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b(accomplished)h(b)-36 b(y)434 b(the)h(follo)-36 b(wing)436 b(theorem)800 10522 y(of)e(A.)g(Cobham)g([)p 0 1 0 0 TeXcolorcmyk(3)p (#cite.Cob) [[159 622 164 634] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)p Black 800 12852 a Fu(Theorem)403 b(2.)p Black 489 w Fj(L)-66 b(et)386 b Ft(\()p Fr(a)12043 13051 y Fq(n)12669 12852 y Ft(\))13175 13051 y Fq(n)p Fp(\270)p Fo(1)15391 12852 y Fj(b)-66 b(e)387 b(an)g(automatic)g(se)-66 b(qenc)g(e)385 b(over)i(an)h(alphab)-66 b(et)387 b Fm(B)40 b Fj(.)572 b(Assume)388 b(that)f(for)g(some)800 14405 y Fr(a)369 b Fm(2)f(B)506 b Fj(the)465 b(limit)e Fr(\261)10307 14604 y Fq(a)11231 14405 y Ft(=)369 b(lim)14418 14604 y Fq(x)p Fp(!1)17269 13882 y Fo(1)p 17240 14099 530 54 v 17240 14863 a Fq(x)17902 14405 y Fm(jf)p Fr(n)h Fm(\267)f Fr(x)g Ft(:)g Fr(a)24004 14604 y Fq(n)24999 14405 y Ft(=)g Fr(a)p Fm(gj)465 b Fj(exists.)596 b(Then)465 b Fr(\261)36609 14604 y Fq(a)37629 14405 y Fj(is)g(r)-66 b(ational.)2128 16750 y Ft(In)434 b([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.All) [[109 566 115 578] [1 1 1 [3 3]] [0 0 1]] pdfm Black(],)h(J.-P)-108 b(.)433 b(Allouc)-36 b(he)434 b(used)f(this)g(to)h (pro)-36 b(v)g(e)433 b(the)g(follo)-36 b(wing)436 b(result.)p Black 800 19080 a Fu(Corollary)602 b(1.)p Black 589 w Fj(The)530 b(p)-66 b(ower)530 b(series)f Fr(f)142 b Ft(\()p Fr(x)p Ft(\))488 b(=)24071 18084 y Fl(P)25473 19468 y Fq(n)p Fp(\270)p Fo(1)27302 19080 y Ft(\()p Fr(\271)p Ft(\()p Fr(n)p Ft(\))368 b(mo)36 b(d)369 b Fr(p)p Ft(\))p Fr(x)35507 18598 y Fq(n)36662 19080 y Fj(is)530 b(tr)-66 b(ansc)g(endental)527 b(over)j Fi(F)50521 19279 y Fq(p)51049 19080 y Ft(\()p Fr(x)p Ft(\))800 20650 y Fj(for)465 b(al)66 b(l)465 b(primes)g Fr(p)p Fj(.)2128 22980 y Ft(Here,)402 b Fr(\271)p Ft(\()p Fr(n)p Ft(\))394 b(denotes)f(the)g(M\304)-650 b(obius-function,)401 b(i.e.,)i(the)393 b(m)-36 b(ultiplicativ)g(e)395 b(function)e(satisfying)i Fr(\271)p Ft(\()p Fr(p)p Ft(\))368 b(=)800 24530 y Fm(\241)p Ft(1)p Fr(;)221 b(\271)p Ft(\()p Fr(p)5007 24048 y Fq(k)5576 24530 y Ft(\))369 b(=)g(0)434 b(for)g(all)g(primes)f Fr(p)h Ft(and)f(in)-36 b(tegers)433 b Fr(k)414 b Fm(\270)369 b Ft(2.)p Black 800 26756 a Fj(Pr)-66 b(o)g(of.)p Black 552 w Ft(Since)480 b Fr(\271)p Ft(\()p Fr(n)p Ft(\))448 b(=)f(0)481 b(if)g(and)e(only)i(if)g Fr(n)f Ft(is)g(divisible)h(b)-36 b(y)480 b(some)h(square)f Fr(a)39517 26274 y Fo(2)40043 26756 y Fr(;)221 b(a)448 b Fm(\270)h Ft(2,)492 b(w)-36 b(e)480 b(see)h(that)e(in)800 