Introduction

T he contentsofthi spaperwerepresented asl ecturesatthe M i ura W i nterSchool on Zeta and L-functi ons hel d i n 2008. T hough the anal yti c conti nuati on of zeta functi ons beyond i ts regi on ofabsol ute convergence i s a fundam entalquesti on,i n generalnot m uch i s know n about the condi ti ons that guarantee a m erom orphi c conti nuati on.Iti sal so i nteresti ng to know how farsuch a functi on can beconti nued, that i s w here the naturalboundary ofanal yti c conti nuati on l i es.

T he choi ce offuncti ons thatare consi dered here are ' arbi trary' ,thati sa m atter ofpersonaltasteand experti se.M ostofthe work reported i son w hatIhavestudi ed or actual l y contri buted to together w i th m y co-authors. T he word ' som e'i n the ti tl e i s to i ndi cate that though the paper i s exposi tory,i t i s not exhausti ve. O nl y outl i nes ofproofs have som eti m es been provi ded.

In the rst part we consi der Eul er products. O ne ofthe m ost i m portant appl icati ons ofzeta functi ons i s the asym ptoti c esti m ati on ofthe sum ofi ts coe ci ents vi a Perron' sform ul a,that i s,the use ofthe equati on

X n x a n = 1 2 i c+ i1 Z c i1 X n 1 a n n s
x s s ds:

To use thi s rel ati on,one usual l y shi fts the path ofi ntegrati on to the l eft,thereby reduci ng the contri buti on ofthe term x s .T hi sbecom espossi bl e onl y i fthe functi on D (s) = P an n s i s hol om orphi c on the new path. In Secti on 3 detai l s of certai n exam pl es from hei ghtzeta functi ons and zeta functi ons ofgroups have been gi ven.

C l earl y al lzeta functi ons do not have Eul er product expansi ons,one i m portant cl ass ofexam pl es bei ng m ul ti pl e zeta functi ons w hi ch have been studi ed often i n recent years. N ot m any general m ethods exi st and here I treat the case of the G ol dbach generati ng functi on associ ated to G r (n),the num ber ofrepresentati ons ofn as the sum ofr pri m es

1 X k1 = 1 1 X kr = 1 (k 1 )::: (k r ) (k 1 + k 2 + + k r ) s = 1 X n = 1 G r (n) n s :
w here i s the cl assi calvon-M angol dtfuncti on.

In al m ost al lexam pl es the naturalboundary,i fi t can be obtai ned,corresponds to thei ntui ti vel y expected boundary and thi scan i n factbeproved i n a probabi l i sti c sense.H oweverone ofthe di cul ti esi n actual l y obtai ni ng the boundary i sthatour anal yses often depend on the di stri buti on ofzeros ofthe R i em ann zeta functi on, and thus on yet unproved hypotheses (see,for exam pl e,T heorem 3 or T heorem 7 bel ow ).

1 Iwoul d l i ke to thank Jean-Pi erreK ahaneforhi scom m entson T heorem 4 and to K ohjiM atsum oto forhonouri ng m e w i th a kanjinam e. Q u' i l s soi enti cirem erci es! 2. E uler products M any D i ri chl et-seri es occurri ng i n practi ce sati sfy an Eul er product and i fthi s producti ssi m pl eweoften getsom ei nform ati on on thedom ai n ofconvergenceofthe D i ri chl etseri es.A m ong such casesi sthe productoveral lpri m esp ofa pol ynom i al i n p s . O ne ofthe ol dest i deas i s due to Esterm ann [START_REF] Sterm Ann | O n certain functions represented by D irichletseries[END_REF]w ho obtai ned a preci se cri teri on for the conti nuati on to the w hol e com pl ex pl ane ofthe Eul er product of an i ntegerpol ynom i ali n p s . H e proved the exi stence ofthe fol l ow i ng di chotom y :

T heorem 1. Let h(X )= 1 + a 1 X + + a d X d = d Y j= 1 (1 j X )2 Z[ X ] then Z (h;s) = Q p h(p s
) is absol utel y convergent for < (s) > 1 and can be m erom orphicall y continued to the hal fpl ane < (s)> 0. Ifh(X )isa productofcycl otom ic pol ynom ial s, i.e.ifj j j= 1 for every j,then and onl y then can Z (h;s) be continued to the whol e com pl ex pl ane. In allother cases the im aginary axis is the natural boundary.

T he strategy ofhi s proofwas to show that every poi nt on the l i ne < s = 0 i s an accum ul ati on poi nt ofpol es or zeros ofZ. Esterm ann' s m ethod was subsequentl y general i sed by m any authors.

D ahl qui st [START_REF] Ahlquist | O n the analytic continuation of E ulerian products[END_REF] , for exam pl e, extended the above case to h bei ng any anal yti c functi on w i th i sol ated si ngul ari ti es w i thi n the uni t ci rcl e. H e used the concept ofvertex num bers and showed that except for the case w here h(p s ) has a ni te num ber of factors of the form (1 p s ) , there i s a naturalboundary of the zeta functi on at < s = 0.

Later, K urokawa [START_REF]O n the m erom orphy of E uler products[END_REF] conti nued on the i dea of Esterm ann to cases w here h depends on the traces of representati ons of a topol ogi cal group and sol ved Li nni k' sprobl em forthe anal yti c conti nuati on ofscal arproductsofthe H ecke-L seri es L(s; i ) w here i are G r osssencharakters (not necessari l y of ni te order) of ni te extensi onsofan al gebrai c num ber el d.H i sresul tcan be stated m ore preci sel y as:

LetF =Q be a ni te extensi on and K i =F be r ni te extensi onsofdegree n i each. T he scal arproductL(s; 1 ;:::; r )hasthe i m agi nary axi sasthe naturalboundary except w hen (n 1 ;:::;n r ) = (1;:::::: ;1;?) or = (1;::::;1;2;2); i n w hi ch case L(s; 1 ;:::; r ) can be conti nued to the w hol e ofC (i bi d. Part II, T heorem 4).

T here i s ofcourse,no reason to bel i eve thatthe naturalboundary woul d al ways be a l i ne. In an exam pl e i nvol vi ng the Eul er-phifuncti on [START_REF]Sur la repartition des z eros de certaines fonctions m erom orphes li ees a la fonction zêta de R iem ann ,th ese de doctorat[END_REF]

Z (s)= 1 X n = 1 1 (n) s = Y p 1 + (p 1) s (1 p s ) 1 ;
the boundary ofconti nuati on i s an open,si m pl y connected,dense set ofthe hal fpl ane < s > 1.

T hequesti on ofanal yti cconti nuati on ofEul erproductsi n severalvari abl esoccur natural l y i n very m any contexts. To ci te just one exam pl e,i n the study ofstri ngs over p adi c el ds [START_REF] Freund | N on-archim edean string dynam ics[END_REF] , products of 5-poi nt am pl i tudes for the open stri ngs are consi dered,w here the am pl i tudes are de ned as p adi c i ntegral s

A p 5 (k i )= Z Q 2 p
jx j k1 k2 jy j k1 k3 j1 x j k2 k4 j1 y j k3 k4 jx y j k2 k3 dxdy:

T he product Q p A p 5 can be anal yti cal l y conti nued to the w hol e ofC ,w hi ch gi ves i nteresti ng rel ati ons ofsuch am pl i tudes w i th realones. W e woul d thusl i ke a m ul ti vari abl e Esterm ann type oftheorem .Forthi swe need som e notati on [START_REF] Ssouabri | M erom orphic C ontinuation of M ultivariable E uler P roducts[END_REF] . Let us consi der n-vari abl e i nteger pol ynom i al s h k and l et h(X 1 ;:::

;X n ;X n + 1 )= 1 + d X k= 0 h k (X 1 ;:::;X n )X k n + 1 :
T heexponentsofthem onom i aloccurri ng i n thi sexpressi on determ i nea pol yhedron i n R n and enabl e usto gi vea descri pti on ofthe dom ai n ofconvergenceforthe Eul er producti n n com pl ex vari abl esZ (h;s 1 ;:::;s n )=

Q

p prim e h(p s1 ;:::;p sn ). T hus we de ne,for 2 R ,

V (h; ): = d \ k= 0 fs 2 C n j< (h ;si)> k + 8 2 E xt(h k )g
w here E xt(h k ) i s the setofthose poi nts w hi ch do notbel ong to the i nteri orofany cl osed segm entofthe N ew ton pol yhedron ofh k . W e show thatthe geom etry ofthe naturalboundary i sthatofa tube overa convex setw i th pi ecew i se l i nearboundary and gi ve a cri teri on for i ts exi stence w hi ch i s anal ogousto T heorem 1. A pol ynom i alh i n severalvari abl esi scal l ed cycl otom i c,i fthere exi stsa ni te set ofnon-negati ve i ntegers m i;j and a ni te set ofi ntegers i ntegers ( j ) j= 1;:::;q such that:

h(X )= q Y j= 1 (1 X m 1;j 1 :::X m n ;j n ) j :
In [START_REF] Ssouabri | M erom orphic C ontinuation of M ultivariable E uler P roducts[END_REF]we prove that ei ther h i s cycl otom i c,or i t determ i nes a naturalboundary of m erom orphy,i . e.

T heorem 2. T he Eul er product

Z (h;s)= Z (h;s 1 ;:::;s n )= Y p h(p s1 ;:::;p sn ) converges absol utel y in the dom ain V (h;1) and can be m erom orphicall y continued to the dom ain V (h;0). M oreover,Z (h;s) can be continued to the whol e com pl ex space C n ifand onl y if h is cycl otom ic. In allother cases V (h;0) is a naturalboundary.

U si ng N ew ton pol yhedra we can w ri te the above as a product ofR i em ann zeta functi ons and a hol om orphi c functi on i n V (h;1=r),forevery naturalnum berr,i . e.

Z (h;s)= Y 1 jm j N r (hm ;si) (m ) G 1=r (s)
w here m i s a n-tupl e ofposi ti ve i ntegers,fN r g an i ncreasi ng sequence ofposi ti ve i ntegers and G (s) an absol utel y convergent Eul er product. W e then treat separatel y the cases w here the set fm : (m ) 6 = 0g i s ni te or i n ni te to show that a m erom orphi c conti nuati on to V (h; ) i s not possi bl e for any < 0.

