional catalysts for toluene hydrogenation[END_REF]. In the same way, the lower deactivation of HBEA-based catalysts could be due by the higher percentage of mesoporous volume compared to HUSY ones.

Effect of Pt location

Figure 1 shows the evolution of the activity per adsorbing site with n Pt /(n Pt +n A ) for Pt/HUSY, Pt/HBEA, Pt/Al 2 O 3 -HUSY and Pt/Al 2 O 3 -HBEA. For a given ratio, the activity per adsorbing site of Pt-impregnated zeolite catalysts is higher than that of Pt/Al 2 O 3 mixed with zeolite and its activity is also more sensitive to n Pt compared to zeolite series mixed with Pt/Al 2 O 3 .

The comparison in Figure 1 accounts the differences on catalysts in terms of total acidity by n A and Pt dispersion by n Pt . In the same way, Pt clusters have been shown to be similar in all catalyst series regarding electronic properties and particle size. Thus, the differences on the activities per adsorbing site cannot be explained by Pt and acid sites intrinsic properties.

As a matter of fact, a pronounced difference between catalysts, that is not being accounted, is the location of Pt: in impregnated catalysts, the Pt is directly dispersed on the zeolite whereas, in mechanical mixtures, the zeolite is Pt-free. Taking into account the dispersion of Pt on HUSY and HBEA, a maximum Pt-acid site distance can be estimated between 30 to 70 nm, by assuming an homogeneous Pt distribution inside the zeolites. On the contrary, for a mechanical mixture composed mainly by zeolite, the maximum Pt-acid site distance can be estimated by Pt/Al 2 O 3 granulometry. For our sample, the (volume weighted) mean diameter mean is 57 μm. Therefore, direct impregnation of Pt on zeolites provided a Pt-acid site distance in the nanoscale whereas only a mesoscale intimacy could be obtained by mechanical mixture.

The detrimental effect of Pt-acid site distance on the activity per adsorbing site agrees with the hypothesis formerly formulated that the rate of H sp supplied to the adsorbing sites is the limiting step on the contribution of acid sites to hydrogenation activity [START_REF] Mendes | Hydrogenating activity of Pt/zeolite catalysts focusing acid support and metal dispersion influence[END_REF]. The higher inter-site distance leads to a longer diffusion pathway, diminishing the rate of H sp supplied to the acid sites and, consequently, the activity per acid site. In the same way, the proximity between Pt clusters and OH groups in HBEA impregnated catalysts was proposed to be also the key parameter to the their higher activity for similar n Pt /(n Pt +n A ) comparing to HUSY ones [START_REF] Mendes | Hydrogenating activity of Pt/zeolite catalysts focusing acid support and metal dispersion influence[END_REF].

Zeolite role on Pt/Al 2 O 3 -based catalysts

We established that the contribution of acid sites is much lower when the Pt-acid site distance is increased, but, nonetheless, zeolites contribute to the hydrogenating activity of Pt sites deposed on Al 2 O 3 . This can be verified by the significantly higher TOF Pt of Pt/Al 2 O 3 -zeolite catalysts comparing to Pt/Al 2 O 3 alone (see Table 3), i.e. similar hydrogenating activity can be attained for lower n Pt when a zeolite is mixed with Pt/Al 2 O 3 (see Figure 2). Focusing on Pt/Al 2 O 3 -HUSY and Pt/Al 2 O 3 -HBEA catalysts, a unique evolution on the activity as function of n Pt seems to occur, irrespectively of the zeolite (see Figure 2). This means that a constant TOF Pt is obtain for Pt/Al 2 O 3 mixed with zeolite which is independent of the n Pt /(n Pt +n A ), contrarily to the results for Pt/zeolite catalysts.

Based on these results, it seems that, when the zeolite is separated from the hydrogenating function (Pt/A 2 lO 3 ), the maximum enhancement of the hydrogenating activity can be achieved for lower n Pt /(n Pt +n A ) ratios. This is an indication that only a part of the acid sites is playing a role, due to the longer H sp diffusion pathway, which can be corroborated by the activities per adsorbing site. Those activities are 1.2 to 5 times lower than those of Pt/zeolite catalysts, at similar hydrogenating to adsorbing sites ratio (see Figure 1). The thermodynamics of H sp diffusion may also amplify the role of inter-site distance: an intrinsic lower H sp diffusion rate on Al 2 O 3 can be expected comparing to that in zeolites.

The diffusion of H sp on insulating materials, like Al 2 O 3 and zeolites, is only possible if the support has defects [START_REF] Prins | Hydrogen Spillover. Facts and Fiction[END_REF]. However, in the case of zeolites, the presence of surface hydroxyls facilitate the surface migration of activated hydrogen [START_REF] Im | Maximizing the catalytic function of hydrogen spillover in platinum-encapsulated aluminosilicates with controlled nanostructures[END_REF].

As the key parameter limiting hydrogenating activity is Pt-acid site intimacy, which does not depend on the zeolite, HUSY and HBEA have a similar contribution to activity, despite their different textural and acidic properties. Furthermore, in such conditions, one may presume that only those acid sites which are near the interface of zeolite with Pt particles have an important contribution, due to its reduced Pt-acid site distance, as proposed before in literature [START_REF] Lin | Hydrogenation of Aromatic Hydrocarbons over Supported Pt Catalysts .III. Reaction Models for Metal Surfaces and Acidic Sites on Oxide Supports[END_REF].

Conclusion

Toluene hydrogenation was studied over catalysts based on Pt/Al 2 O 3 mechanically mixed or not with either HUSY or HBEA zeolites.

The turnover frequency of platinum is significantly higher for intimate mixtures of zeolites with Pt/Al 2 O 3 comparing to Pt/Al 2 O 3 individual catalyst. This improved activity per metal site was proposed to be caused by the hydrogenation of reactant that is adsorbed on these additional acidic sites through spilled over hydrogen (H sp ).

The mixture of either HBEA or HUSY zeolites with Pt/Al 2 O 3 showed a unique evolution on the activity with n Pt and so a similar contribution of both zeolites to hydrogenating activity. Therefore, for a fixed Pt-acid site distance, the relevance of textural and acidic properties is low when the acid sites contribution is limited by the H sp supply rate.

Comparing to Pt/zeolite impregnated catalysts, the activity per adsorbing site is lower when the Pt is deposed on Al 2 O 3 , for a same metal to adsorbing sites ratio. This is proposed to be due to a detrimental effect of longer Pt-acid site distances on the rate of H sp with which acid sites of zeolites are supplied.

In summary, the hydrogenating activity of zeolite-based Pt catalysts seems to be mostly a function of the accessible metal atoms, the total acid sites and the distance between Pt clusters and acid sites. In particular, the contribution of acid sites decreases with increased Pt-acid site distances. Therefore, the toluene hydrogenation is specially fitted to establish the degree of intimacy of bifunctional catalysts.

Figure 1 :

 1 Figure 1: Activity per total adsorbing sites as function of n Pt /(n Pt +n A ) for HBEA ( ) and HUSY ( ) zeolites mixed with Pt/Al 2 O 3 in toluene hydrogenation. Pt/HUSY ( ) and Pt/HBEA ( ) series were published elsewhere [14].

Figure 2 :

 2 Figure 2: Catalytic activity as function of accessible Pt sites for Pt/Al 2 O 3 -HBEA ( ), Pt/Al 2 O 3 -HUSY ( ) and Pt/Al 2 O 3 ( ) catalysts in toluene hydrogenation.

Table 1 :

 1 . Commercial Pt/Al 2 O 3 in powder form was supplied by Sigma-Aldrich with 0.8% Pt measured by ICP-MS. Catalysts were prepared by manual milling of Pt/Al 2 O 3 with either HBEA or HUSY zeolites in different proportions (see Table 1) resulting in an intimate mixture of both powders with a particle size under 200 μm. Catalyst composition on Pt/Al 2 O 3 and zeolite, overall Pt content and Pt/zeolite ratio.