28306 y(the)433 b(notation)h(of)g(Theorem)p 0 1 0 0 TeXcolorcmyk 434 w(2)p (#Theo.2) [[209 462 215 474] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)8336 31178 y Fr(\261)8912 31377 y Fo(0)9806 31178 y Ft(=)672 b(lim)11187 31975 y Fq(x)p Fp(!1)13997 30279 y Ft(1)p 13953 30872 740 54 v 13953 32089 a Fr(x)14825 31178 y Fm(jf)p Fr(n)369 b Fm(\267)g Fr(x)h Ft(:)f Fm(9)p Fr(a)g Fm(\270)g Ft(2)p Fr(;)221 b(a)25351 30629 y Fo(2)25877 31178 y Fm(j)p Fr(n)p Fm(g)370 b Ft(=)e(1)296 b Fm(\241)31710 29916 y Fl(Y)32322 32706 y Fq(p)33629 29305 y Fl(\265)34607 31178 y Ft(1)f Fm(\241)37278 30279 y Ft(1)p 37014 30872 1179 54 v 37014 32089 a Fr(p)37667 31705 y Fo(2)38325 29305 y Fl(\266)39672 31178 y Ft(=)368 b(1)296 b Fm(\241)43459 30279 y Fr(\274)44245 29797 y Fo(2)p 43459 30872 1312 54 v 43790 32089 a Ft(6)44903 31178 y Fr(;)800 34670 y Ft(hence,)454 b(the)449 b(limit)i(exists)g(and)e(is)h(irrational.)629 b(So)450 b(b)-36 b(y)450 b(Theorem)p 0 1 0 0 TeXcolorcmyk 450 w(2)p (#Theo.2) [[370 405 375 417] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)k(the)c(sequence)g(\()p Fr(\271)p Ft(\()p Fr(n)p Ft(\))590 b(\(mo)36 b(d)442 b Fr(p)p Ft(\)\))50972 34869 y Fq(n)p Fp(\270)p Fo(1)800 36219 y Ft(is)434 b(not)f(automatic.)41278 b Fh(\244)2128 38445 y Ft(Alb)36 b(eit)454 b(short)g(and)f(ingenious,) 459 b(the)453 b(pro)36 b(of)455 b(has)e(the)h(disadv)-72 b(an)-36 b(tage)454 b(that)f(it)h(is)g(di\261cult)f(to)h(apply)g(to)800 40342 y(other)386 b(situations)h(for)h(t)-36 b(w)g(o)387 b(reasons.)563 b(First,)396 b(it)387 b(requires)g(the)f(ev)-72 b(aluation)388 b(of)38865 39345 y Fl(Q)40120 40729 y Fq(p)40870 38867 y Fl(\263)41664 40342 y Ft(1)295 b Fm(\241)44302 39819 y Fo(1)p 44070 40036 935 54 v 44070 40800 a Fq(p)44544 40548 y Fg(2)45138 38867 y Fl(\264)46318 40342 y Ft(and)386 b(a)h(pro)36 b(of)800 42312 y(that)380 b(the)f(result)h(is)g (irrational.)561 b(This)381 b(is)f(equiv)-72 b(alen)-36 b(t)381 b(to)f(Euler's)g(ev)-72 b(aluation)381 b(of)f Fr(\263)100 b Ft(\(2\))381 b(and)e(the)h(fact)g(that)800 43862 y Fr(\274)1586 43380 y Fo(2)2587 43862 y Ft(is)477 b(irrational.)706 b(In)476 b(our)g(case,)487 b(these)476 b(are)g(w)-36 b(ell)477 b(kno)-36 b(wn,)487 b(y)-36 b(et)476 b(non-trivial)h(facts.)706 b(Ho)-36 b(w)g(ev)g(er,)488 b(in)475 b(other)800 45412 y(cases)510 b(there)f(migh)-36 b(t)509 b(b)36 b(e)510 b(no)f(kno)-36 b(wn)510 b(form)-36 b(ula)510 b(for)g Fr(\261)26747 45611 y Fq(a)27302 45412 y Ft(.)806 b(The)510 b(second,)528 b(more)510 b(fundamen)-36 