A resul t si m i l ar to the above theorem can al so be obtai ned for Eul er products of anal yti c functi ons on the uni t pol y-di sc P (1) i n C n rather than pol ynom i al s (op. ci t. T heorem 4). H owever T heorem 2 i s i n generalnot enough to treat Eul er products of the form Q p h(p;p s ) w hi ch occur,for exam pl e,i n zeta functi ons of groups and hei ght zeta functi ons. In certai n cases authors have been abl e to nd naturalboundari esofsuch Eul erproductsw hi l e even foran apparentl y si m pl e case l i ke f(s) = [START_REF] Sautoy | Zeta functions of groups and natural boundaries[END_REF]we m i ght be unabl e to provi de a com pl ete answer (see the next secti on).

Q p 1 + p s + p 1 2s
In fact i t does not su ce to prove that each poi nt i s a l i m i t poi nt ofpol es or zeros ofthe si ngl e factors,si nce pol es and zeros coul d cancel . In certai n si tuati ons i t i s possi bl e to nd condi ti ons w hi ch ensure that too m uch cancel l ati on am ong potenti alsi ngul ari ti es i s i m possi bl e and thereby get i nform ati on on seri es l i ke the one just ci ted. For i nstance,i n [START_REF]J-C Schlage-P uchta,N aturalB oundariesofD irichletSeries[END_REF]we obtai n : T heorem 3. A ssum e the R iem ann -function has in nitel y m any zeros o the l ine

1 2 + it. Suppose thatf isa function ofthe form f(s)= Q 1 ( (s 1 2 )+ 1 2
) n where the exponents n are rationalintegers and the series P n 2 converges absol utel y for every > 0. T hen f is hol om orphic in the hal fpl ane < s > 1 and has m erom orphic continuation in the hal f pl ane < s > 1 2 . D enote by P the set of prim e num bers p, such that n p > 0, and suppose that for all > 0 we have P ((1 + )x) P (x)

x p 5 1 2
l og 2 x. T hen the l ine = s = 1 2 is the naturalboundary of f; m ore precisel y, every pointofthis l ine is accum ul ation pointofzeros off.

To getthi sresul twe need som e com bi natori algeom etry on the l i nesofD ahl qui st [START_REF] Ahlquist | O n the analytic continuation of E ulerian products[END_REF] . T he fol l ow i ng i s a sketch ofthe argum entto getthe above naturalboundary. B y assum pti on ofthe fal si ty ofthe R i em ann hypothesi s,forevery > 0 and every t therei sa zero = + iT of ,such thatP (T=t) P (T=((1+ )t)) (T=t) l og 2 (T=t), w here = p 5 1

2 . Instead ofshow i ng that thi s parti cul ar cannotbe cancel l ed out by pol es or zeros of other factors, , we show that not al l zeros can be cancel l ed out. If 1=2 p + 1 2 i s not a zero off for any p 2 P and any T p 2 [ t;(1 + )t] ,usi ng com bi natori alargum entswe reach the contradi ctory concl usi on that < p 5 1

2 . So i n every square ofthe form fs :< s 2 [ 1 2 ; 1 2 + ] ;= s 2 [ t;t+ ] g,there i sa zero off. C oncerni ng generalEul er products ofpol ynom i al s i n p and p s ,there exi sts a conjecture [START_REF] Sautoy | Zeta functions of groups and rings[END_REF] .

C onjecture 1. Let W (x;y) = P n ;m a n ;m x n y m be an integral pol ynom ial with W (x;0) = 1. T hen D (s) = Q p W (p;p s
) is m erom orphicall y continuabl e to the whol e com pl ex pl ane ifand ifonl y ifitis a nite product ofR iem ann -functions. M oreover,in the l atter case if = m axf n m :a n ;m 6 = 0g,then < s = is the natural boundary ofD .

T hough al lknow n exam pl es con rm thi s i t i s sti l lfar from bei ng resol ved. In fact we bel i eve that any re nem ent ofEsterm ann' s m ethod i s not enough to prove thi s conjecture [START_REF]J-C Schlage-P uchta,E ssentialsingularities ofE uler products[END_REF] .

W e de ne an obstructi ng poi nt z to be a com pl ex num ber w i th < z = ,such that there exi sts a sequence ofcom pl ex num bers z i ,< z i > ,z i ! z,such that D hasa pol e ora zero i n z i for al li. O bvi ousl y,each obstructi ng poi nt i s an essenti al si ngul ari ty for D ,the converse not bei ng true i n general .

Si nce D m ay not be convergenton the hal f-pl ane < s > ,to conti nue i t m erom orphi cal l y i t i s w ri tten as a product ofR i em ann -functi ons and a functi on R (s) hol om orphi c,zero-free,and bounded on every hal f-pl ane < s > + . T hus there exi st i ntegers c n ;m such that

D (s)= Y n ;m (ns+ m ) cn ;m R (s):
W hen approxi m ati ng D (s) by a product ofR i em ann -functi ons,the m ai n contributi on com esfrom m onom i al sa n ;m x n y m w i th n m = . W e col l ectthese m onom i al s together i n W that i s,we have

W (x;y)= W (x;y)+ X n ;m a n ;m x n y m ;
w here P m eanssum m ati on overal lpai rsn;m w i th n m < (i n [START_REF] Sautoy | Zeta functions of groups: zeros and friendly ghosts,A m er[END_REF]theterm i nol ogy ' ghost pol ynom i al 'i s used).

W e can cl assi fy such pol ynom i al s i nto excl usi ve,non-em pty cases as fol l ow s :

(1) W = W and W i scycl otom i c;i n thi scase,D i sa ni te productofR i em ann -functi ons; (2) W i s not cycl otom i c; i n thi s case, every poi nt of the l i ne < s = i s an obstructi on poi nt;

(3) W 6 = W , W i s cycl otom i c,and there are i n ni tel y m any pai rs n;m w i th a n ;m 6 = 0 and n m < < n + 1 m ;i n thi s case, i s an obstructi on poi nt; (4) W 6 = W , W i s cycl otom i c, there are onl y ni tel y m any pai rs n;m w i th a n ;m 6 = 0 and n m < < n + 1 m ,but there are i n ni tel y m any pri m es p such that the equati on W (p;p s ) = 0 has a sol uti on s 0 w i th < s 0 > ;i n thi s case every poi nt ofthe l i ne < s = i s an obstructi on poi nt;

(5) N oneoftheabove;i n thi scase,no poi nton thel i ne< s = i san obstructi on poi nt.

In the thi rd case we need an understandi ng of the zeros of the R i em ann-zeta functi on to have i nform ati on about the m erom orphi c conti nuati on and as we w i l l see i n the nextsecti on thatthi sm ay onl y gi ve condi ti onalanswers.H oweveri n the l ast case we m i ght be abl e to say nothi ng about the anal yti c conti nuati on as we w i l lsee i n the exam pl e ofD (s)= Q p (1 p 2 s + p s ). W e woul d need som e real l y new i deas to understand Eul er products ofpol ynom i al s i n p and p s .

1.

A random series. From a probabi l i sti c poi nt of vi ew , i t i s usualto study random D i ri chl etseri esand show thatal m ostsurel y they have naturalboundari es. Such generi c condi ti ons com fort us i n the bel i ef that for a D i ri chl et seri es there shoul d be m erom orphi c conti nuati on up to an expected dom ai n.

O ften i n thede ni ti on ofa random seri esthecoe ci entsarerandom (forexam pl e i n K ahane [START_REF] Ahane | Som e random series of functions[END_REF]or Q u e el ec [START_REF]P ropri et es presque sûres des s eries de D irichlet et des produits d'E uler[END_REF] ). In the fol l ow i ng [START_REF]J-C Schlage-P uchta,N aturalB oundariesofD irichletSeries[END_REF]we use random vari abl es i n the exponent to resem bl e the Eul er products W (p;p s ) di scussed before.

W e cal la functi on regul ari n a dom ai n i fi ti sm erom orphi cup to a di screte setof branch poi nts i n the dom ai n,that i s,i t i s hol om orphi c w i th the excepti on ofpol es and branch poi nts. W e can now state the fol l ow i ng probabi l i sti c resul t :

T heorem 4. Let
l i m i nf ! 1 m ax x2 R P ( = x)= 0;
and suppose thatfor > h the series

1 X = 1 j c + j 2 a + b
converges al m ostsurel y. T hen with probabil ity 1 the function

Z (s)= 1 Y = 1 (a s+ b ) c +
is regul ar in the hal f-pl ane < s > h and has the l ine < s = h as its natural boundary.

To gi ve an i dea ofthe argum entsused i n the proofwe l ets 0 = h + itbe a poi nt on the supposed boundary w i th t6 = 0 rati onal ,and consi derthe square S w i th si de l ength 2 n centred i n s 0 . For > 0 gi ven,we show that w i th probabi l i ty > 1 the functi on Z i s ei ther not m erom orphi c on S,or has a zero or a pol e i n S. T hen for a sui tabl y chosen i ndex we consi der

Z (s)= 1 Y 6 = (a s+ b ) c + :
such thati fZ i sm erom orphi con S,so i sZ .LetD 1 be thedi vi soroftherestri cti on ofZ to S,and l et D 2 be the di vi sor of (a s+ b ) restri cted to S. W e show that D 1 + (c + )D 2 i s non-tri vi alw i th probabi l i ty > 1

. T he num ber ofzeros of (a s+ b )i n S equal sN (T + h) N (T ),w here N denotesthe num berofzerosof w i th i m agi nary part T ,and T and h arecertai n realnum berssati sfyi ng T 1000 and h 6. U si ng a cl assi calesti m ate [START_REF] Acklund | U ber die N ullstellen der R iem annschen Zetafunktion[END_REF] ,we can show that D 2 i s non-tri vi al . W e note that i n the i ni ti alstatem ent ofthe above theorem ,the term ' hol om orphi c'appeared i nstead of' regul ar'(T heorem 3,i bi d. ). In fact,as poi nted out by J-P.K ahane,the ni te product of -functi ons dom i nati ng the behavi our ofZ i n a hal f-pl ane< s > h + can yi el d branch poi ntsatal lpol esand zerosofthe i nvol ved -functi ons.