	Catalyst

series Pt/Al2O3 (wt.%) Zeolite (wt.%) Pt content (wt.%) Pt/zeolite (10 -2 wt/wt) Pt/Al2O3 100 - 0.8 - Pt/Al2O3 - HUSY

		11	89	0.09	0.10
		33	67	0.26	0.39
		46	54	0.37	0.69
		11	89	0.09	0.10
	Pt/Al2O3 -HBEA	33	67	0.26	0.39
		46	54	0.37	0.69

Table 3 :

 3 Pore volumes by N 2 adsorption, concentrations of Brønsted (BAS) and Lewis (LAS) acid sites by pyridine thermodesorption followed by FTIR, Pt dispersion by H 2 -O 2 titration and wavenumber of the FTIR band for Pt 0 CO species.

	Sample	Vmicro (mL g -1 )	Vmeso (mL g -1 )	nBAS (µmol g -1 )	nLAS (µmol g -1 )	nA (µmol g -1 )	D (%)	nPt (µmol g -1 )	νCOmax (cm -1 )
	Pt/Al2O3	0.01	0.44	-	96	96	44	18.1	2094
	HBEA	0.19	0.53	220	352	572	-	-	-
	HUSY	0.33	0.20	202	87	289	-	-	-

Table 3 :

 3 Hydrogenating to adsorbing sites ratio, activity, turnover frequency and activity per total adsorption sites at t=1 min, and deactivation rate at t=60 min.

	Catalyst series	Pt content	nPt/ (nPt+nA)	A (mmol h -1 g	-1 )	TOFPt (10 3 h -1 )	A/(nPt+nA) (h -1 )	Deactivation @ 60 min (%)
		(wt.%)
	Pt/Al2O3	0.8	0.158	15.0		0.83	131	60
		0.09	0.008	1.8		0.86	7	90
	Pt/Al2O3 -HUSY	0.26 0.37	0.026 0.040	9.8 14.1		1.65 1.69	42 68	83 92
		0.09	0.004	3.0		1.48	6	77
	Pt/Al2O3 -HBEA	0.26 0.37	0.015 0.024	10.3 12.0		1.67 1.40	25 34	52 43

Acknowledgments

Authors would like to thank M. Rivallan and E. Soyer for all the work performed on FTIR characterization techniques and Fundação para a Ciência e Tecnologia for financial support through CQE (Project UID/QUI/00100/2013) and PhD grant (SFRH/BD/87927/2012).


Cécile Dartyge

András Sárközy

On pseudo-random subsets of the set of the integers not exceeding N

Keywords: pseudo-random, subset. 2000 Mathematics Subject Classification: 05A05, 11Z05

The notion of pseudo-randomness of subsets of {1, 2, . . . , N } is defined, and the measures of pseudo-randomness are introduced. Then three special examples are studied. In two cases it turns out that the subset in question possesses strong pseudo-random properties, while the third example is a negative one.

Introduction

In many applications (cryptography, simulation, etc.) we need a random subset R of the positive integers not exceeding a certain fixed integer N . (E. g. in [START_REF] Mauduit | Construction of pseudorandom binary sequences by using the multiplicative inverse[END_REF] and [START_REF] Sárközy | On pseudorandomness in families of sequences derived from the Legendre symbol[END_REF] large families of finite binary sequences with strong pseudo-random properties are presented with potential use in cryptography, and in both constructions we start out from polynomials f (n) of form f (n) = a∈A (n -a) where A is a random subset of a given size of {1, 2, . . . , p} for some prime p.) In most cases we replace the random subset by a pseudo-random (=PR) subset, i. e., by a subset which is of "random type", which "looks random", and which has been constructed by a suitable algorithm. But when is a subset a "good" PR subset, when it is "of random type"? Clearly, the subset R which consists of the even integers not exceeding N , or, in case of subsets containing half of the elements of the given set, subset R which never contains n, n + 1 and n + 2 simultaneously cannot be considered as a "good" PR subset; if we end up with such a subset then it must be discarded. Note that in both examples the special "non-random type" structure is related to the ordering of the integers and, indeed, the starting point of the study of pseudo-randomness of subsets of finite ordered sets must be their ordering. Clearly, it suffices to study subsets of {1, 2, . . . , N }, the study of subsets of other finite ordered sets can be reduced to this case. So we will study properties of subsets of {1, 2, . . . , N } which are related to the ordering of the integers. Clearly, to do this, we will have to use number theoretic tools intensively.

In the last 20 years numerous papers have been written on random structures. In particular, random (Bollobás [START_REF] Bollobás | Random Graphs[END_REF]), pseudo-random (Thomason [START_REF] Thomason | Pseudo-random graphs[END_REF], [START_REF] Thomason | Random graphs, strongly regular graphs and pseudo-random graphs[END_REF]) and quasi-random graphs (Chung, Graham and Wilson [START_REF] Chung | Quasi-random graphs[END_REF], [START_REF] Chung | Quasi-random graphs[END_REF], Simonovits and T. Sós [START_REF] Simonovits | Szemerédi partitions and quasi-randomness[END_REF]), pseudo-random (Haviland and Thomason [START_REF] Haviland | Pseudo-random hypergraphs[END_REF], [START_REF] Haviland | On testing the "pseudo-randomnes" of a hypergraph[END_REF], [START_REF] Thomason | Pseudo-random graphs[END_REF], [START_REF] Thomason | Random graphs, strongly regular graphs and pseudo-random graphs[END_REF]) and quasi-random hypergraphs (Chung and Graham [START_REF] Chung | Quasi-random classes for hypergraphs[END_REF], [START_REF] Chung | Quasi-random hypergraphs[END_REF], Kohayakawa, Rödl and Skokan, [START_REF] Kohayakawa | Hypergraphs, quasi-randomness and conditions for regularity[END_REF]), quasi-random set systems (Chung and Graham [START_REF] Chung | Quasi-random set systems[END_REF]), quasi-random subsets of Z n (Chung and Graham [START_REF] Chung | Quasi-random subsets of Z n[END_REF]) are studied. In these papers typically structures without ordering are considered, correspondingly, combinatorial tools dominate. The study of quasi-random subsets of Z n [START_REF] Chung | Quasi-random subsets of Z n[END_REF] is closest to our subject, we will return to this question in a subsequent paper.

Pseudo-randomness of finite binary sequences has also been studied intensively, mostly in connection with cryptography (see e. g., [START_REF] Menezes | Handbook of Applied Cryptography[END_REF]); since the elements of sequences are ordered, thus this field is closer to our subject. Usually the pseudo-randomness of algorithms generating sequences ("pseudo-random generators") is studied and not that of individual sequences, and the tools of computational complexity are used (see, e. g. [START_REF] Goldwasser | Mathematical Foundations of Modern Cryptography: Computational Complexity Perspective[END_REF]). This approach has certain limitations and weak points which were analyzed in [START_REF] Sárközy | On finite pseudorandom binary sequences and their applications in cryptography[END_REF]. Thus in [START_REF] Mauduit | On finite pseudorandom binary sequences, I. Measure of pseudorandomness, the Legendre symbol[END_REF] Mauduit and Sárközy proposed another, more constructive approach. Later this approach was used and extended in numerous papers (see the survey paper [START_REF] Sárközy | On finite pseudorandom binary sequences and their applications in cryptography[END_REF]). In this field the number theoretic tools dominate (but combinatorial, probabilistic, algebraic and analytic tools are also used). In this paper our goal will be to show that the tools introduced in [START_REF] Mauduit | On finite pseudorandom binary sequences, I. Measure of pseudorandomness, the Legendre symbol[END_REF] and extended by Hubert and Sárközy in [START_REF] Hubert | On p-pseudorandom binary sequences[END_REF] can be adapted easily to study the pseudo-randomness of subsets of {1, 2, . . . , N }. First in section 2 we will introduce the measures of pseudo-randomness of subsets of {1, 2, . . . , N }. In the rest of the paper we will study special examples; both positive and negative examples will be presented.

2. The measure of pseudo-randomness of subsets of {1, 2, . . . , N }.

In [START_REF] Mauduit | On finite pseudorandom binary sequences, I. Measure of pseudorandomness, the Legendre symbol[END_REF] Mauduit and Sárközy introduced the following measures of pseudo-randomness of finite binary sequences.