b(tal)509 b(problem)g(is)800 46961 y(that)433 b(in)h(man)-36 b(y)433 b(cases)h Fr(\261)12518 47160 y Fq(a)13443 46961 y Ft(=)15302 46438 y Fo(1)p 14956 46656 1162 54 v 14956 47419 a Fp(jB)24 b(j)16684 46961 y Ft(for)434 b(all)h Fr(a)p Ft(,)e(so)h(Theorem)p 0 1 0 0 TeXcolorcmyk 434 w(2)p (#Theo.2) [[333 294 339 306] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(cannot)f(b)36 b(e)434 b(applied.)2128 48674 y(The)390 b(aim)g(of)g(this)f(note)h(is)g(to)f(giv)-36 b(e)391 b(another)e(pro)36 b(of)390 b(of)g(Corollary)i(1.)564 b(In)389 b(fact,)399 b(w)-36 b(e)390 b(ha)-36 b(v)g(e)390 b(the)f(follo)-36 b(wing)800 50224 y(more)434 b(general)g(result.)p Black 800 52554 a Fu(Theorem)511 b(3.)p Black 559 w Fj(L)-66 b(et)474 b Ft(\()p Fr(a)12309 52753 y Fq(n)12935 52554 y Ft(\))13441 52753 y Fq(n)p Fp(\270)p Fo(1)15743 52554 y Fj(b)-66 b(e)474 b(an)g(automatic)g(se)-66 b(quenc)g(e.)624 b(Assume)475 b(that)f(for)g(some)g(letter)g Fr(a)g Fj(and)g(for)800 54104 y(every)482 b(inte)-66 b(ger)480 b Fr(k)528 b Fj(ther)-66 b(e)481 b(exists)h(an)h(inte)-66 b(ger)480 b Fr(n)j Fj(such)g(that)g Fr(a)30389 54303 y Fq(n)31416 54104 y Ft(=)402 b Fr(a)33513 54303 y Fq(n)p Fo(+1)35743 54104 y Ft(=)f Fm(\242)221 b(\242)g(\242)402 b Ft(=)g Fr(a)41204 54303 y Fq(n)p Fo(+)p Fq(k)43477 54104 y Ft(=)f Fr(a)p Fj(.)650 b(Then)482 b(ther)-66 b(e)800 55653 y(is)549 b(a)g(c)-66 b(onstant)548 b Fr(c)524 b(>)h Ft(0)549 b Fj(such)h(that)e(for)h(an)g(in\257nite)d (numb)-66 b(er)548 b(of)h(inte)-66 b(gers)547 b Fr(x)i Fj(we)g(have)g Fr(a)45327 55852 y Fq(n)46478 55653 y Ft(=)524 b Fr(a)549 b Fj(for)f(al)66 b(l)800 57203 y Fr(n)369 b Fm(2)g Ft([)p Fr(x;)221 b Ft(\(1)296 b(+)f Fr(c)p Ft(\))p Fr(x)p Ft(])p Fj(.)2128 59533 y Ft(Other)433 b(criteria)h(in)-36 b(v)g(olving)435 b(strings)e(of)i(rep)36 b(eated)433 b(v)-72 b(alues)434 b(can)f(b)36 b(e)434 b(found)f(in)g([)p 0 1 0 0 TeXcolorcmyk(4)p (#cite.Yao) [[436 181 442 193] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)21571 62195 y(2.)651 b Fn(Main)501 b(resul)-89 b(ts)2128 64520 y Ft(Before)435 b(pro)-36 b(ving)434 b(our)f(theorem,)g(w)-36 b(e)434 b(\257rst)f(giv)-36 b(e)435 b(some)e(corollaries.)p Black 800 66850 a Fu(Corollary)603 b(2.)p Black 591 w Fj(L)-66 b(et)529 b Ft(\()p Fr(q)12685 67049 y Fq(i)13061 66850 y Ft(\))13567 67049 y Fq(i)p Fp(\270)p Fo(1)15675 66850 y Fj(b)-66 b(e)530 b(a)i(se)-66 b(quenc)g(e)529 b(of)h(p)-66 b(ositive)530 b(inte)-66 b(gers)529 b(such)i(that)40861 65854 y Fl(P)42263 67237 y Fq(i)43144 66327 y Fo(1)p 42993 66544 