E xam ples

3. 1. Zeta function of a sym plectic group. T he l ocalzeta functi on associ ated to the al gebrai c group G i s de ned as

Z p (G;s)= Z G + p jdet(g)j s p d w here G + p = G(Q p )\ M n (Z p )
, j : j p denotes the p-adi c val uati on and i s the norm al i sed H aar m easure on G(Z p ). In [START_REF] Sautoy | Zeta functions of groups: zeros and friendly ghosts,A m er[END_REF] ,du Sautoy and G runewal d prove that the naturalboundary of the zeta functi on Z (G;s) of the sym pl ecti c group G Sp 6 gi ven by [START_REF] Igusa | U niversalp-adic zeta functions and their functionalequations[END_REF] Z (s=3) = (s) (s 3) (s 5) (s 6)

Y p 1 + p 1 s + p 2 s + p 3 s + p 4 s + p 5 2s
hasa naturalboundary at< s = 4 3 .To show thatevery poi nton the boundary i san accum ul ati on poi nt ofzeros,the authors consi der the parti alderi vati ves at (0; 1) ofthe equati on 1 + (1

+ V + V 2 + V 3 )U + V 3 U 2 = 0 w here V = p 1 ; U = p 4 s .
T he Im pl i ci t Functi on T heorem then guaranteesthe exi stence ofa sol uti on for the above equati on i n U = 1 + V + (V ),w here for p l arge enough (V ) contai ns term s sm al l er than p 2 . T hus for every i nteger n there i s a sol uti on as t ! 1 . In the above,C i s a non zero constant and r the rank ofthe Pi card group of V . T here i s a further conjecture,due to Peyre [START_REF] Peyre | H auteurs etm esuresde Tam agaw a surlesvari et es de Fano D uke M ath[END_REF] ,on the constant C rel ati ng i t to the Tam agawa m easure. W e w i l lconcentrate on P n ,the projecti ve n-space overthe el d Q w i th the cl assi calnorm al i sed hei ght functi on H n :P n 1 ! R > 0 de ned by H (x) = m ax i fj x i j g, w here gcd(x 1 ;:::;x n + 1 )= 1.(O therde ni ti onsofthe hei ghtexi stbutwe shal lnot treat them here. T he i nterested reader coul d see,for exam pl e, [START_REF] Ssouabri | M anin's conjecture on toric varieties w ith di erent heights -preprint arX[END_REF] ).

s = 4 l og(1 p 1 + (p 1 )) l ogp + ( 2n
W e now gi ve detai l s ofanal yti c conti nuati on and boundari es ofa few zeta functi ons i n the above context w hi ch have Eul er products i n severalvari abl es.

2. 1.

A cubic surface. For studyi ng the above case,i t i s possi bl e to rst choose the anti -canoni call i ne bundl e and assum e that i t be am pl e. T hi s then determ i nes a projecti ve em beddi ng ofthe desi ngul ari sed m odelofthe vari ety usi ng a fan decom posi ti on i nto ni tel y m any si m pl i cali ntegralcones(fordetai l ssee,forexam pl e, [START_REF]C om pter des points d'une vari et e torique[END_REF]or [START_REF] Salberger | Tam agaw a m easures on universal torseurs and points of bounded heights on Fano varieties A st[END_REF] ). T he zeta functi on i s then de ned,for < s l arge enough,as

Z U (s)= X x2 U 1 H (x) s :
D e l a B reteche and Sw i nnerton-D yer [START_REF] Sir | Sw innerton-D yer,Fonction zêta des hauteurs associ ee a une certaine surface cubique[END_REF]proved that the zeta functi on associ ated to si ngul ar cubi c surfaces has a natural boundary at < s = 3=4. W e fol l ow the treatm ent of [START_REF]Fonctions zêta des hauteurs[END_REF]to gi ve a sum m ary ofthei r ori gi nalproofw here they study the m ul ti -vari abl e functi on Z (s 1 ;s 2 ;s 3 )=

X x1 x2 x3 = x 3 4 gcd(x1 ;x2 ;x3 )= 1 1 x s1 1 x s2 2 x s3
3 outsi de the uni on ofthree l i nes i n the hyper-surface x 4 = 0. T he Eul er product of thi s functi on i s gi ven by

Y p 1 + P (1 p 3si )p (2sj + s k ) p (3s1 + 3s2 + 3s3 ) (1 p 3s1 )(1 p 3s2 )(1 p 3s3 )
w here i n the sum each ofi;j;k take the val ues 1;2;3. T he above i s then w ri tten w i th the hel p of functi ons ' conveni ent'for < s > 3 4 and F (s) w hi ch i nvol ves the Eul er product ofa pol ynom i ali n two vari abl es W (x;y)= 1 + (1 x 3 y)(x 6 y 2 + x 5 y 1 + x 4 + x 2 y 2 + xy 3 + y 4 ) x 9 y 3 w i th x = p 1=4 ; y = p 3=4 s . H ere agai n the authors succeed i n establ i shi ng that every poi nt on the assum ed boundary i s the l i m i t poi nt ofa subset ofzeros ofthe functi on

F (s)= Q p W (p 1=4 ;p 3=4 s ) Q 4 j= 2 (1 p (1+ j(s 1)
). Fora xed pri m e p, the num ber ofzeros w i th < s > 3 4 ofW ,i . e.

F (s)= Y k;k 0 k k 0 =4+ k 0 =N > 1 (k + k 0 (s 1)) b(k;k 0 ) Y p W N (p 1=4 ;p 3=4 s ) w here W N (p 1=4 ;p 3=4 s )= W (p 1=4 ;p 3=4 s ) Y k;k 0 k k 0 =4+ k 0 =N > 1 (1 p (k+ k 0 (s 1)) ) b(k;k 0 ) :
T he zeros of W N (p 1=4 ;p 3=4 s ) and W (p 1=4 ;p 3=4 s ) are the sam e. Further for every real one can constructa sub-sequence ofi ts zerosw hi ch converge to 3 4 + i and w hi ch are not pol es of

Y k;k 0 k k 0 =4+ k 0 =N > 1 (k + k 0 (s 1)) b(k;k 0 ) :
T hese zeros are agai n the zeros ofF (s) and therefore no conti nuati on i s possi bl e beyond the assum ed boundary.

For w hat concerns the asym ptoti cs,i t i s know n that j x 2 U :H (x) t j= tQ (l ogt)+ O (t 7=8+ ) w here the degree ofQ i s 6 and the l eadi ng coe ci enti s 1 6

Q p f(1 1=p) 7 (1+ 7=p+ 1=p)g.

2. 2.

A n n-fol d product. In [START_REF] Ssouabri | M erom orphic C ontinuation of M ultivariable E uler P roducts[END_REF] ,we consi der i nstead an i m pl i ci t projecti ve em beddi ng determ i ned by a ni te set ofequati ons and do not need a fan decom posi ti on.

Let X be a tori c vari ety and A d;n = A a d n i nteger m atri x al lofw hose row sum s are zero.T he rati onalpoi nts ofthe tori c vari ety are de ned by

X (A ): = f(x 1 ; ;x n )2 P n 1 (Q ): Y i:aj;i 0 x aj;i i = Y i:aj;i< 0 x aj;i i 8jg
and the m axi m altorus U (A ) com pri ses ofthose el em ents ofX (A ) the product of w hose coordi nates i s non-zero. Each poi nt i n the m axi m altorus corresponds to a uni que n-tupl e ofco-pri m e posi ti ve i ntegers w hi ch we denote by (m 1 ;

;m n ). W e de ne a m ul ti vari abl e zeta functi on,for< s i > 1,com parabl eto the one used for tori c vari eti es i n [START_REF]C om pter des points d'une vari et e torique[END_REF]as

Z A (s)= X m i2 N F A (m 1 ; ;m n ) m s1 1 :::m sn n ;
w here

F A (m 1 ; ;m n ) = 1 i f gcd (m 1 ; ;m n )= 1; Y i m aj;i i = 1 8j; = 0 otherw i se.
T he de ni ng equati ons are m ul ti pl i cati ve and we thus get an Eul er product expansi on ofan anal yti c functi on i n n com pl ex vari abl es N ow the functi on h A (X ) i s expressed as a rati onalfuncti on Y

Z A (s)=
2 K (1 X ) c( ) W (X )
forposi ti ve i ntegersc( ),a ni te i ndex setK and an i ntegern-vari abl e pol ynom i al W . W e can prove,usi ng T heorem 2,thatZ A (s)hasa naturalboundary.In fact,i t i s possi bl e to expl i ci tl y descri be the w hol e boundary ofanal yti c conti nuati on (see [START_REF] Ssouabri | M erom orphic C ontinuation of M ultivariable E uler P roducts[END_REF] ,T heorem 6). T he descri pti on ofthe anal yti c conti nuati on ofthi s zeta functi on can now be used to deduce the asym ptoti c properti esofthe hei ghtdensi ty functi on on U (A ) because ofthe equati on

j x 2 U (A ):H (x)= m ax i j m i (x)j t j= C (A ) X m i t F A (m 1 ; ;m n ) w here C (A ) i s a com putabl e constant.
A sa speci alcase,we getasym ptoti c resul tsforthe num berofn-fol d productsof rel ati vel y pri m e posi ti ve i ntegers that equalthe nth power ofan i nteger. B atyrev and T schi nkel [START_REF] Schinkel | M anin's conjecture for toric varieties[END_REF] showed that thi s probl em i s equi val ent to the asym ptoti c descri pti on ofthe hei ght densi ty functi on on the m axi m altorus ofthe hyper-surface x 1

x n = x n n + 1 . N ow that there i s onl y one equati on i nvol ved,i . e. d = 1,we use the m atri x A n = (1;

;1; n) and the rati onalpoi nts are

U (A n ): = f(x 1 ; ;x n + 1 )2 P n (Q ):x 1 x n = x n n + 1 ;; x 1 x n 6 = 0g:
To express h A n (X ) preci sel y as a rati onalfuncti on on the uni t pol y-di sc P (1) we noti ce that i f

h A n (X )= X X for 2 N n + 1 0
,to sati sfy the de ni ti on ofF A n we requi rethatA n ( )= 0.T husfora n-tupl e r,we use the notati on j rjfori tswei ght,i . e.the sum ofi tsn coordi natesand for al lr such that j rj =n i s a non-negati ve i nteger,we l et l(r) = (r 1 ;:::;r n ;j rj =n). Further we ensure that the condi ti on ofcopri m al i ty ofthe p i i s m et and obtai n

h A n (X )= ( n Y i= 1 (1 X n i X n + 1 ) 1 ) X jrj=n 2 N 0 X r1 1 :::X rn n X jrj=n n + 1 :
T hesum i n theexpressi on abovei snotcycl otom i cand thi sgi vesthenaturalboundary of

Z A n (s)= R Y p ( X r2 D n 1 p hl(r);si )
for R a ni te product ofR i em ann zeta functi ons,to be

V (0)= fs 2 C n + 1 j< (hl(r);si)> 0 8 r 2 D n g:
U si ng the above anal yti c properti esand a m ul ti vari abl e Tauberi an theorem [START_REF]C om pter des points d'une vari et e torique[END_REF]we prove that T heorem 5. T here exists > 0 such that

j x 2 U (A n ):H (x) t j= tQ n (l ogt)+ O (t 1 )
where Q n (l ogt) is a non-vanishing pol ynom ialofdegree

d n = 2n 1 n n 1.
A ctual l y we can descri be the l ast pol ynom i alrather preci sel y for al ln 3 ([ 4] , T heorem 7).