Consider a finite binary sequence

(2•1) E N = (e 1 , . . . , e n ) ∈ {-1, 1} N .
Then the well-distribution measure of E N is defined as

(2•2) W (E N ) = max a,b,t t-1 j=0 e a+jb ,
where the maximum is taken over all a, b, t ∈ N such that 1 a a + (t -1)b N , while the correlation measure of order k of E N is defined as

(2•3) C k (E N ) = max M,D M n=1 e n+d 1 e n+d 2 • • • e n+d k ,
where the maximum is taken over all D = (d 1 , . . . , d k ) and M such that O d 1 < . . . < d k N -M . Then the sequence is considered as a "good" pseudo-random sequence if both these measures W (E N ) and C k (E N ) (at least for "small" k) are "small" in terms of N (in particular, both are o(N ) as N → ∞.) Indeed Cassaigne, Mauduit and Sárközy [START_REF] Cassaigne | On finite pseudorandom binary sequences VII: the measure of pseudorandomnes[END_REF] showed that for a "truly random" E N ∈ {-1, +1} N , both W (E N ) and, for fixed k, C k (E N ) are around N 1/2 with "near 1" probability (see also [START_REF] Alon | Measure of pseudorandomnes for finite sequences: typical values[END_REF]). Thus for a "really good" PR sequence we expect the measures (2•2), (2•3) to be not much greater than N 1/2 . The pseudo-randomness of a sequence of form (2•1) can be interpreted in the following way: suppose ξ is a random variable distributed according to the law (2•4) P (ξ = 1) = P (ξ = -1) = 1 2 ; e. g., we obtain such a ξ if we toss a coin and put +1 if it shows head, -1 if it is tail. Now suppose that ξ 1 , ξ 2 , . . . , ξ N are independent random variables each of distribution (2•4), e. g., we toss the coin N times. We record the outcome of each coin toss, and let e i denote the value of ξ, i. e., e i = +1 if the i-th coin toss is head and e i = -1 if it is tail. In this way we get a binary sequence of form (2•1). The question is: what can we say about a "typical"

sequence (e 1 , . . . , e N ) obtained in this way, what are its most important properties? We say "important" in two senses: important in the applications, and also important in the sense that our sequence possesses it with probability "near 1". Definitions (2•2) and (2•3), and the theorems of Cassaigne, Mauduit and Sárközy described above can be considered as the answer to these questions. Hubert and Sárközy [START_REF] Hubert | On p-pseudorandom binary sequences[END_REF] generalised this model and this notion of pseudo-randomness in the following way.

Replace ξ in (2•4) by a random variable which again may assume two values only, but now they are not equally probable. Suppose they occur with probability p, resp. 1 -p, and by technical reasons, let ξ be defined so that its mean value is 0. In other words, replace (2•4) by, say, (2•5)

P (ξ = 1 -p) = p, P (ξ = -p) = 1 -p,
so that, once more, the expected value M (ξ) verifies

(2•6) M (ξ) = 0.
Then by (2•6), one may define the notion of p-pseudo-randomness (pseudo-randomness with respect to the distribution in (2•5)) again by formulas (2•2) and (2•3). In other words, Hubert and Sárközy [START_REF] Hubert | On p-pseudorandom binary sequences[END_REF] define the notion of p-pseudo-randomness in the following way. Consider a finite binary sequence

E N = (e 1 , . . . , e N ) ∈ {1 -p, -p} N .
Then the p-well-distribution measure of E N is defined as

(2•7) W (E N , p) = max a,b,t t-1 j=0 e a+jb ,
while the p-correlation measure of order k of E N is defined as

(2•8) C k (E N , p) = max M,D M n=1 e n+d 1 e n+d 2 • • • e n+d k .
(The maximum in (2•7) and (2•8) is taken in the same way as in (2•2) and (2•3), respectively.) Then again the sequence is considered as a "good" p-pseudo-random sequence if both these measures W (E N , p) and C k (E N , p) (at least for "small " k) are small in terms of N . Again, this terminology is justified by the fact that, as it is proved in [START_REF] Hubert | On p-pseudorandom binary sequences[END_REF], for a p-random E N both W (E N , p) and C k (E N , p) are around N 1/2 . Clearly, this notion of p-pseudo-randomness generalises the notion of pseudo-randomness.

The definition of p-pseudo-randomness can be adapted easily to define the pseudorandomness of subsets of {1, . . . , N }. Suppose we want to check a subset R ⊂ {1, . . . , N } for pseudo-randomness. Let |R| = h. Then a random integer n ∈ {1, . . . , N } belongs to R with probability h N , which corresponds to the p = h/N = |R|/N case of pseudorandomness (but it is not identical with it!) Thus defining the sequence

(2•9) E N = E N (R) = (e 1 , e 2 , . . . , e N ) ∈ 1 - |R| N , - |R| N N by (2•10) e n = 1 -|R| N for n ∈ R -|R| N for n ∈ R (n = 1, 2, . . . , N ),
we may define the well-distribution measure and the correlation measure of order k of the subset R by formulas (2•7), and (2•8) respectively:

(2•11) W (R, N ) = W E N (R), |R| N = max a,b,t t-1 j=0 e a+jb and (2•12) C k (R, N ) = C k E N (R), |R| N = max M,D M n=1 e n+d 1 • • • e n+d k
where E N (R) is defined by (2•9) and (2•10).

One would expect and might like to show that these measures are "small" (are around N 1/2 ) for a "truly random subset" R of {1, . . . , N }; this fact does not follow from the analogous results on p-pseudo-randomness, and there are difficulties in adapting their proofs in [START_REF] Hubert | On p-pseudorandom binary sequences[END_REF]. We remark that here the natural definition of "truly random subset" would be to take every R ⊂ {1, . . . , N } with uniform probability 2 -N . However, in the applications (e. g., in [START_REF] Mauduit | Construction of pseudorandom binary sequences by using the multiplicative inverse[END_REF] and [START_REF] Sárközy | On pseudorandomness in families of sequences derived from the Legendre symbol[END_REF]) h = |R| is fixed, thus it is better to show the smallness of the P R measures in the sharper form that we fix h (with h → +∞, N -h → +∞) and then we consider the h-element subsets of {1, . . . , N } with uniform probability N h -1 . We will return to this question in a subsequent paper.

We emphasize that the notions of "p-pseudo-randomness" and "pseudo-randomness of subset" of {1, . . . , N } are very close but not identical. Indeed, we use the same tools and formulas but, on the other hand, in the first case we study binary sequences while in the second subsets. Besides, in the first case p is fixed, and then for a "good" p-pseudo-random binary sequence the frequencies of the two elements need not be exactly p, resp. 1 -p, it is enough if they are near these values; on the other hand, in the second case first we consider a subset R ⊂ {1, . . . , N }, and then the associated p value is exactly p = |R|/N , so that the proportion of the elements selected is exactly p. Finally, in the first case we typically consider a fixed p with ε < p < 1 -ε (for some ε >), while in the second case typically we are interested also in subsets R with |R| = o(N ) so that now |R|/N (which corresponds to p) is o(1).

Subsets formed by mod p residues of polynomials

Let p be a prime number, F p the field of modulo p residue classes, and Fp its algebraic closure.

The first "good " pseudo-random sequence studied by Mauduit and Sárközy in [START_REF] Mauduit | On finite pseudorandom binary sequences, I. Measure of pseudorandomness, the Legendre symbol[END_REF] was the sequence defined by the Legendre symbol:

e n = n p for 1 n p -1.
They showed that this sequence has good PR properties, the well-distribution measure (2•2) and the correlation measure of order k (2•3) are O k ( √ p log p) (with a good and explicite control of the dependance in k). In [START_REF] Mauduit | On finite pseudorandom binary sequences II. The Champernowne, Rudin-Shapiro, and Thue-Morse sequences: a further construction[END_REF] they extended these results to sequences of the form

e n = f (n) p if f (n) ≡ 0 (mod p) 1 if f (n) ≡ 0 (mod p),
for 1 n p where f is a permutation polynomial whose unique zero in F p has odd multiplicity. A permutation polynomial f ∈ F p [X] is a polynomial whose associated polynomial function x → f (x) is a permutation of F p . For example if (k, p -1) = 1, the monomial x k is a permutation polynomial (see [START_REF] Mauduit | On finite pseudorandom binary sequences II. The Champernowne, Rudin-Shapiro, and Thue-Morse sequences: a further construction[END_REF] for other examples).