772 54 v 42993 67308 a Fq(q)43413 67443 y Ff(i)44389 66850 y Fr(<)490 b Fm(1)p Fj(.)796 b(Assume)800 68587 y(that)460 b(for)h(al)66 b(l)461 b Fr(k)414 b Fm(\270)369 b Ft(1)p Fj(,)462 b(ther)-66 b(e)459 b(ar)-66 b(e)460 b(indic)-66 b(es)459 b Fr(i)21576 68786 y Fo(1)22102 68587 y Fr(;)221 b(:)g(:)g(:)j(;)d(i)25459 68786 y Fq(k)26488 68587 y Fj(such)462 b(that)e Ft(\()p Fr(q)33175 68786 y Fq(i)33495 68942 y Ff(l)33873 68587 y Fr(;)221 b(q)35032 68786 y Fq(i)35352 68897 y Ff(m)36189 68587 y Ft(\))369 b(=)g(1)461 b Fj(for)f Ft(1)369 b Fm(\267)h Fr(l)398 b(<)369 b(m)f Fm(\267)i Fr(k)45 b Fj(.)595 b(F)-100 b(or)800 70137 y(al)66 b(l)518 b(inte)-66 b(gers)515 b Fr(n)466 b Fm(\270)f Ft(1)p Fj(,)530 b(set)517 b Fr(a)14676 70336 y Fq(n)15768 70137 y Ft(=)465 b(0)p Fj(,)530 b(if)516 b(ther)-66 b(e)516 b(exists)g(some)h Fr(i)g Fj(such)h(that)e Fr(q)37660 70336 y Fq(i)38036 70137 y Fm(j)p Fr(n)p Fj(,)530 b(and)517 b Fr(a)43369 70336 y Fq(n)44460 70137 y Ft(=)465 b(1)518 b Fj(otherwise.)800 71687 y(Then)464 b(the)h(se)-66 b(quenc)g(e)463 b Ft(\()p Fr(a)12815 71886 y Fq(n)13441 71687 y Ft(\))13947 71886 y Fq(n)p Fp(\270)p Fo(1)16240 71687 y Fj(is)i(not)g(automatic.)p Black Black eop %%Page: 3 3 3 2 bop Black 0 TeXcolorgray Black 11805 886 a Fk(A)444 b(CRITERION)d(F)-28 b(OR)444 b(NON-A)-28 b(UTOMA)-85 b(TICITY)440 b(OF)k(SEQUENCES)10492 b(3)p Black Black 800 3542 a Fj(Pr)-66 b(o)g(of.)p Black 552 w Ft(Assume)407 b(that)g(the)g(sequence)g(\()p Fr(a)21127 3741 y Fq(n)21753 3542 y Ft(\))22259 3741 y Fq(n)p Fp(\270)p Fo(1)24495 3542 y Ft(is)g(automatic,)414 b(and)406 b(let)i Fr(k)452 b Ft(b)36 b(e)407 b(a)h(giv)-36 b(en)408 b(in)-36 b(teger.)569 b(Cho)36 b(ose)800 5092 y(indices)316 b Fr(i)5397 5291 y Fo(1)5923 5092 y Fr(;)221 b(:)g(:)g(:)i(;)e(i)9279 5291 y Fq(k)10165 5092 y Ft(as)316 b(in)g(the)f(Corollary)-108 b(.)541 b(By)317 b(the)e(Chinese)h(remainder)g(theorem,)339 b(there)315 b(is)i(some)f(in)-36 b(teger)800 6642 y Fr(n)496 b Ft(solving)g Fr(n)338 b Ft(+)f Fr(l)503 b Fm(\264)474 b Ft(0)591 b(\(mo)36 b(d)442 b Fr(q)16631 6841 y Fq(i)16951 6997 y Ff(l)17329 6642 y Ft(\),)511 b(that)494 b(is,)512 b Fr(a)24016 6841 y Fq(n)p Fo(+)p Fq(l)26139 6642 y Ft(=)473 b(0)496 b(for)g(1)474 b Fm(\267)g Fr(l)504 b Fm(\267)474 b Fr(k)45 b Ft(.)763 b(Hence,)511 b(the)495 b(assumptions)f(of)800 8192 y(Theorem)p 0 1 0 0 TeXcolorcmyk 531 w(3)p (#Theo.3) [[130 643 135 655] [1 1 1 [3 3]] [0 0 1]] pdfm Black 531 w(are)531 b(satis\257ed,)556 b(and)530 b(w)-36 b(e)531 b(deduce)f(that)g(there)h(exist)g(some)g