U nlucky cases.

In the l ast two subsecti ons we coul d gi ve sati sfactory descri pti onsofthe anal yti c behavi ourofthe Eul erproducts. T hi sneed notal waysbe possi bl e. In the fol l ow i ng we can onl y show the exi stence ofa condi ti onalnatural boundary [START_REF]J-C Schlage-P uchta,N aturalB oundariesofD irichletSeries[END_REF] .

P roposition 1. Suppose thatthere are in nitel y m any zeros of o the l ine 1 2 + it. T hen the function

f(s)= Y p 1 + p s + p 1 2s
has m erom orphic continuation to the hal fpl ane < s > 1 2 ,and the l ine < s = 1 2 is the naturalboundary off.

T hi s i s an exam pl e ofcase (3) ofour cl assi cati on ofthe previ ous secti on. W e noti ce that the realparts ofthe zeros off(s) are exactl y 1 2 and thus we can not construct a sub-sequence ofzeros or pol es w hi ch woul d converge on each poi nt of the presum ed naturalboundary < s = 1 2 . T he condi ti onalresul t above i s attai ned by expressi ng f(s) as a product offuncti ons ' conveni ent'for < s > 1 2 and Y m 1 ((4m + 1)s 2m ) ((4m + 3)s 2m 1) ;

w hi ch i s ofthe type consi dered i n T heorem 3. W e next consi der the Eul er product D (s) = Q p (1 p 2 s + p s ) w hi ch can be w ri tten as

D (s)= Y p (1 p 2 s ) Y p (1 + p s 1 p 2 s )= (s 2)D (s):
W e expect a naturalboundary at < s = 2,or at l east an essenti alsi ngul ari ty at s = 2,and our onl y m ethod to prove thi s i s to approach thi s poi nt from the ri ght. B ut for < s = > 2 we esti m ate the second product as

X p X n 1 j p 2n (2n + 1)s j X p p 2 X n 1 p n ( 2) X p p 2 1 1 2 2 :
So the productforD convergesabsol utel y i n the hal f-pl ane < s > 2,i n parti cul ar, D does not have any zeros or pol es i n thi s hal f-pl ane. T hi s exam pl e fal l s under case (5) ofthe cl assi cati on m enti oned. It i s worse than Proposi ti on 1 w here we coul d not uncondi ti onal l y prove the exi stence of zeros or pol es cl usteri ng on the assum ed boundary w hereas here such zeros or pol es do not even exi st.

N o E uler products

T here are num erouscontextsi n w hi ch we com e acrosszeta functi onsthatdo not have an Eul er product. W e ci te just two exam pl es. T he rst,m enti oned because i tcom es from a contextqui te di erentfrom the otherexam pl es we treated,i s that ofD i ri chl et seri es generated by ni te autom ata.

R oughl y speaki ng,a sequence (u n ) w i th val ues i n a ni te set i s d-autom ati c i f we can com pute the n-th term ofthe sequence by feedi ng the base d representati on ofn to a ni te state m achi ne. O ne ofthe best know n am ong 2-autom ati c cases i s the T hue-M orse sequence, 01 10 1001 10010110 generated by the substi tuti on m aps0 ! 01; 1 ! 10.T he D i ri chl etseri es

P 1 n = 0 un n s
correspondi ng to a d-autom ati c sequence can be m erom orphi cal l y conti nued to the w hol ecom pl ex pl ane.A m ong consequencesi ti sproved [START_REF] End Es France | A utom atic D irichlet series[END_REF]thatautom ati csequences have l ogari thm i c densi ti es. It woul d be i nteresti ng to know how D i ri chl et seri es associ ated to non autom ati c sequences (l i ke the i n ni te Fi bonacciword generated by the substi tuti ons 0 ! 01; 1 ! 0) behave. T he second exam pl e i s i n severalvari abl es. T he Eul er-Zagi ersum de ned as

r (s 1 ; ;s r )= 1 X m 1 = 1 1 X m r = 1 m s1 1 (m 1 + m 2 ) s2 (m 1 + + m r ) sr
has been studi ed w i th m uch enthusi asm . T hi s functi on can be anal yti cal l y conti nued to the w hol e C r space. M atsum oto i ntroduced the general i sed m ul ti pl e zeta functi on

r ((s 1 ; ;s r );( 1 ; ; r );(w 1 ; ;w r ))= r X j= 1 1 X m j = 0 r Y i= 1 ( i + m 1 w 1 + m i w i ) si
w here w i ;m i are com pl ex param etersw i th branchesofl ogari thm ssui tabl y de ned. T hi s too can be conti nued as a m erom orphi c functi on to the w hol e C r pl ane. W e do not w i sh to el aborate on thi s subject but the i nterested reader can nd detai l s el sew here (see,for exam pl e,the exposi tory paper [START_REF]A nalytic properties of m ultiple zeta-functions in several variables, N um ber T heory D ev[END_REF]for references).

4. 1. G oldbach zeta function. H ere we consi der the num ber G r (n); r 2, of representati onsofn asthe sum ofr pri m es. Egam iand M atsum oto [START_REF]C onvolutions of von M angoldt functions and related D irichlet series[END_REF]i ntroduced the generati ng functi on

r (s)= 1 X k1 = 1 1 X kr = 1 (k 1 )::: (k r ) (k 1 + k 2 + + k r ) s = 1 X n = 1 G r (n) n s
usi ng the von M angol dtfuncti on .T hi sseri esi sabsol utel y convergentfor< s > r, and has a si m pl e pol e at s = r. It i s cl ear that to study the anal yti c properti es i n thi s context i t i s necessary to have i nform ati on on the zero-free regi on of ,the R i em ann zeta functi on, and the presence of even one zero of m ay prevent us from havi ng usefuli nform ati on. A l lresul ts that we w i l ltal k about w i l ltherefore be under the assum pti on ofthe R i em ann H ypothesi s (R H ). W e can show that from the anal yti c poi nt ofvi ew ,under R H , r i s determ i ned by the case r = 2 [START_REF] Schlage | M erom orphic continuation of the G oldbach generating function[END_REF] .

T heorem 6. Suppose that the R iem ann H ypothesis is true. T hen for any r 3 there existpol ynom ial s f r (s);g r (s);h r (s),such that r (s)= f r (s) (s r+ 1)+ g r (s) 0 (s r+ 1)+ h r (s) 2 (s r+ 2)+ R (s); where R (s) is hol om orphic in the hal f-pl ane < s > r 1 and uniform l y bounded in each hal f-strip ofthe form < s > r+ 1,T < = s < T + 1,with T > 0.

T hi s i s done by com puti ng the functi on usi ng the ci rcl e m ethod w hi ch gi ve the three m ai n term s. A bound (under R H ) for

X n x (n)e 2 i n X n x e 2 i n
gi ves an error term oforder O (x r 1 ) for som e posi ti ve for al lbut the above three term s. It i s thus i m portant to consi der the si tuati on of r = 2. W e recal l that thi s case occursi n the consi derati on ofthe G ol dbach conjecture thatevery even i nteger l arger than 2 i s the sum oftwo pri m es. To study thi s probl em often i t i s natural to consi der the correspondi ng probl em for and try to show that G 2 (n)> C p n. N ow ,assum i ng the R H ,the authors i n [START_REF]C onvolutions of von M angoldt functions and related D irichlet series[END_REF]descri bed the anal yti c conti nuati on of 2 and for obtai ni ng a naturalboundary they used unproved assum pti ons on the di stri buti on ofthe i m agi nary parts ofzerosof . In thi s contextwe denote the set ofi m agi nary parts ofnon-tri vi alzeros of by . T he bel i efthat the posi ti ve el em entsi n are rati onal l y i ndependenti sfol kl ori c and Fuji i [ 21]used the fol l ow i ng speci alcase : C onjecture 2. Suppose that 1 + 2 = 3 + 4 6 = 0 with i 2 . T hen f 1 ; 2 g = f 3 ; 4 g.

In [START_REF]C onvolutions of von M angoldt functions and related D irichlet series[END_REF]an e ecti ve versi on ofthe above conjecture i s form ul ated,i . e. C onjecture 3. T here is som e < 2 ,such that for 1 ;:::; 4 2 we have either f 1 ; 2 g = f 3 ; 4 g,or j ( 1 + 2 ) ( 3 + 4 )j exp (j 1 j+ j 2 j+ j 3 j+ j 4 j ) ;

and i t i s proven that:

T heorem 7. Suppose the R iem ann hypothesis hol ds true. T hen 2 (s)can be m erom orphicall y continued into the hal f-pl ane < s > 1 with an in nitude ofpol es on the l ine 3 2 + it. Ifin addition C onjecture 3 hol dstrue,then the l ine < s = 1 isthe natural boundary of 2 . M ore precisel y,the setofpoints 1+ i with l i m & 1 j ( + )j= 1 is dense on R .

In [ i bi d. ] the authors conjectured that under the sam e assum pti ons the dom ai n ofm erom orphi c conti nuati on of r shoul d be the hal f-pl ane < s > r 1. N oti ce that a di rect consequence ofT heorem 6 con rm s the fol l ow i ng :

T heorem 8. Ifthe R H hol ds true,then r (s)has a naturalboundary at< s = r 1 for allr 2 ifand onl y if 2 (s) has a naturalboundary at< s = 1.

In [START_REF] Schlage | M erom orphic continuation of the G oldbach generating function[END_REF]i ti sal so show n thati fthe R H and C onjecture 2 hol d true,then 2 (s)does have a naturalboundary at < s = 1 and a si ngul ari ty can be descri bed preci sel y as T heorem 9. Ifthe R H hol ds true, then 2 has a singul arity at 2 1 , where 1 = ( 1)j 2 (2 1 + )j> 0:

T hi sl astresul thel psusobtai n an -resul tforG r (n).W econsi dertheosci l l ati ng term H r (x)= r X

x r 1+ (1 + ):::(r 1 + ) w here the sum m ati on runs overal lnon-tri vi alzeros of . T he generati ng D i ri chl et seri es for

X n x G r (n) 1 r! x r H r (x)
has a si ngul ari ty at 2 1 + r 1,w hi ch gi ves the fol l ow i ng :

C orollary 1. Suppose thatR H hol ds true. T hen we have

X n x G r (n)= 1 r! x r + H r (x)+ (x r 1 ):
In fact the qual i ty of the error term does not i m prove w i th i ncreasi ng r. W e m enti on a few hi stori calfacts about the error term . Fuji i [START_REF] Fujii | A n additive problem ofprim e num bers[END_REF]

obtai ned under the R H X n x G 2 (n)= x 2 =2 + O (x 3=2 )
w hi ch he l ater i m proved [START_REF] Fujii | A n additive problem ofprim e num bers,II[END_REF] ,by expl i ci tl y w ri ti ng the osci l l ati ng term ,to

X n x G 2 (n)= x 2 =2 + H 2 (x)+ O ((x l ogx) 4=3 ):
Further,i n [START_REF] Schlage | M ean representation num ber of integers as the sum of prim es[END_REF]we used the di stri buti on ofpri m es i n short i nterval s to esti m ate exponenti alsum s cl ose to the poi nt 0 and proved that T heorem 10. Suppose thatthe R H is true. T hen we have ) ); as announced i n [ i bi d. T heorem 1C ] .