In this section we generalize this construction to power residues. Let d|p -1 and f be a permutation polynomial. The equation f (x) = 0 has a unique solution x 0 in F p , and in Fp we have the factorization:

f (x) = (x -x 0 ) r 0 (x -α 1 ) r 1 • • • (x -α s-1 ) r s-1 ,
where α 1 , . . . , α s ∈ F p . We suppose:

(3•1) (d, r 0 ) = 1.
We will study the pseudo-random properties of the following set V

(3•2) V := {x ∈ F p , ∃y ∈ F p {0} : f (x) ≡ y d (mod p)}.
The cardinality of V is (p -1)/d. The associated sequence E(V ) = {e n } 1 n p defined by (2•9) and (2•10) satisfies:

(3•3) e n = 1 -α if n ∈ V -α if n ∈ V , with (3•4) α = card V p = p -1 dp .
We will show that V has strong PR properties:

Theorem 3.1. Under (3•1), we have

(3•5) W (V, p) 1 + 9 d -1 d s √ p log p,
and for k 2,

(3•6) C k (V, p) k + 9 d -1 d k 1 + 1 p k ks √ p log p.
First we prove (3•5). Recall that

W (V, p) = max a,b,t b+a(t-1) p t-1 j=0 e aj+b .
By definition of the sequence E(V ) we have:

(3•7) t-1 j=0 e aj+b = (1 -α) t-1 j=0 aj+b∈V 1 -α t-1 j=0 aj+b ∈V 1.
Next in (3•7) we use the formula

t-1 j=0 aj+b ∈V 1 = t - t-1 j=0 aj+b∈V 1,
and we obtain:

(3•8) t-1 j=0 e aj+b = t-1 j=0 aj+b∈V 1 -tα.
We introduce character sums to detect d-power residues. Let χ 0 denote the principal character modulo p. When (x, p) = 1, we have

(3•9) χ d =χ 0 χ(x) = d if ∃y ∈ F p {0} : x = y d , 0 otherwise.
Thus we have:

t-1 j=0 e aj+b = 1 d χ d =χ 0 0 j t-1 χ(f (aj + b)) -αt.
The contribution of χ 0 is t/d if aj + b ≡ x 0 (mod p) for all 0 j t -1 and t/d -1 otherwise. We easily check that

1 d t-1 j=0 χ 0 (f (aj + b)) -αt 1.
Thus we have:

(3•10) t-1 j=0 e aj+b 1 + 1 d χ d =χ 0 χ =χ 0 0 j t-1 χ(f (aj + b)) .
To evaluate the character sum we will use the following lemma: Lemma 3.2. Suppose that p is a prime number, χ is a non-principal character modulo p of order d (so that d|p-1), f

(x) ∈ F p [X] has the factorization f (X) = b(X -x 1 ) d 1 • • • (X - x s ) d s (where x i = x j for i = j) in Fp with (d, d 1 , . . . , d s ) = 1.
Let X, Y be real numbers with 0 < Y p. Then

X<n X+Y χ(f (n)) < 9s √ p log p.
This is Lemma 2 in [START_REF] Sárközy | A finite pseudorandom binary sequence[END_REF], it is a slightly modified form of Theorem 2 in [START_REF] Mauduit | On finite pseudorandom binary sequences, I. Measure of pseudorandomness, the Legendre symbol[END_REF], and it was derived from A. Weil's theorem [START_REF] Weil | Sur les courbes algébriques et les variétés qui s'en déduisent[END_REF]. By Lemma 3.2 we obtain:

t-1 j=0 e aj+b 1 + 1 d χ d =χ 0 χ =χ 0 9s √ p log p 1 + 9 d -1 d s √ p log p,
this ends the proof of (3•5). Now we study the correlation measures. Let k 2. We have to compute:

C k (V, p) = max
e n+d 1 • • • e n+d k = k j=1 (1 -α) 1 d χ d =χ 0 χ(f (n + d j )) -α 1 - 1 d χ d =χ 0 χ(f (n + d j )) . = k j=1 1 d χ d =χ 0 χ(f (n + d j )) -α = k j=1 1 d χ d =χ 0 χ =χ 0 χ(f (n + d j )) + 1 dp = 1 d k k j=1 χ d =χ 0 χ =χ 0 χ(f (n + d j )) + 1 p .
If there exists some j, 1 j k such that n + d j ≡ x 0 (mod p) then this j is unique and

e n+d = -α if = j 1 d χ d =χ 0 χ =χ 0 χ(f (n + d )) + 1 p if = j, and
e n+d 1 • • • e n+d k - 1 d k M n=1 k j=1 χ d =χ 0 χ =χ 0 χ(f (n + d j )) + 1 p | -α + 1 dp | 1.
There exists at most k integers n M such that n + d j ≡ x 0 (mod p) for some 1 j k.

Thus we have (see also [START_REF] Mauduit | On finite pseudorandom binary sequences II. The Champernowne, Rudin-Shapiro, and Thue-Morse sequences: a further construction[END_REF] section 8):

(3•11) M n=1 e n+d 1 • • • e n+d k - 1 d k M n=1 k j=1 χ d =χ 0 χ =χ 0 χ(f (n + d j )) + 1 p k.
We define

Z = 1 d k M n=1 k j=1 χ d =χ 0 χ =χ 0 χ(f (n + d j )) + 1 p .
We develop the above product:

Z = 1 d k k r=0 1 p k-r 1 j 1 <...<j r k χ j 1 =χ 0 χ d j 1 =χ 0 • • • χ j r =χ 0 χ d j r =χ 0 M n=1 χ j 1 (f (n + d j 1 )) • • • χ j r (f (n + d j r )).
To evaluate this sum we do the same operations as in [START_REF] Sárközy | A finite pseudorandom binary sequence[END_REF] p. 382-384. We will not give all the details. The only differences arise from the permutation polynomial f . Since F * p is cyclical we may write each χ j like χ j = χ δ where χ is a character of order p -1.

Let δ = (δ 1 , . . . , δ r ), and δ i = δD i for 1 i r. Since χ d j i = χ 0 , we have dδ i ≡ 0 (mod p -1). Thus (3•12) δ ≡ 0 (mod (p -1)/d).

We write χ * = χ δ . It is proved in [START_REF] Sárközy | A finite pseudorandom binary sequence[END_REF] [START_REF] Mauduit | On finite pseudorandom binary sequences, I. Measure of pseudorandomness, the Legendre symbol[END_REF], that χ * = χ 0 , more precisely, the order D of χ * is D = (p -1)/(p -1, δ) and in our case, D|d. The computations p. 383 of [START_REF] Sárközy | A finite pseudorandom binary sequence[END_REF] yield to:

M n=1 χ j 1 (f (n + d j 1 )) • • • χ j r (f (n + d j r )) = M -1 n=0 χ * f (n + d j 1 ) D 1 • • • f (n + d j r ) D r .
We apply Lemma 3.2 with χ * instead of χ and with the polynomial

F (n) = f (n + d j 1 ) D 1 • • • f (n + d j r ) D r .
In Fp we have

F (x) = r i=1 (x + d j i -x 0 ) r 0 D i s-1 =1 r i=1 (x + d j i -α ) r D j i .
Since (r 0 , d) = 1 and (D 1 , . . . , D r ) = 1, we have (d, r 0 D 1 , . . . , r 0 D r ) = 1, and since by the assumptions of the theorem and the definition of x 0 the α's do not belong to F p , the condition of Lemma 3.2 is satisfied. By Lemma 3.2 we obtain:

(3•13). M -1 n=0 χ j 1 (f (n + d j 1 )) • • • χ j r (f (n + d j r )) 9ks √ p log p
We apply this upper bound in Z, and by (3•11) we end the proof of the theorem:

Z 9ks √ p log p k r=0 1 p k-r k r = 9ks √ p log p 1 + 1 p k .