Fr(c)k(>)f Ft(0)d(and)f(arbitrarily)i(large)800 9741 y(in)-36 b(tegers)434 b Fr(x)g Ft(suc)-36 b(h)434 b(that)g Fr(a)13223 9940 y Fq(n)14219 9741 y Ft(=)369 b(0)435 b(for)g(all)g Fr(n)370 b Fm(2)g Ft([)p Fr(x;)221 b Ft(\(1)297 b(+)e Fr(c)p Ft(\))p Fr(x)p Ft(].)580 b(On)434 b(the)f(other)h(hand,)g(w)-36 b(e)434 b(can)h(b)36 b(ound)433 b(from)800 11291 y(b)36 b(elo)-36 b(w)500 b(the)g(n)-36 b(um)g(b)36 b(er)498 b(of)i(in)-36 b(tegers)500 b Fr(n)482 b Fm(2)f Ft([)p Fr(x;)221 b Ft(\(1)341 b(+)f Fr(c)p Ft(\))p Fr(x)p Ft(])500 b(suc)-36 b(h)499 b(that)g Fr(a)34472 11490 y Fq(n)35579 11291 y Ft(=)482 b(0)500 b(in)f(the)g(follo)-36 b(wing)502 b(w)-36 b(a)g(y)-108 b(.)777 b(Let)800 12841 y Fr(\262)405 b(>)f Ft(0)455 b(b)36 b(e)454 b(giv)-36 b(en,)461 b(and)454 b(let)g Fr(K)550 b Ft(b)36 b(e)454 b(some)h(constan)-36 b(t)454 b(with)29253 11844 y Fl(P)30655 13228 y Fq(i>K)33124 12318 y Fo(1)p 32973 12535 772 54 v 32973 13299 a Fq(q)33393 13434 y Ff(i)34282 12841 y Fr(<)404 b(\262)p Ft(.)641 b(Let)454 b Fr(L)h Ft(b)36 b(e)454 b(the)g(least)h(common)800 14578 y(m)-36 b(ultiple)401 b(of)i Fr(q)7889 14777 y Fo(1)8414 14578 y Fr(;)221 b(:)g(:)g(:)j(;)d(q)11904 14777 y Fq(K)12816 14578 y Ft(.)568 b(Then)401 b(the)g(set)h(of)g(all)h (in)-36 b(tegers)401 b Fr(n)h Ft(suc)-36 b(h)401 b(that)g Fr(q)36763 14777 y Fq(i)37507 14578 y Fm(6)369 b(j)p Fr(n)403 b Ft(for)f(all)g Fr(i)369 b Fm(\267)g Fr(K)497 b Ft(is)402 b(p)36 b(erio)g(dic)800 16548 y(with)457 b(p)36 b(erio)g(d)457 b Fr(L)p Ft(,)463 b(and)457 b(has)g(densit)-36 b(y)457 b Fr(d)19648 16747 y Fq(K)20968 16548 y Fm(\270)22410 15552 y Fl(Q)23665 16936 y Fq(i)p Fp(\267)p Fq(K)25850 15074 y Fl(\263)26643 16548 y Ft(1)296 b Fm(\241)29200 16025 y Fo(1)p 29050 16243 V 29050 17006 a Fq(q)29470 17141 y Ff(i)29954 15074 y Fl(\264)30747 16548 y Ft(,)464 b(with)457 b(equalit)-36 b(y)458 b(if)g(and)e(only)i(if)g(the)e Fr(q)50233 16747 y Fq(i)51066 16548 y Ft(are)800 18445 y(pairwise)434 b(coprime.)579 b(Note)434 b(that)20068 21127 y Fr(d)20744 21326 y Fq(K)22024 21127 y Fr(>)23783 19467 y Fp(1)23405 19865 y Fl(Y)23492 22665 y Fq(i)p Fo(=1)25323 19254 y Fl(\265)26301 21127 y Ft(1)296 b Fm(\241)28859 20229 y Ft(1)p 28708 20822 953 54 v 28708 22038 a Fr(q)29285 22237 y Fq(i)29793 19254 y Fl(\266)31140 21127 y Fr(>)369 b Ft(0)p Fr(;)800 24241 y Ft(that)433 b(is,)h Fr(d)5963 24440 y Fq(K)7308 24241 y Ft(can)g(b)36 b(e)433 b(b)36 b(ounded)432 b(a)-36 b(w)g(a)g(y)435 b(from)f(0)g(indep) 36 b(enden)-36 b(tly)432 b(of)i Fr(K)95 b