X n x G 2 (n)= 1 2 x 2 + H 2 (x)+ O (x l og 5 x); and X n x G 2 (n)= 1 2 x 2 + H 2 (x)+ + (x l
G 2 (n)= 1 (q) X n x G 2 (n)+ O (x 1+ o( 1

C onsequences

O ne ofthe am usi ng consequences ofthe exi stence ofa naturalboundary i s to suggestthatthere i sa ' natural 'l i m i tto w hatcan be achi eved forasym ptoti cresul ts associ ated to D i ri chl etseri esby usi ng com pl ex anal ysi s.A naturalboundary coul d show the non-exi stence ofcertai n asym ptoti c resul tsi nvol vi ng errorterm sand thus i m pl y the exi stence ofan i nverse resul t,i . e.an -term . U sual l y w hen provi ng an -resul t we rst deri ve an expl i ci t form ul a w i th osci l l ati ng term s and then show that these term s cannot canceleach other out for al lchoi ces ofthe param eters.In [START_REF]J-C Schlage-P uchta,N aturalB oundariesofD irichletSeries[END_REF]we show thateven i fwe al l ow fori n ni te osci l l atory sum sto be partofthe m ai n term s,we sti l lget l ower bounds for the error term s. T hus a naturalboundary at < s = precl udes the exi stence of an expl i ci t form ul a w i th m ai n term s over the zeros ofthe R i em ann zeta functi on and an error term O (x ). W e state thi s m ore preci sel y as : P roposition 2. Leta n be a sequence ofcom pl ex num bers such thatthe generating D irichl et-series has a naturalboundary at < s = h . T hen there does not exist an expl icitform ul a ofthe form

A (x): = X n x a n = X 2 R c x + O (x )
for any sequence c with j c j (1+ j j ) c and j R \ fs :< s > ;j = sj< T gj T c and for any < h . In particul ar,for any sequence i ; i ,1 i k and any > 0 we have A (x)= X i x i + (x h ):

In practi ce,thei ntegraltaken overtheshi fted path need notal waysconvergeand we m ay not be abl e to obtai n an expl i ci t form ul a. T hi s can happen even w hen the seri esi s m erom orphi c i n the enti re pl ane,for exam pl e,the age-ol d di vi sor probl em w here we have an -esti m ate ofsi ze x 1=4 though the correspondi ng D i ri chl et-seri es 2 (s) i s m erom orphi c on C . H owever, i n certai n cases we can actual l y obtai n expl i ci t form ul ae i f we nd boundson thegrow th oftheD i ri chl et-seri es.W econsi dera caseofan Eul erproduct Q p W (p;p s ) w hi ch we have al ready encountered as the p-adi c zeta functi on of G Sp 6 .T hi scan be i nterpreted asa counti ng functi on [START_REF]C ounting Functions ofC lassicalG roups[END_REF]by establ i shi ng a bi jecti on between ri ghtcosetsof2t 2tsym pl ecti c m atri cesand sub m odul es of ni te i ndex ofZ 2t w hi ch are equalto thei r dual s and cal l ed pol ari sed.

T heorem 12. D enote by a n the num ber ofpol arised sub m odul es ofZ 6 oforder n. T hen we have for every > 0 where runs over all zeros of , and the coe cients c 1 , c 2 , c 3 , and are num ericall y com putabl e constants. M oreover, the error term cannot be im proved to O (x 4=3 ) for any xed > 0.

T he i nterpretati on above al l ow s us to use the zeta functi on Z (G Sp 6 ;s) as the generati ng functi on for a n . A ppl yi ng the M el l i n transform we obtai n

A (x)= 1 2 i 3+ i1 Z 3 i1
Z (s) (s)x s ds:

For and > 0 xed, we have ( + it) e ( 2 )t . W e now choose a path (fol l ow i ng Tur an [ 31,A ppendi x G ] ) to shi ft the i ntegrati on. T he i ntegralon thi s new path i s bounded above by x 4=3+ . H ence,we obtai n the form ul a A (x)= X < > 4=3+ ( )x res s= Z (s)+ O (x 4=3+ ); w here runsoverthe pol esofZ (s),and al lcom pl ex num bers4=3+ =6.W e al ready saw that< s = 4 3 i sthe naturalboundary forZ (s)and asi n Proposi ti on 2,we now get an -resul t.

T he m oralofthe story i s notnecessari l y to getthe bestpossi bl e -resul tbut to show that a non-tri vi alresul t i s obtai nabl e by thi s m ethod.

R eferences

U ni versi t e de Li l l e 1, Laboratoi re PaulPai nl ev e, U . M . R .C N R S 8524, 59655 V i l l eneuve d' A scq C edex, France. bhow m i k@ m ath. uni v-l i l l e1. fr

1 b a .

 1a (a );(b );(c ) be realsequences, such that a ;b ! 1 , and set h = l i m sup ! Let be a sequence ofindependentrealrandom variabl es,such that

  1) i l ogp : N ow for a l arge pri m e p and a xed poi nt A w i th < s = 4 on the boundary we can nd a sequence ofi ntegers n p such that (2n p 1) log p ! = (A ). Further the fact that l og(1 p 1 + (p 1 )) l ogp > 0 for l arge enough p m eans that Z (s=3) cannot be conti nued beyond i ts assum ed boundary < s = 4. N oti ce that thi s i s an exam pl e of the ' l ucky' si tuati on we encountered i n the fourth case ofthe cl assi cati on of Q W (p;p s ). 3. 2. H eight zeta functions. Severalpeopl e i n the recent past have studi ed the anal yti c properti es ofhei ght zeta functi ons associ ated to counti ng rati onalpoi nts on al gebrai cvari eti es.O fparti cul ari nteresti sthe caseofa vari ety w i th am pl e anticanoni calbundl e (cal l ed a Fano vari ety) V over a num ber el d k w hose k-rati onal poi nts are Zari skidense i n V ,for a hei ght functi on H de ned natural l y over the anti -canoni calsheaf. H ere an i m portantm oti vati on i s M ani n' s conjecture that,for U a sui tabl y de ned open subset ofV , j fx 2 U (k):H (x) tgj C t(l ogt) r 1

  arge.N ow for< s > 3 4 + 1 N ,there exi stsui tabl y chosen ni te num ber ofi ntegers b(k;k 0 ) such that

h

  A (p s1 ;:::;p sn ):

1 2 +

 2 14: 1347:::iis the rstrootof . M oreover, l i m & 0

( 1 ) 1 a 6 +

 116 A (x): = X n n e n =x = c 1 x 7=3 + c 2 x 2 + c 3 x 5=3 + X x + 8 O (x 4=3+ );

  ogl ogx): w hi ch con rm s the conjecturalval ue ofthe error term[ 18,C onj.2. 2] . R ecentl y G ranvi l l e[START_REF] Ranville | R e nem ents ofG oldbach's C onjecture,and the G eneralized R iem ann H ypothesis[END_REF]used theerrorterm O ((x l ogx) 4=3 )to obtai n a new characteri sati on ofthe R H ,i . e.forthe ' tw i n pri m e constant'C 2 ,the R i em ann H ypothesi s i s equi val ent to the esti m ate X U si ng the G R H one coul d si m i l arl y nd bounds for the exponenti al sum s i n questi on w hi ch woul d gi ve T heorem 11. T he G eneral ised R iem ann H ypothesisforD irichl etL-functionsL(s; ), m od q,is equival entto the estim ate X

	n x	G 2 (n) nC 2	Y pjn	p 1 p 2	x 3=2+ o(1) :

n x; qjn


Carole Brandon

Marc Veyrat

Alberto Burri

Cretto Di Burri

I-Real Gibellina

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M+i eSPACE o-F DESIRE : the question of the female presence HyperUrbain 7 / Rio De Janeiro / Brasil

The city is often measured to its limits. Urbi et orbi. Finally the boundaries that separate a body from within, ®-IDENTIFIED, urban, from a body outside treated as strange. However, with the advent of digital technology and our social relations made more complex by these different threatening clouds that overhang us, urbanity is first and foremost a matter of networks: our physical signatures, our systemic presences are now diluted in an interlacing of hybrid movements and of memory flows. TOWARDS i: Gibellina, small town of Sicily located forty five minutes from Palermo, disappears in an earthquake in 1968. Several artists are then invited by the mayor Ludovico Corrao -murdered in 2011 -to imagine works echoing this disaster, around the new city, ®-CONSTRUCTED a few kilometers away. Alberto Burri (-! 1915 -1995! -), influenced by the dry salt lakes of California, signs il Cretto di Burri (-! 1984 -1989! -), a gigantic labyrinth, a huge veil of white concrete. The visitor wanders between and in, and in the gaping crevasses, experiencing this sacrificial earth / body scarcely covered, like so many scars of an absence present in WALL >< WALL murmurings. We will develop this communication from two colors -WHITE i-MACULATED and YELLOW CANARIO -to try to explain the stakes of the game between i-&-U of this co-presence of the female body and the surveillance eSPACES BETWEEN [BODIES / MACHINES].1 1/ WHITE i-MACULATED White is the basis of the Middle Ages color scheme with black and red, distinguishing even matt white from bright white. In fact, the Inuit have 50 different terms to describe the variations of this astonishing color. As Michel Pastoureau explains, it becomes synonymous with hygienist terms over time but remains the only color to keep its symbolism forever. 2 Here it interests us first of all because it is at the same time process of enlightenment of a shade and lighting but mainly because the expressions which use it confers the peculiarity to open spaces and spaces-time suspended : "Make a sleepless night, have a white, a white voice, a blank check ..." Thus, it designates a high brightness i+M/PRESSION and, like the work Cretto di Burri, perhaps even a form of blindness, diving into his divine roots. It is the whiteness of the cycle of life, of innocence, of the shroud, of the bride's veil, of the angel as well as the paper, but also on the margin that ®-CONNECT all pages ... Associated with racial theories, fantasies of Suprematism, it becomes a marker, signals a physical barrier, a limit of the i-MACULATED, the focal point of the in/acceptable ... This i-DEAL would therefore seem more than necessary to introduce our first harm (-! to male !-) of the female body in the public eSPACE, the questions of a becoming Other, eternally ®-QUESTIONNED from the territory, its borders and its lighting.