A construction using the index

In this section we will give a construction for subsets with strong PR properties which will be based on the notion of index (discrete logarithm) and which is a variant of the construction given in [START_REF] Sárközy | A finite pseudorandom binary sequence[END_REF]. Thus we will refer to [START_REF] Sárközy | A finite pseudorandom binary sequence[END_REF] repeatedly, and we will leave some details to the reader.

Let p be an odd prime, g a fixed primitive root, and let ind a denote the modulo p index (discrete logarithm) of a to the base g so that g ind a ≡ a (mod p) and, to make the index unique, 1 ind a p -1.

Theorem 4.1. Let h, be integers with 0 h < h + p -1, and define the subset R of {1, . . . , p -1} by

R = {n : 1 n p -1, h < ind n h + }.
Then we have

(4•1) |R| = , (4•2) W (R, p -1) < 2 √ p(log p) 2
and, for all k ∈ N, k < p,

(4•3) C k (R, p -1) < 9k2 k √ p(log p) k+1 .
Proof. The equality (4•1) is trivial. The proof of (4•2) is based on the Pólya-Vinogradov inequality:

Lemma 4.2. If p is a prime number, χ a non-principal character modulo p and X, Y are real numbers with X < Y , then we have

X<n Y χ(n) < √ p log p.
(See, e. g., [START_REF] Davenport | Multiplicative Number Theory[END_REF], p. 135 for a proof.) Assume that e aj+b = 0 j<t h<i h+ g i ≡a+jb (mod p)

1 -p -1 = 0 j<t h<i h+ g i ≡a+jb (mod p) 1 - t p -1 .
By the formula

1 p -1 χ χ(a)χ(b) = 1 if a ≡ b (mod p)
and (a, p) = 1 0 otherwise, here we have

0 j<t h<i h+ g i ≡a+jb (mod p) 1 = 1 p -1 χ t-1 j=0 h+ i=h+1 χ(a + jb)χ(g i ).
The contribution of the principal character is

1 p -1 t-1 j=0 h+ i=h+1 1 = t p -1 .
Thus it follows from (4•5) that

(4•6) t-1 j=0 e aj+b = 1 p -1 χ =χ 0 t-1 j=0 h+ i=h+1 χ(a + jb)χ(g i ) = 1 p -1 χ =χ 0 t-1 j=0 χ(a + jb) h+ i=h+1 χ i (g) 1 p -1 χ =χ 0 t-1 j=0 χ(aj + b) h+ i=h+1 χ i (g) .
The first inner sum can be estimated by Lemma 4.2 as in [START_REF] Chung | Quasi-random subsets of Z n[END_REF] in [START_REF] Sárközy | A finite pseudorandom binary sequence[END_REF], and we obtain

(4•7) t-1 j=0 χ(aj + b) < √ p log p,
and as in [START_REF] Chung | Quasi-random graphs[END_REF] in [START_REF] Sárközy | A finite pseudorandom binary sequence[END_REF], the second inner sum is

(4•8) h+ i=h+1 χ i (g) < 2 |1 -χ(g)| .
By (4•7) and (4•8) we get from (4•6) that

(4•9) t-1 j=0 e aj+b 2 √ p log p p -1 χ =χ 0 1 |1 -χ(g)| .
By [START_REF] Davenport | Multiplicative Number Theory[END_REF] in [START_REF] Sárközy | A finite pseudorandom binary sequence[END_REF] the last sum is

(4•10) χ =χ 0 1 |1 -χ(g)| < (p -1) log p.
It follows from (4•9) and (4•10) that

t-1 j=0 e aj+b < 2 √ p(log p) 2 .
This holds for every a, b, t satisfying (4•4) which, by (2•11), completes the proof of (4•2). The proof of (4•3) is based on Lemma 3.2 stated in the previous section. In order to prove (4

1 -p -1 = M n=1 k j=1 1 p -1 χ j h+ i j =h+1 χj (n + d j )χ j (g i j ) -p -1 = M n=1 k j=1 1 p -1 χ j =χ 0 h+ i j =h+1 χj (n + d j )χ j (g i j ) = 1 -1) k χ 1 =χ 0 . . . χ k =χ 0 M n=1 k j=1 χj (n + d j ) k j=1 h+ i j =h+1 χ j (g i j ) 1 (p -1) k χ 1 =χ 0 . . . χ k =χ 0 M n=1 k j=1 χ j (n + d j ) k j=1 h+ i j =h+1 χ j (g i j ) .
This expression, apart from a missing factor 2 k , is of nearly the same form as the upper bound in [START_REF] Haviland | Pseudo-random hypergraphs[END_REF] in [START_REF] Sárközy | A finite pseudorandom binary sequence[END_REF] and, by using Lemma 3.2, it can be estimated in the same way. Thus we end up with an upper bound less by a factor 2 k than the one in [START_REF] Sárközy | A finite pseudorandom binary sequence[END_REF]:

M n=1 e n+d 1 • • • e n+d k < 9k2 k p 1/2 (log p) k+1
which, by (2•12) completes the proof of (4•3).

Subsets obtained by sifting

In this last section we will study a subset of {1, . . . , N } obtained by sifting: the subset of the square free numbers. Let Q(N ) be the set of the square free numbers less than N . The cardinality of Q(N ) is (see for example [START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF] section I.3.7)

(5•1) Q(N ) = 6 π 2 N + O( √ N ).
We define the rate

q N = card Q(N ) N .
We have

(5•2) q N = 6 π 2 + O 1 √ N .
By (2•9) and (2•10), the corresponding sequence is E N (Q(N )) = (e 1 , . . . , e N ) with

(5•3)

e n = 1 -q N if µ 2 (n) = 1, -q N if µ(n) = 0 = (1 -q N )µ 2 (n) -q N (1 -µ 2 (n)) = µ 2 (n) -q N .
It is easy to see that this sequence does not have good PR properties. First the welldistribution measure is large, for example for each n N/4, we have e 4n = -q N . Thus we have

(5•4) W Q(N ), N [N/4] n=1 e 4n N q N 4 -1.
More generally we can see by elementary means that this sequence is not well-distributed in every arithmetic progression of modulus > 1:

Lemma 5.1. Let a 2, b, t such that b + a(t -1) N . We have:

t-1 j=0 e aj+b = 6t π 2 (F (a, b) -1) + O √ N , with F (a, b) =    0 if there exists p such that p 2 |(a, b), p|a 1 -1 p 2 -1 p|(a,b) a ≡0 (mod p 2 ) 1 -1 p otherwise.
By this lemma we see that W (Q(N ), N ) is given by the arithmetic progression modulo 4.

Corollary 5.2. For N 2, we have:

W (Q(N ), N ) = 3N 2π 2 + O( √ N ).
Proof of Corollary 5.2. By Lemma 5.1, we have

W (Q(N ), N ) = max b+a(t-1) N 6t π 2 |F (a, b) -1| + O √ N = max a,b (N -b) a 6 π 2 |F (a, b) -1| + O( √ N ).
By (5•2) and (5•4), to prove Corollary 5.2, it is sufficient to show that

(5•5) |F (a, b) -1| 1 a 1 4 .
By the definition of F , it is clearly the case when F (a, b) = 0. We see that the possible other large values of |F (a, b) -1| are obtained for b = 1 and we have 0

F (a, 1) -1 a ( π 2 6 -1) 1 a
which is less than 1/4 for all a 3 and we have (F (2, 1) -1)/2 = 1/6. This proves that Corollary 5.2 is a consequence of that Lemma 5.1.

Proof of Lemma 5.1. We have by (5•3)

t-1 j=0 e aj+b = t-1 j=0 µ 2 (aj + b) -tq N .
Next we use the formula

µ 2 (n) = d 2 |n µ(d),
and exchange the order of summations:

t-1 j=0 e aj+b = t-1 j=0 d 2 |aj+b µ(d) -tq N = d 2 a(t-1)+b µ(d) 0 j t-1 aj+b≡0 (mod d 2 )
1 -tq N .

The congruence aj + b ≡ 0 (mod d 2 ) has a solution if and only if (a, d 2 )|b and we have

t-1 j=0 e aj+b = d 2 a(t-1)+b (d 2 ,a)|b µ(d)t(a, d 2 ) d 2 - 6t π 2 + O( √ N ).
It remains to compute the sum over d:

d 2 a(t-1)+b (d 2 ,a)|b µ(d)(a, d 2 ) d 2 = (d 2 ,a)|b µ(d)(a, d 2 ) d 2 + O d> √ at+b µ(d) 2 (a, d 2 ) d 2 .
The error term above is clearly less than O( a/t). Let

f (d) = µ(d)(a,d 2 ) d 2 if (d 2 , a)|b 0 otherwise.
This function is multiplicative and we have

f (p) =      -1 if p 2 |(a, b) -1 p 2 if (p, a) = 1 -1 p if p a and p|b 0 otherwise.
We have

(d 2 ,a)|b µ(d)(a, d 2 ) d 2 = d f (d) = p (1 + f (p)) = 6 π 2 F (a, b).
This ends the proof of (Lemma 5.1).

The correlation measures are also large for every k.

Lemma 5.3. For all k 2, we have For z 2 we define D z = p<z p 2 . We take d j = jD z for j = 1, . . . , k so that for p < z, u(p, d 1 , . . . , d r ) = 1. We have This ends the proof of Lemma 5.3. We may also ask if in general, the subset obtained by sifting condition has poor PR properties. This is clearly the case for the distribution if we sieve with small numbers. For example the set of the integers which are sums of two squares is poorly distributed modulo p for p ≡ 3 (mod 4) and the correlation are not small either.

C k Q(N ), N k N.

e

  n+d 1 • • • e n+d k , with M and D = (d 1 , . . . , d k ), 0 d 1 . . . d k such that M + d k p.If n ≡ x 0 -d j (mod p) for any 1 j k, we have by(3•9)

(4• 4 ) 1

 41 a a + (t -1)b p -1, and define E p-1 = (e 1 , . . . , e p-1 ) by (2•9) and (2•10) where now |R| = by (4•1), and N = p -1. Then the sum in (2

Proof. Let M and d 1 µ 2 µ 2 Lemma 5 . 4 . 2 ,

 122542 , . . . d k be some positive integers such that 0 d 1 < d 2 < . . . < d k and M + d k N . We have to evaluateC(M, d 1 , . . . , d k ) = M n=1 e n+d 1 • • • e n+d k .By (5•3) we have:C(M, d 1 , . . . , d k ) = (n + d j ) -q N .We take the term-by-term product:C(M, d 1 , . . . , d k ) = (-q N ) k M + (n + d j i ).This sum will be estimated with this following result of Tsang[START_REF] Tsang | The distribution of r-tuples of squarefree numbers[END_REF]: Let d 1 , . . . , d r be distinct integers such that |d i | cx for 1 i r, r log x/(25 log log x), and c an absolute constant. For any x 3, we have(5•6) n x µ 2 (n + d 1 ) • • • µ 2 (n + d k ) = A(d 1 , . . . , d r )x + O c (x 7/11 (log x) 8 ),where A(d 1 , . . . , d r ) = p 1 -u(p, d 1 , . . . , d r ) p and u(p, d 1 , . . . , d r ) is the number of distinct residue classes modulo p 2 represented by the numbers d 1 , . . . , d r . The implied constant in the error term in (5•6) depends only on c. This is a slightly weaker form of Theorem 2 of Tsang [27]. By Lemma 5.4 we have for d k cM : C(M, d 1 , . . . , d k ) = (-6) k π 2k M + M 1 r k (-6) k-r π 2(k-r) 1 j 1 <...<j r k A(d j 1 , . . . , d j r ) + O(k r M 7/11 (log M ) 8 ).

A(d 1

 1 , . . . , d r ) = Finally we obtain with this choice of d 1 , . . . , d r : C(M, d 1 , . . . , d k

  •3) consider any D = (d 1 , . . . , d k ) with non-negative integers d 1 < . . . < d k and positive integer M with M + d k n+d 1 e n+d 2 • • • e n+d k =

	(4•6), the sum in (2•12) is
	M	e M	k
	n=1	n=1	j=1	h<i h+
				g i ≡n+d j (mod p)
				p -1. Then, as in (4•5) and

*

Research partially supported by the Hungarian National Foundation for Scientific Research, Grant T043623 and T049693, and by French-Hungarian EGIDE-OMKFH Á exchange program Balaton F-2/03.


Shraman N Goswami
email: shraman.goswami@honeywell.com

Prof M Govardhan

Effect of Lean on Performance of an Axial Compressor Rotor with Circumferential Casing Grooves

Keywords:

Axial Compressors used in gas turbine engines are susceptible to stall. This phenomenon is more prevalent in aero gas turbine engines, due to requirement of varied operating conditions based on flight envelop. There are a number of methodologies in use for increasing stall margin of compressor and hence the operating range of the engines. One of such most widely used techniques is circumferential casing grooves. Circumferential casing grooves helps in increasing stall limit, but generally with a penalty on efficiency, by breaking down tip vortices. In order to increase the efficiency and pressure ratio, rotor blades are designed to reduce secondary losses. Designing blades with lean and sweep are one of the techniques to reduce secondary flow losses. A number of literatures are available in public domain, giving detailed understanding of effect of circumferential groove and 3D blade features like lean and sweep. In this current work, an effort is made to understand the interaction effect of lean and circumferential grooves, using computational fluid dynamics (CFD). In order to use CFD as a tool to understand flow physics and predict performance, a thorough validation is carried out. CFD results, both performance parameters as well as span-wise distribution of different flow variables, are compared with available experimental data of NASA Rotor 37.

Current investigation starts with generation of a baseline rotor, having no lean and no sweep. The rotor geometry is created using hub and tip profiles of NASA Rotor 37. Flow path used for this rotor is same as that of NASA Rotor 37. The profiles are stacked along a radial line through the center of gravities of the profiles. This has resulted in desired geometry, without lean and sweep. Slight modification is made in terms of stagger angle of the tip profiles, to get comparable performance as NASA Rotor 37. Five circumferential casing grooves between leading edge and trailing edge of the rotor are created as per industry standard. Meshing and modeling are done according to the best practices developed while validating CFD methodology. Analyses of baseline rotor are carried out from choke to stall, with solid shroud as well as grooved shroud. Grooves shroud has resulted in higher stall margin, as expected. In order to study the interaction effect of casing grooves with lean, a number of different rotors geometries are generated with varying amount of lean as well as different span location from where lean starts. Results obtained from these numerical simulations are presented in this paper. Performance and flow features are compared with baseline rotor, with and without grooves, in an attempt to understand global as well as localized effect of interaction between casing grooves and lean.