Let's remember David Lynch's obsession with troubled female figures, like the interplay of light and shadow that he paints patiently in his films. The latter oscillate, fluid, between spaces illuminating each take a new way of considering the territory. The light draws, reverses, bypasses, detours, sparkles, opacifies, interferes, brushes, disturbs, splashes ... It confirms the affections of David Lynch for disturbing hybridizations of spaces and time where the double characters, split, inverted, indelimitable, with shapeless contours disintegrate to question the join of spaces between. And the female figure without the possibility of conquering a territory, it seems, shows us this new way to experience them. The crystal velvet Jimmy Scott where his particular voice hesitates, takes over from the light. In Twin Peaks, the light turn beacon in the red, then winding with the sound of the saxophone she pierces black bottoms; pulsating facets of jerky and tragic images, multiplying the possibilities of infinite spaces and shimmering surfaces. If this is no longer a reference to Caravaggio, they are gaps in an unstable cartography. And the light also becomes laser, perforating the spaces to open new ones; causing such Modulateur-Espace-Lumière 3 of Lászlò Moholy-Nagy variations, dispersions, infinities of possibilities to consider and live the same space. Virginia Woolf, in A Room of One's Own, 4 suggests that the prerequisites for women to write are "to have five hundred pounds a year and a room with a door locked." She does not question a binary opposition of the private space and the public space (-! the writing of its own Journal proves it well !-), but the inseparable necessity to bind them. 5 As much material as spiritual, they must remain indissociable. "For such will be our ruin, if, in the immense abstract space of your public life, you forget the intimate image; or if we forget in the immensity of our intimate emotions, the outside and public world. Our two houses will be destroyed, the public building and the private house, the material and the spiritual, because they are inseparably linked". 6 The Room of One's Own does'nt only designate a domestic space, it is also a free space of selfexperimentation, in itself. A new space of lights, a liquid space, a space between. About Jacob's Room, Virginia Woolf still writes in her Journal on Thursday, April 15, 1920, "I think the most important point is that she should be free. But what about the form? Suppose the House will do everything together. The intensity of life compared to immobility. Experiences. Change your style at will".7

Carole Brandon, Canario Box, teaser vidéo 02, 2015, 01'13" https://youtu.be/Swm-kfsyxo8

The author proposes white empty spaces, spaces of silence. 8 These lighting devices allow fragmentation, shimmer, uncertainties. A Sycamore Trees written by David Lynch for Lost Highway, speaks precisely of the uncertainty of seeing: "And I'll see you in the branches that blow In the breeze I'll see you in the trees" The affection of the Impressionists for these oscillations of surfaces, expresses the new palpitations of the world of their time (-! new sounds, new fluidities, new transparencies !-); it is not a question of freezing these moments in the painting but of revealing all the hydridations of spaces announced by the glass, the metal, the fumes, the speeds ... and the place of the women. Perhaps, the notion of border in the public eSPACE, would seem to us a concept eminently masculine and ancestral.9 It would now be ®-QUESTIONED. The first fruits set out by the place and roles of women in the public space since this industrial 19th century. Something mixes up and announces a form of overflow "marked by a growing individualisation and exacerbated mobility, a fluid (Marzloff, 2005) and even fluid modernity (Baumann, 2000) would be emerging. [...] From then on, any attempt to define social sets and thus to establish borders (-! whether geographical or symbolic !-) proves to be futile due to the short-term instability of these sets. By trying to frame fluid or liquid sets, the boundary concept loses its solidity and therefore its relevance or scope". 10 First, the man ®-LOOK. Sentinel i-MOBILE, it becomes in / by its conquests, what we could name as the coefficient ®-BUILDER of a variable with fuzzy contours: the private. This concept is possible only through the stage of the transition to surveillance and, through it, we will try to understand how a woman is discovered in other brutally deterritorialized ?+) Always projected to the dawn of a tragedy, like Helen of Troy, ultimately invisibilised and finally dispossessed of any imaginary of territory -because in the game between i-&-U of all territories -and definitely above ground.

The question of clothing -its ®-KNOWLEDGE protocols and the identification of a Other Woman but also of all LGBTQIAP+ minority minorities -thus becomes a producer in the public eSPACE, of the affirmation of a crossing, a setting at the bottom of the borders through a white i-MACULATED; a process of blindness but also of revelation (-! like the veil of Veronica existing only because it carries the image of Jesus, and through him the image of God, the father !-) ; i-MACULATED because necessarily associated with a window / victim targeted and potentially affirmed by this fault -the crossing of the public eSPACE -consecutive of this deterritorialization. On March 31, 2008, Pippa Bacca was brutally raped and murdered during her performance: crossing countries in wedding dress, from Milan to Jerusalem ... Pippa is everywhere, Pippa is in us, her commitment to the power of art to cross the borders. Pippa clashes with the ignoble human reality. Pippa is here,11 Pippa is in Turkey. in 2010, the director Bilgos Elmas turns the documentary My letter to Pippa. In the latter, the filmmaker, dressed in a black dress, travels the same path as the Italian artist, from where she was found dead to the Syrian border. Pippa is in France. In 2012, filmmaker and visual artist Joël Curtz pays homage to this committed woman. In The Bride, he reveals the unpublished images shot by Pippa Bacca during his journey. Pippa with Nathalie Léger's La Robe Blanche returns in 2018. 12 The book resembles a long complaint, a line floating in an uncertain fog where the white neither halo nor virginity, terrorizes its light, petrifies its goodness. The women artists (Marina Abramovic, Niki de Saint Phalle, Gina Pane, Ana Mendieta ...) seem to find in the white color, unexpected forces of unequaled power. The white, symbol of peace, immunizes and humanizes ?+) Nevertheless it is for better the task that they dress of white. Nathalie Léger explains us that at the beginning, there are two dresses, a guarantor, the other witness. The witness of what should have been a message of peace, has become proof of rape and murder. The change was so sudden. The passage of the borders finally closed.

In front of this No Limite, this existential rage, this search for the drunkenness imagined by Pippa in the very concept of his work, the response of the GUARD barrier can only be brutal: death is inevitable. Now, what becomes particularly interesting with regard to this question of deterritorialization is of course that this i-STORY can continue to ignore the borders -in a beyond this disappearance, this negation -by social networks and the Internet. Pippa is no longer but interferes in a memory instance between territory and digital: an eSPACE BETWEEN [BODY/MACHINE], a form of PARADE-i White i-MACULATED, a line of sound, abstract, the in/visible of a song, the sadness of anger. Denn es gehet dem Menschen wie dem Vieh. 13 In fact, the fate of man and that of the brute is the same. An excerpt from this song is played by Kathleen Ferrier in the music of Claude Chabrol's film Que la bête meure. 14 The film was released in theaters in 1969. And that's not a coincidence. Because this one already questions intrinsic concepts of what will become our daily digital through a learned process of nesting. Two men, a woman. Helen is Paul's sister. Charles lost his son, crushed crossing the street by Paul's car, which did not stop. Another border business. Between life and death. The street of good and evil. At the end of the film, when we learn with Charles (-! That we see intermittently writing the scenario of the killing of Paul in the film's scenario! -) and Hélène (-! Movie starlet of a blond i-MACULATED by the beast reminding us through his name the permanence of the tragedy !-) that Paul, the, the beast, is dead, it is through a television in black and white. Already, the information can only be trusted, in the public eSPACE and for ourselves, through a screen. There is a gap between the act and its ®-TRANSMISSION. Enshined as a relic that takes only the distance from the i+M/PORTANCE; in a way, in a form of programmed revelation, each path of i can only live on / by its expansion on the networks, abstracted by its own i-REAL. From all of our facial recognition surveillance points, the death of our becoming subject is programmed by the public space. Our survival seems to take place in the disconnected ubiquitous fluidity … in the uncertain interspaces.

Another scene of revelation: Charles trains Paul on a small sailboat in an outing at sea. To announce that his end is near. Here the i+M/PERMANENCE (-! The trace of the material information perceived in its impermanence -) of the tragedy in the boat box is ®-PLAYED -like the wedding dress of Pippa -by the white veil. The sail slams in the wind. White is i-MACULATED by the menis, a form of existential rage. "Sing to us, goddess, the anger of Achilles, the son of Peleus, whom he conceived angry because of the daughter of Chryses, a detestable anger, which to the Achaeans brought innumerable misfortunes." Once again the Other, Other than masculine is unseen, as absent, but makes noise. And the erasure by the crossing produces an anger that causes in a disappearance ®-TURN. This abstraction of a call for help is the sail for the bride's veil. The exposure, the crossing which engraves a without ®-TURN possible. The public eSPACE can only produce a process of blindness, deletion or transfiguration of the female body, which would only be revealed through the crossing of these eSPACEs. Claude Chabrol, after the events of 1968 and a few years before the event of the 343 sluts, produced here a reflection on what we could mean as a dynamic of the Fresh Widow. 15 Decidedly, the beast can die only on the condition of an experience of the feminine, that is to say the Other definitely Other ®-AFFIRMING ceaselessly a No Limit in the conquest of the eSPACE ... Au price of a metaphysical crossing. What are the headwinds ?+( The machine and its program -its rules, its laws -can only be translated into a machination, a ®-SETTING UP (-! By means of the computer !-). is the subtlety of the work of Ian Hamilton Finlay In his Scotland garden, Little Sparta, 16 the phrase of Saint-Just, the present order is the disorder of the future is engraved on stones, or rather slabs (-! Steles ?-) separated, a little as if the line (-! of force !-) had been broken (-! of force !-), crossed by a look, a bottle, fabric and the alcohol that would drag us behind the horizon, beyond the barrier of our gaze. The white corrector.