INTRODUC TION

Prior research work has demonstrated that axial compressor rotor blades with 3D design features, such as lean and sweep, improves overall performance of a rotor. When carefully designed, these design features increase or redistribute blade loading and decrease secondary flow impacts on the main flow. This results in higher pressure ratio, efficiency as well as stall margin in many of the cases. A detailed investigation on effect of swept blade on a compressor cascade is reported by Williams [START_REF] Godwin | Effect of Sweep on Performance of Compressor Blade Sections, as Indicated by Swept-Blade Rotor, Unswept-Blade Rotor and Cascade Tests[END_REF]. In another cascade experiment by Breugalmans [START_REF] Breugalmans | Investigation of Dihedral Effects in Compressor Cascades[END_REF], effect of lean on flow field is studied. It is reported that lean helps in unloading the end walls and hence achieves decrease in secondary flows. Sasaki et al. [START_REF] Sasaki | Comparison of Sweep and Dihedral Effects on Compressor Performance[END_REF] studied the effect of sweep and dihedral. They have reported the presence of two counter rotating vortices inside the rotor passage which energizes the end wall boundary layers and delays corner stall. Inoue et al. [START_REF] Inoue | Controlled-Endwall-Flow Blading for Multistage Axial Compressor[END_REF] also reported increased performance by controlled end wall blading; which is due to reduced secondary flows. In a more recent study, Govardhan et al. [START_REF] Govardhan | Study of Sweep and Induced Dihedral Effects in Sub-Sonic Axial Flow Compressor Passages Part 1: Design Considerations -Changes in Incidence, Deflection and Streamline Curvature[END_REF] have done a detailed numerical investigation of sweep and induced lean. The sweep in axial direction induces lean and it is reported that the flow streamlines gets deflected towards hub and shroud due to induced lean. In order to increase stall margin of axial compressors, one of the most commonly used passive mechanism is casing treatment. In an early study conducted by Everett [START_REF] Bailey | Effect of Grooved Casing Treatment on the Flow Range Capability of a Single-Stage Axial-Flow Compressor[END_REF], it is reported that a five groove configuration produced highest stall margin. A range of different number of casing grooves is considered in this study. Location, depth and width of grooves are varied to find out impact on performance. It has been reported that grooves around mid-chord results in highest stall margin. Donald et al. [START_REF] Urasek | Effect of Casing Treatment on Performance of an Inlet Stage for a Transonic MultiStage Compressor[END_REF] has carried out experimental study to understand the impact of casing grooves placed on the inlet stage of a multi stage axial compressor. They have considered blade-angled slots, circumferential slots and axially skewed slots in this study. It is reported that axially skewed slot results in best performance. They have reported that both rotor and stage performance have increased as well as a significant increase in stall margin. In a more recent study Fan Lin et al. [START_REF] Lin | Aerodynamics of Compressor Casing Treatment: Part I: Experiment and Time-Accurate Numerical Simulation[END_REF] have studied the flow phenomena in presence of slot type axial grooves. They have placed axial slots such that leading edge of the blade passes through the mid chord of the slots, which are inclined from the radial direction. The study shows different flow pattern in the slots at low speed and near design speed. At low speed the slots helps in flow exchange between high momentum and low momentum flow region, whereas at near design speed the slots work as a reservoir such that fluid is stored and released. In either case the fluid exchange results in reduced tip leakage flow and hence delays separation and stall. Khan at al [START_REF] Khan | Effect of Circumferential Grooves and Tip Recess on Stall Characteristics of Transonic Axial Compressor Rotor[END_REF] have carried out a study with casing treatment and tip recess. They have reported that the effect of tip recess is minimal as compared to circumferential casing grooves on stall margin. It is reported that the circumferential grooves helps to breakdown the tip vortex and results in increased operating range. The first two grooves were said to have played major role in enhancing the operating range. This current study is conducted to understand change in flow field in the presence of lean and circumferential grooves, and their effect on performance and stall margin. NASA Rotor 37 is considered for validation of the numerical process and best practices are established. A baseline rotor is designed by removing lean and bow from the NASA Rotor 37 blades. A number of different leans with five circumferential casing grooves are numerically investigated. It is to be noted that this study considers only aero design aspects. Structural analysis might put limitations, and needs to be considered while doing actual hardware design.

VALIDATION OF NUMERICAL METHOD

The numerical methodology to be used for this study is validated with NASA Rotor 37 experimental data, which was originally designed at NASA Lewis Research Center by Reid & Moore [START_REF] Reid | Design and Overall Performance of Four Highly Loaded, High-Speed Inlet Stages for an Advanced High-Pressure-Ratio Core Compressor[END_REF][START_REF] Reid | Experimental Study of Low Aspect Ratio Compressor Blading[END_REF]. The rotor has been extensively investigated by many other researchers [START_REF] Hah | A Viscous Flow Study of Shock-Boundary Layer Interaction, Radial Transport and Wake Development in a Transonic Compressor[END_REF][START_REF] Suder | Experimental and Computational Investigation of the Tip Clearance Flow in a Transonic Axial Compressor Rotor[END_REF][START_REF] Hah | Development of Hub Corner Stall and Its Influence on the Performance of Axial Compressor Blade Rows[END_REF][START_REF] Chima | Calculation of Tip Clearance Effects in a Transonic Compressor Rotor[END_REF][START_REF] Gerolymos | Tip-Clearance and Secondary Flows in a Transonic Compressor Rotor[END_REF][START_REF] Yamada | Unsteady Three-Dimensional Flow Phenomena due to Breakdown of Tip Leakage Vortex in a Transonic Axial Compressor Rotor[END_REF]. This rotor being highly loaded and having shock near the tip at design speed, is a good candidate for validation. Fully hexahedral, structured meshes are generated using Numeca AutoGrid 5® version 9.0-2. Mesh sensitivity study is carried out to get mesh independent results. Mesh count used in this study is 3.5 millions. Figure 1 shows some views of the final mesh used for this validation. Numerical simulations (steady, RANS, single passage) are carried out using Ansys CFX® version 15.0. Qualitative descriptions of different boundary conditions [START_REF] Reid | Design and Overall Performance of Four Highly Loaded, High-Speed Inlet Stages for an Advanced High-Pressure-Ratio Core Compressor[END_REF][START_REF] Reid | Experimental Study of Low Aspect Ratio Compressor Blading[END_REF] are shown in Figure 2. As turbulence models play a major role, results with three different turbulence models, namely BSL (baseline k-omega), SST (shear stress transport) and SST-Reattachment [START_REF]Ansys CFX®15.0 User Manuals[END_REF], are compared to test data [START_REF] Biollo | Aerodynamic Behavior of A Novel Three-Dimensional Shaped Transonic Compressor Rotor Blade[END_REF] and presented in figure 3. It is observed that results obtained with BSL turbulence model has a good match with test data and is used for all subsequent analyses. A very good match between numerical and test data is obtained for the averaged total pressure ratio from choke to stall. The speed line comparison of total pressure ratio is plotted in

BASELINE ROTOR DESIGN

In order to understand the effect of lean, a baseline rotor is to be designed, devoid of any lean or sweep. This baseline rotor is designed using the Rotor37 blade profiles at hub and at tip. The two profiles are CG stacked in a radial line, which has resulted in a rotor without any lean or sweep. It is observed that the performance of the baseline rotor is lower than the Rotor37 performance as well as choke flow is low. Hence tip profile of the baseline rotor is adjusted and comparable performance with respect to Rotor37 is achieved. Figure 8 shows a comparative picture of Rotor37 and the Baseline Rotor.

BASELINE ROTOR WITH CASING GROOVES

Five circumferential casing grooves are created on the shroud and meshed using 1:1 matching mesh with the main flow passage. The grooves are designed as per industry standard. Figure 9 shows 1:1 matching mesh used for the Baseline Rotor with casing grooves. A series of analyses were carried out to compare the performance of baseline rotor performance with and without casing treatment.

ROTORS WITH LEAN

A number of Rotor geometries are generated to study the effect of lean on flow field in presence of grooved casing treatment. As the baseline rotor stacking axis is radial, the leaned geometries are created by rotating the CG of the blade elements in the R-Theta plane, as shown in Figure 12. As a first step baseline rotor tip profile is leaned by ±5%, ±10% and ±25% of rotor pitch angle, where leaning started from 75% span. Rotor being rotating in the counter clockwise direction, +ve leaning is in the opposite direction to direction of rotation. As +ve lean resulted in better stall margin (10% lean), two more rotors were designed with +ve lean; one 15% lean starting from 75% span and another 10% lean starting from 50% span. The groove geometries are maintained same for all the cases. Figure 13 shows the leading edge view of different leaned geometries.

RESULTS AND DISCUSSIONS

Numerical simulation results obtained from different simulations are presented in Figure 14 through Figure 16. Choke flow reduces for all the grooved cases as compared to solid shroud case, in line with findings by some other researchers ( [START_REF] Huang | CFD Investigation on the Circumferential Grooves Casing Treatment of Transonic Compressor[END_REF], [START_REF] Huang | An analysis of the Circumferential Grooves Casing Treatment for transonic compressor flow[END_REF]). A comparison of spanwise variation of Mass Flux (Density x Meridional Velocity) at trailing edge is shown in Figure 17. It can be observed that mass flux is lower near the tip region for the grooved shroud case, resulting in a decrease in choke flow. Figure 18 shows the mass flux contour at 99.75% span, where lower mass flux region is easily identified.

It is observed that near the design point maximum efficiency is predicted for "-25%Lean, 75%Span" case. Efficiency for this case is higher than the baseline cases (both solid shroud and grooved) by ~0.5%. Increase in efficiency for this rotor has resulted in loss of stall margin by ~4%.

Figure 19 shows variation of stall margin with changing lean. It is noted that "+10%lean,75%span" geometry has resulted in highest stall margin of ~20%, with an increase of ~7% with respect to baseline rotor with solid shroud. Increase in stall margin with respect to baseline rotor without lean is ~2.6% stall margin. The stall margin goes down if lean is either increased or decreased, indicating existence of an optimum lean for stall margin improvement.

EFFECT OF LEAN IN THE LOW FLOW SIDE

In order to find the effect of lean on flow field and hence the stall margin, two close mass flow points are considered (Figure 20). The two case compared are no lean and "+10%Lean,75%Span". The blade loading plot for these two cases at 99% span is shown in Figure 21. It is noticed that the pressure side loading does not change much, whereas the suction side of the leaned blade is getting unloaded downstream of the shock. It can also be observed that the shock strength is lesser in case of leaned blade than the baseline blade.

This unloading can be attributed to the leaning of the blade, which allows the flow to escape through the tip gap more readily, as shown in Figure 22. Blade leaning has resulted in lesser flow turning (Figure 23), above 60% span. Due to the lower loading near the trailing edge (Figure 21) of leaned blade the trailing edge wake of the leaned blade is thinner than the no lean geometry (Figure 24). This is also reflected in higher pressure ratio for the leaned blade. Figure 25 shows blade loading at span 50% to 99.35%. It is observed that loading is higher for leaned blade till ~90% span, above which loading decreases. This increased loading at the lower span has resulted in higher pressure ratio. The decrease in loading near the tip of the blade has helped in reaching deeper in stall and hence an increased stall margin. Total pressure in stationary frame at blade tip is plotted in Figure 26. The points chosen for comparison are marked in Figure 20. It can be observed that for the leaned blade the total pressure in the passage is higher than the no lean case. This indicates a healthier flow in the leaned blade passage, which has resulted in a higher stall margin.

Figure 26: Velocity Vector at Blade Tip Flow inside the grooves is shown in Figure 27. It is observed that the flow inside the first and second grooves is more distorted than the next three grooves; the flows in these two grooves are not entirely circumferential (also reported by Khan et al. [START_REF] Biollo | Aerodynamic Behavior of A Novel Three-Dimensional Shaped Transonic Compressor Rotor Blade[END_REF]). This can be attributed to the location of shock inside the passage. This has resulted in a region of low momentum fluid in these two grooves. In case of the no lean blade, the extent of low momentum region is slightly more than the blade with lean.

EFFECT OF LEAN ON STARTING SPAN

"+10%Lean,75%Span" and "+10%Lean,50%Span" are two geometries having the same lean but lean starts at different spans. Performance parameters are presented in Figure 29 and Figure 30. The Pressure ratio for these two cases are comparable but efficiency and stall margin for "+10%Lean,50%Span" is lower than "+10%Lean,75%Span" case. A closer look at the blade loading charts (Figure 31) at different span reveals that case "+10%Lean,50%Span" has higher loading till 65% span. Above 65% span loading is lower and then again above 95% loading is more than "+10%Lean,75%Span". This redistribution of loading has helped "+10%Lean,75%Span" to throttle deeper into stall.

CONCLUSIONS

The following conclusions can be drawn from this study of effect of lean in presence of casing grooves.

1. In rotors with circumferential grooves, mass flow is lower near tip region, as compared to that of solid shroud rotor; resulting in decreased choke flow for grooved rotors. 2. Leaning in the opposite direction of rotor rotation (+ve leaning for the current study) results in increased stall margin. 3. There exists an optimum lean amount for maximum stall margin gain. 4. Shock strength in case of leaned blade is lower in the suction side near tip region. 5. Positive leaning has resulted in a lower flow turning near the tip as compared to that of rotors with negative lean. 6. Negative lean results in increased adiabatic efficiency at the cost of reduced stall margin. 7. Decreased tip loading, in case of positive lean blade, results in increased stall margin. 8. Trailing edge wake for positive lean case is thinner than the no lean case. 9. Grooves which are closer to the shock location, has more distorted flow inside them. 10. The separation line on suction side of the blade is downstream for the negative lean blade as compared to positive lean blade, for the similar mass flow rate conditions. 11. When the lean is started from a lower span, the blade loading increases near hub, then it decreases and finally again increases near the tip. 12. For the case where lean starts from lower span the stall margin is reduced.

Figure 1 :

 1 Figure 1: Mesh Used for Validation

Figure 2 :

 2 Figure 2: Boundary Conditions Used in Numerical Model

Figure 3 :

 3 Figure 3: Comparison of Turbulence Models

Figure 4 .

 4 In order to validate the flow inside the passage, predicted by numerical simulation, spanwise distribution of different flow parameters are compared. For this comparison, flow rate corresponding to design flow is considered. The Hub to shroud variation of Total Pressure Ratio, Total Temperature Ratio and Efficiency are plotted in Figure5through Figure7. It is observed that numerical results compare reasonably well with the experimental data, and hence the same methodology is used for studying the flow field with a good level of confidence.

Figure 4 :

 4 Figure 4: Total Pressure Ratio Comparison

Figure 6 :

 6 Figure 6: Spanwise Total Temperature Ratio Comparison

Figure 7 :

 7 Figure 7: Spanwise Adiabatic Efficiency Comparison

Figure 8 :

 8 Figure 8: Comparison of Rotor37 and Baseline Rotor Geometry

Figure 9 :Figure 10 :

 910 Figure 9: 1:1 Matching Mesh Between Main Passage and GroovesThe comparison of total pressure ratio and efficiency of baseline rotor with and without casing grooves are plotted in Figure10and Figure11. The pressure ratio and efficiency are almost same between with and without groove cases. A stall margin improvement of 3.72% is achieved with the casing grooves. The stall margin is defined as follows.

Figure 11 :

 11 Figure 11: Baseline Rotor With and Without Groove, efficiency

Figure 12 :Figure 13 :

 1213 Figure 12: Generation of Leaned Blades

Figure 14 :

 14 Figure 14: Total Pressure Ratio

Figure 15 :

 15 Figure 15: Total Temperature Ratio

Figure 17 :Figure 18 :

 1718 Figure 17: Mass Flux Comparison at Rotor Trailing Edge

Figure 19 :

 19 Figure 19: Effect of Lean on Stall Margin

Figure 20 :

 20 Figure 20: Points Considered for Comparison of Effect of Lean

Figure 21 :

 21 Figure 21: Blade Loading at 99% Span, With and Without Lean

Figure 23 :

 23 Figure 23: Comparison of Flow Turning

Figure 24 :

 24 Figure 24: Trailing Edge Wake Comparison

Figure 25 :

 25 Figure 25: Blade Loading Comparison

Figure 27 :

 27 Figure 27: Velocity Vectors Inside the Grooves

Figure 28 :

 28 Figure 28: Skin Friction and Streamlines on Blade Suction Side

Figure 29 :

 29 Figure 29: Total Pressure Ratio

Table 1 .

 1

	Lean	Span	Stall	Change in
			Margin	Stall Margin
	Baseline, Smooth Shroud	13.26%
	Baseline, Grooved Shroud	17.53% 3.72%
	5	75	17.95% 4.69%
	10	75	20.17% 6.92%
	15	75	17.41% 4.16%
	25	75	12.06% -1.20%
	-5	75	11.17% -2.09%
	-10	75	14.57% 1.31%
	-25	75	9.09%	-4.17%
	10	50	16.95% 3.69%

Table 1 :

 1 Effect of Lean on Stall Margin

ACKNOWLEDGMENTS

The authors wish to acknowledge and thank Honeywell Aerospace management for allowing the publication of this work.