From the Hausmanian transformations of Paris to compensate for the anonymity of social spaces created by the capitalist organization of work, Walter Benjamin remarks "the emergence of a new concept of experience [...] that refers to the life of the particular individual".17 This is accompanied by an interest in the interior and the private houses in which the woman shelters, at home. Heart seems to be a system, it is dedicated exclusively to household chores. And at the same time, this modern organization of the public space (-! which is not yet eSPACE !-) creates new places of consumption, department stores and passages. Junction between domestic space and public space, they allow the honest woman to conquer a new mobility. "Such an active behavior of representation on the mercantile mode makes them join the structural position of the prostitute, although this role is accompanied by another role, foreign to the prostitute: that of the consumer".18 The two bodies of one and the other meet within these particular spaces. In the pictorial (-! and literary !-) representation, this zenemy is found in the composition and the place of the bodies on the surface of the canvases. The public sphere, forbidden to women by ancestral tradition, is allowed little by little in painting and urban space. The house and the room persist as double and reversible spaces, opening new doors (-! the bathroom, the balcony, the garden !-). And the freest and liberating figure of the emergence of this intermediate space (-! urban and pictorial !-) is the prostitute.

In this patriarchal cartography the misogynist nineteenth century installed moralizing social codes and the legal incapacity of women in the civil code of 1804 instituted by Napoleon 1st. Michelle Perrot studies the gender of the city and explains how "migrations have introduced a certain confusion of spaces and sexes.

From where progressively, the desire to order the city by circulation of the flows and specializations of the spaces". 19 While the public space remains a bitter and brutal battleground for women's reappropriation, it has helped modernize social relations. The woman embodies the conquering figure of this confusion of spaces of a controlled body that emancipates itself by becoming multiple in the image. Indeed, the passages and department stores according to Walter Benjamin, inaugurate intermediate spaces, no street, no room, no inside or outside. Transitions, they change the gestures and the postures, the women mix and join spaces of metamorphoses and meetings. They will reveal differences and differences by linking the possibilities of being a woman: the range widens between married women or not, between women and men consumers.

There is no longer any question of a white horse or Napoleonic costume, Madame Réamier's opaque white dress, portraits of the Serziat or L'enlèvement des Sabines of a David borrowed from certainties and praise of a normed Greek beauty where the white opacify and impose itself as a value of power. It is the tremors of a turban on the face of a Negress, a powerful naked body in the bed of an Olympia, the stroller who seeks the kaleidoscopic flights of the city. Faced with the black and gray reorganisations of public spaces (-! Eiffel Tower, paved roads, railroad ... !-) gas lighting contributes to this artificiality veiling all possibilities to interfere by fragment, by piece. And our smartcities are of the same order. Our public spaces that have become eSPACEs no longer need metal projections to build power because they now have surveillance cameras, hyperconnections, recordings of everything we do. If in the nineteenth century transparency evoked games of reflection, abstract inter-spaces, adventures of seeing, today smart opaque means ease, the ease offered to monitor us, to scrutinize us, to record us individually and continually , from public space to private space; the smart is an invisibilisation of these spaces between, spaces become spaces of algorithmic computations to constrain and categorize the bodies. The smart is the total opacity of the white, it is the frontal lighting of Bentham's Panopticon20 described by Michel Foucault "the full light and the look of a supervisor captures better than the shadow, which finally protected. Visibility is a trap".21 What makes it possible to ensure the automatic functioning of power continues: "The Panoptic is a machine to dissociate the couple to see and be seen, without ever seeing; in the central tower we see everything without ever being seen". 22 Now migrating into the public space, it is the impossibility of breathing, wandering, new experiences of the collective space. Street art, showmen, skaters, acrobats, now have their dedicated spaces. The street is no longer a space to live but to undergo to go from one point to another under surveillance. In lyon then in other cities, a collective has created a collaborative mapping to identify the locations of surveillance cameras23 : their purpose is to allow us to identify areas of unsupervised areas of shadow of spaces of freedom. These areas appear on the map as uncategorizable gaps. Whose boundaries are not imposed by "a certain concerted distribution of bodies, surfaces, lights, looks". 24 Certainly, we are still far from Edouard Manet's Déjeuner sur l'herbe 25 where the naked, scandalous body of this woman in the foreground -near this inverted fruit basket -escapes by the repetition of the body of this other woman in the background, draped in white, for all to lead us to the light, the gap between the trees ... The information society brings us back to opaque white, a barrier material. Like the Facebook wall or museum wall what could be a passage, highlighted, we finally partition into the opacity of white ... separates us again and again through questions of color.

2/ YELLOW CANARIO

Whence the WALL >< WALL murmurings evoked in 2015 compared to the Carole Brandon artwork Canario Box 26 which took place in 3 cities: Chambéry, Curitiba and Dubrovnik. With this interactive street art piece, silhouettes in white fabric at the scale of one of the participants are stuck in the city center. Each silhouette has a QRcode: it allows to file a very short voice message. This message once saved gives access to two boxes: the previous message left by another user before him and a map where are geolocated all the silhouettes in the city. Each silhouette thus visible on the map of the application ®-PRESENT a special association of Chambéry helping the poor. Each association is ®-LINKED to a Facebook page. To be able to like the page, the user must first have left a message, agree to be geolocated at the location of the fabric silhouette and therefore of it on the map. This work works in closed circuit. You have to be in the street and walk the public eSPACE to discover the work, which does not work or is visible or accessible directly on the internet.

Canario Box thus composes a Russian dolls game where the eSPACES are nested and interconnected in each other. There are first tangible ones with the city to go physically: the points of i+M/PACT, places of our presentation to the world confronted with the places of the ®-PRESENTATION by the question of the traces, those registered of our in the city, those of the silhouettes on the walls, a sensitive cartography of the territory also drawn by the silhouettes in the streets ... But there are also the places of the intangible where we will cause links, the QRcodes that open on a new eSPACE (-! in/visible, in fact visible only through the screen but all the time !-): the web, the Canario Box site and its dedicated application. Finally, leave a message ®-SEND not only to the i+D to abandon a small fugitive trace (-! eventually replaceable term !-) in the public eSPACE but also to open a new dimension of this last in order to ®-CREATE the social bond, individually, claimed, signaled and signed through these WALL >< WALL murmurings / thinkable wall, facing the hubbub of the city. By drawing a ®-CONNECTION with the other messages, the map, the silhouettes, the identified pages of these associations present in the city and ®-PRESENTED on the social network Facebook, each Canario Box becomes PART-i of the public eSPACE.

With the hypermedia artwork, these eSPACES thus fit in an ordered, superimposed and hierarchical cut: mathematical, quantifiable and comprehensible. Each voluntarily isolated fragment becomes a box of secrecy which functions a priori in autonomy and in complete independence of the city and the Internet. But to activate and appropriate the whole of the work it is necessary to cross all the eSPACEs nested in each other in order to have access to the data and to understand the complex interconnected devices that ultimately articulate the public eSPACE. If this characteristic nesting is not trivial and recalls some video and / or interactive installations such as Dan Graham's Present Continuous Past(s), [START_REF] Graham | Present Continuous Past(s), 1974, installation vidéo circuit fermé, 1 caméra noir et blanc, 1 moniteur noir et blanc, 2 miroirs, 1 microprocesseur[END_REF] unlike closedcircuit installations, this process of image-making here is not not the purpose of the work. The spectator must patiently peel, deconstruct, appropriate the different strata that structure the public eSPACE -City >< Web >< social network Facebook -who closes it if he wants to join this community, communicate, break the WALL >< WALL barrier, validate, even amplify its connections with the public eSPACE that surrounds it. In itself, if this device does not allow to communicate directly or visibly, it proposes by the dismantling of this in/visibility, potential exchanges outside profiling and monitoring. Obviously, in this operating system of ®-LOOK without being seen, all this is reminiscent of the very concept of moucharabieh. [START_REF] Brandon | 100 Notions pour l'Art Numérique[END_REF] From the muffled secret, from the individual rustling of anger, from a rising clamor, from a form of sizzle -recalling, of course, the brutal question of the dazibao or the sizzling backs of Jacques-Louis David behind Madame Récamier [START_REF] David | Portrait de Madame Récamier[END_REF] or La Mort de Marat [START_REF] David | La mort de Marat[END_REF] -who are impressed on the place of the public affair, this res publica of the street can only be echoed with the successive acts of the gilets jaunes; a form of exuberant overflowing a priori language, which is reminiscent of the slogans situationist thrown on the walls during the month of May 1968. But it is especially another way to open the eSPACE around usno not by dragging us behind -but like Lucio Fontana's concetto spaziale, [START_REF] Fontana | concetto spaziale[END_REF] affirming an iconoclastic gesture to reflect and make us think about the eSPACE by the trace performance. Each one of them, at the same time signature and sign of the anonymous (-! Of the Other !-) occupies the eSPACE with us and even between our absence (-! Or from the interest of the ghost silhouettes !-), reveals it, translates it, affirms a ça a été of a être public contre d'-i as an extension of our bubble bodies burst on the walls, a shared vertigo.

The yellow i+D/sign, 32 strange, also resonates in our memories because it is also that of the STRANGE-®, a fixative disorder. "With Christianity, the Middle Ages degrades the value associated with yellow. After the failure of the Crusades, we look for enemies inside the country. We create exclusion codes, infamous clothes to stigmatize. Judas is first represented with red hair, then wears a yellow dress from the twelfth century. Yellow too, the red or star that refers to the Jews from the thirteenth century in European cities; a sign that the Nazis will resume in the 1930s. Yellow becomes the color of ostracism. Medieval symbolism makes yellow the color of cheaters, traitors, felons, lies, sickness". 33 To associate it also with madness will not make yellow much more attractive in the public eSPACE, unless of course to use it as we mentioned before, as a reflective signal (-! that it is not necessarily yellow !-) with the use of the safety triangle 34 ...

But what seems even more surprising with these tweets resonating in the public eSPACE is always the presence / absence of the good / bad female role. In Allegory and Effects of Good and Bad Government, set of frescoes painted by Ambrogio Lorenzetti between 1338 and 1339 on the walls of the Sala dei Nove (-! the Hall of Nine !-) or Sala della Pace (-! Peace Hall !-) of the Siena Palazzo Pubblico, we will focus more particularly on Effects of bad government in the city and the countryside. Indeed three women ®-PRESENT alone all the harmful effects of a hypothetical bad government, which could be put in place from a problem of political mediation of the Communal State. In the center of a long platform planted opposite us, we find Tyrannia, who, under a demonic aspect, eats the Justice at his feet. Above them hang three women who are evil: Avaritia, an old woman dressed with bat wings; Superbia, Pride, unsheathed sword and crooked yoke and Vanagloria, Vain Glory, who delights in a mirror. True Memento M-O+O-R_i this last replaces the Lust traditionally present. In fact, misery, abuse, destruction and famine come. And finally the only potential craftsman of a semblance of device under construction is then the blacksmith who makes weapons !+) If, at the time when this fresco is ordered, Siena is in the grip of famine, death and insurrections, the historical reference to the work of Ambrogio Lorenzetti aims at a problematic of occupation and political mediation of the Communal State, where the trouble gaining the public eSPACE (-! Greek kairo, that is to say finally, the decisive tipping point opened by the WALL >< WALL !-) questions the very notion of the common good continually torn between res publica and underlying desire, in/visible, to put us in a supreme authority, or even to suffer the yoke of tyranny. Thus, every system of order or disorder proposed au Jour du i, questioned here by the work Canario Box, reflect this necessary awareness of the link urbi et orbi, of the city and its contado.

Like the Dead Drops 35 whose initiative is driven by the artist Aram Bartholl in 2010 in New York -USB keys are sealed in the gaps of the urban space and allow off Internet networks to exchange digital data; the anchor points are mapped on his site, the work is participative [START_REF]D'autres artistes les déposent en hacking urbain sans forcément les géolocaliser[END_REF] -Canario Box invites us to create our own bubble eSPACE in the public eSPACE. If leaving and saving a message allows access to other messages recorded before us in the same place, access to the map that geolocates all the silhouettes, affirms our support for local associations to help the most vulnerable people in society.

In the line of Traces, [START_REF]L'application iPhone créée en 2014 par l'entreprise Ripple[END_REF] an application only for iphone released the same year, drop messages in the city protect them, and as in a safe makes them valuable. However unlike the commercial application that geolocates and keeps the messages for a lucrative use, with the artwork Canario Box, the messages are left on our private server, closer to the prayers filed in the Wailing Wall in Jerusalem [START_REF]Considérés comme l'endroit le plus saint pour la prière Le Mur des Lamentations est le terme occidental pour désigner le mur de soutenement de l'esplanade du temple de Jérusalem, dans le quartier juif et de l'autre côté, du Mont Du Rocher où se situe la Mosquée Al Aqsa de Jérusalem-est. An 17 ap JC. Les milieux juifs et arabes le nomment plutôt El Bourak du nom de la monture de Mahomet qui s'y est élevé au ciel lors de son voyage nocturne. Pour les juifs la place est spéarée en deux d'un côté les Hommes et de l'autre côté, plus petit pour les femmes[END_REF] or the Ema 39 in Japanese temples. Indeed on the fifty registered messages, users considered it as filing a secret. To confide is perhaps to lighten up body? But the participants also played with the message itself, hidden anonymously therefore carrying fictions, lies, fantasies. Unfortunately, the center of Chambéry, classified city heritage of art and history, freezes and forbids any unexpected proposals. The silhouettes were stuck for one hour for some and fifteen days for others (-! because on empty stores or nooks already tagged !-) We could not let the messages proliferate and study the virality, the impact and even the use that users could make of them. Gwen Rigal, lecturer at the Cave of Lascaux, in "Le temps sacré des cavernes", 41 evokes the i+D of a myth of Genesis, a myth of the origins, a participating cave (-! André Leroi-Gourhan !-): "At the beginning of time, humans and animals were not yet clearly differentiated between them and lived together underground. And one day, for one reason or another, they decide to rise to the surface following the cracks in the rock. And finally emerge through the entrance porches of the caves. And at the same time that on this occasion sprung life, obviously, death appeared. As all this happened at a time when the rock was still soft, and well the walls of the caves have kept until today the trace of this original passage. From where perhaps the fascination of the prehistoric men for the reliefs. Perhaps they were looking for clues to this original passage ... as if something was happening on the other side of the wall". 42

In this i-REAL 43 we would therefore be au Jour du i -the new day of our immersion in mixed realitieswithout looking for these cracks guaranteeing with our bodies and machines of vision, our co-presence in the world. A slippery, soft, ever more abstract world, we i+M/POSING oscillating movements between a body of pre-historical touch (-! the body below, animal, buried in the earth !-) and a blinding light i-MATERIEL (-! the programs and their digital avatars !-) projected ever further in a hyper-urban environment.

Carole

Carole Brandon ,

 Brandon KISS & FLY, stop motion de la totalité des images enregistrées par les webcams 2ème semaine de l'exposition Jeux d'artistes au Musée Château d'Annecy 15 décembre 2012 / 30 mars 2013 https://youtu.be/Bw-Ow2wtyvk

  Canario Box, like Kiss & Fly -another hypermedia installation created by Carole Brandon for the exhibition "Artists' Games" in 2013 at the Musée-Château d'Annecy -designates this strange eSPACE, suspended between tangible and intangible -definitely unlikely -between the physical entry of our body and this supernatural dimension of the city or airport as a system of order (-! the Stargate !-), a Co-MOTION taking us beyond this physical projection to another world, the place of a common being in an afterlife what our ME &-i remains attached to the world ... Marc Veyrat, i-REAL Monde 01, Société i Matériel / 89/92 R&D / Pixelpirate, 2019 Avec Matthieu Quiniou, Samuel Da Silva, Véronique Gay-Rosier, Jonathan Juste, Lise Missillier, Roberto Barbanti Un projet KITCHEN XR / Carole Brandon & Salar Shahna "When the viewer passes in front of the ©box (-! virtual gallery, sculpture-space and multiple created in 2003 !-), webcams record his image, which makes visible the traces of his passage in a landscape which exists only by the movements of his body around the machine (-! the ©box !-). His movements reveal on the video-projected image both the computer program that generates the interactivity, a pre-recorded narrative panoramic landscape (-! videograms !-), the body positions around the ©box, the recording of webcams and the reception area (-! the museum !-). The image of the viewer makes the whole process visible in a parallel life form. Like painting, he contemplates a landscape-program that runs in front of him but can only be seen with him in the device". 40 This hybrid form between digital material and projection screen, ®-GIVEN to the visible, would therefore depend on a machinery ensuring, with the body, the Co-MOTION between tangible and intangible worlds, a BETWEEN[BODY/MACHINE]. And might even define the SUR-NATURAL conditions of a possible immersion.

  Brandon / Marc Veyrat, 2019

41

Gwen Rigal, Le temps sacré des cavernes, José Corti, 2016. 42 https://www.franceculture.fr/emissions/la-methode-scientifique/la-methode-scientifique-emission-du-jeudi-25-avril-2019 43 i-REAL est une oeuvre d'art numérique et hypermédia de la Société I Matériel (-! co-production 89/92 R&D / Pixelpirate !-) qui mixe des environnements en VR déclenchées à l'aide de cartes sur un plateau de JE(U). Des indices disséminés sur les cartes, le plateau, les réseaux sociaux et dans les environnements VR pourront à terme déclencher une blockchain... Sur lequel règne un maître du JE(U) : une IA également en cours de développement... https://youtu.be/3NPY511eGzQ

Carole Brandon, L'ENTRE [CORPS/MACHINE], La Princesse et son MAC, thèse en Arts & Sciences de l'Art,

2016, sous la direction de Françoise Parfait, Université Paris 1 Panthéon -Sorbonne, https://hal.archives-ouvertes.fr/tel-02111485/. 2 Pastoureau, M. et Simonnet, D., Le Petit Livre des Couleurs, Point Histoire, 2014. 3 Moholy-Nagy, Lászlò, Modulateur-espace-lumière, 1930, 151 x 70 x 70 cm, métaux divers, plastique et bois. http://www.fam1500.uqam.ca/fresque/1930_moholynagy_01.htm 4 Woolf, Virginia, Une chambre à soi, trad. Clara Malraux, (collection 10/18, 2001).5 Henry James parlera de "House of fiction" pour désigner un habitat mental ouvert sur le monde protégeant l'expérience. In Henry James, La création Littéraire (the Art of the Novel), (Denoël/Gonthier, 1980).

Woolf, Virginia, Trois guinées, trad. de l'anglais par Vivian Forrester, (10/18, 2002), p.227-228.

Woolf, Virginia, Oeuvres romanesques, (Gallimard, bibliothèque de la Pleiade, NRF, 2012), p.1396.

Woolf, Virginia, Diary, tomeIV, 1931-1935, 17 juillet 1935, (Haverst Books, 1983). Ces espaces sont étudiés par Edward L. Bishop, "Mind the gap : the spaces in Jacob's Room", (Woolf Studies Annual, vol.10, 2004), p.31-49.

Raibaud, Yves, 2015. La ville faite par et pour les hommes, https://hal.archives-ouvertes.fr/hal-00974265/file/Une_ville_faite_par_et_pour_les_hommes.pdf

 10 Belton Leslie, De la permanence du concept de frontière. Les liens entre travail et vie privée à La Défense, Espaces et sociétés, 2009/3 (n° 138), p. 99-113. DOI : 10.3917/esp.138.0099. https://www.cairn.info/revue-espaces-et-societes-2009-3-page-99.

htm 11 https://www.

pippabacca.it/ 12 Brandon, Carole, https://www.facebook.com/photo.php?fbid=403476043542996&set=a.118435082047095&type=3&theater

Brahms, Johannes, Vier ernste Gesänge (Quatre chants sérieux), op. 121 (1896), Source Wikipédia.

Chabrol, Claude, Que la bête meure, 1969, (France / Italie), 01h53'.

Duchamp, Marcel, Fresh Widow, 1920 -1964, bois peint, cuir, 79,2 x 53,2 x 10,3 cm. Inscriptions : S.D. au verso sur la tablette : Marcel Duchamp 1964 N. INSC. au verso sur la tablette sur la plaque de cuivre : Marcel Duchamp 1964 / Ex / Rrose / FRESH WIDOW, 1920, EDITION GALERIE SCHWARZ, MILAN, Collection Centre Georges Pompidou, Paris.

Ian Hamilton Finlay, Little Sparta, http://www.ianhamiltonfinlay.com/ian_hamilton_finlay.html

Gagnebin, Jeanne-Marie, Histoire et narration chez Walter Benjamin, (L'Harmattan, 1994), p.92.

Buchet-Rogers, Nathalie, Fictions du scandale, corps féminin et réalisme Romanesque au dix-neuvième siècle, Perdue University Press, West Lafayette, Indiana,

1998), p.40.

Perrot, Michelle, Le genre de la ville, in Communications, 1997, volume 65, numéro 1, p.151.

Foucault, Michel, Surveiller et Punir,Tel, Gallimard, 1975, p.233

Ibid. p.234.

Site de la cartographie et présentation du projet : https://lyon.sous-surveillance.net/

Ibid. p.235

Manet, Edouard, Le Déjeuner sur l'herbe, 1863, huile sur toile, 207 x

cm, Musée d'Orsay, Paris. 26 Réalisé avec Arnaud Burgniard (développement de la carte) et les étudiants du Master 1 Hypermédia Communication promo 2013-2014.