Ft(.)579 b(No)-36 b(w)434 b(for)g Fr(x)369 b Fm(!)g(1)p Ft(,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)6555 26472 y Fm(jf)p Fr(n)369 b Fm(2)g Ft([)p Fr(x;)221 b Ft(\(1)296 b(+)f Fr(c)p Ft(\))p Fr(x)p Ft(])369 b(:)h Fr(a)18378 26671 y Fq(n)19373 26472 y Ft(=)e(1)p Fm(gj)1108 b(\270)f(jf)p Fr(n)369 b Fm(2)g Ft([)p Fr(x;)221 b Ft(\(1)296 b(+)f Fr(c)p Ft(\))p Fr(x)p Ft(])369 b(:)g Fm(8)p Fr(i)g Fm(\267)g Fr(K)464 b Ft(:)369 b Fr(q)42654 26671 y Fq(i)43399 26472 y Fm(6)g(j)p Fr(n)p Fm(gj)p Black 5192 w Ft(\(1\))p Black 28285 28697 a Fm(\241)29539 27435 y Fl(X)29545 30254 y Fq(i>K)31680 28697 y Fm(jf)p Fr(n)g Fm(2)g Ft([)p Fr(x;)221 b Ft(\(1)296 b(+)f Fr(c)p Ft(\))p Fr(x)p Ft(])369 b(:)g Fr(q)43396 28896 y Fq(i)43772 28697 y Fm(j)p Fr(n)p Fm(gj)23544 31937 y(\270)1107 b Fr(d)26360 32136 y Fq(K)27271 31937 y Fr(cx)295 b Fm(\241)g Fr(L)h Fm(\241)f Fr(\262cx)g Fm(\241)h(jf)p Fr(i)368 b Ft(:)i Fr(q)39303 32136 y Fq(i)40047 31937 y Fm(\267)f Ft(\(1)295 b(+)g Fr(c)p Ft(\))p Fr(x)p Fm(gj)23544 33874 y(\270)1107 b Ft(\()p Fr(d)26866 34073 y Fq(K)28072 33874 y Fm(\241)295 b Fr(\262)p Ft(\))p Fr(cx)g Ft(+)g Fr(o)p Ft(\()p Fr(x)p Ft(\))p Fr(;)800 36105 y Ft(since)434 b Fm(jf)p Fr(i)369 b Ft(:)g Fr(q)7140 36304 y Fq(i)7884 36105 y Fm(\267)g Ft(\(1)296 b(+)e Fr(c)p Ft(\))p Fr(x)p Fm(gj)369 b Ft(=)g Fr(o)p Ft(\()p Fr(x)p Ft(\),)434 b(for)g(otherwise)g (the)f(series)33301 35109 y Fl(P)35208 35582 y Fo(1)p 35057 35800 772 54 v 35057 36563 a Fq(q)35477 36698 y Ff(i)36395 36105 y Ft(w)-36 b(ould)434 b(div)-36 b(erge.)578 b(In)434 b(fact,)g(if)19658 39422 y(lim)221 b(sup)20445 40477 y Fq(x)p Fp(!1)23998 38523 y Fm(jf)p Fr(i)369 b Ft(:)g Fr(q)27151 38722 y Fq(i)27895 38523 y Fr(<)g(x)p Fm(gj)p 23998 39117 7051 54 v 27154 40333 a Fr(x)31550 39422 y(>)g Ft(0)p Fr(;)800 42325 y Ft(there)433 b(exists)h(some)g (constan)-36 b(t)433 b Fr(k)414 b(>)369 b Ft(0)434 b(suc)-36 b(h)432 b(that)17229 45103 y(lim)221 b(sup)17995 46159 y Fq(n)p Fp(!1)21569 44205 y Fm(jf)p Fr(i)369 b Ft(:)g(2)24795 43723 y Fq(n)25791 44205 y Fm(\267)g Fr(q)27770 44404 y Fq(i)28514 44205 y Fr(<)g Ft(2)30545 43723 y Fq(n)p Fo(+1)32374 44205 y Fm(gj)p 21569 44798 11838 54 v 26248 46015 a Ft(2)26898 45631 y Fq(n)p Fo(+1)33909 45103 y Fr(>)f(k)45 b(;)800 48177 y Ft(th)-36 b(us,)4139 47181 y Fl(P)5541 48565 y Fq(i)6421 47654 y Fo(1)p 6271 47872 772 54 v 6271 48635 a Fq(q)6691 48770 y Ff(i)7702 48177 y Ft(=)527 b Fm(1)p Ft(.)857 b(By)527 b(Theorem)p 0 1 0 0 TeXcolorcmyk 526 w(3)p (#Theo.3) [[248 283 253 295] [1 1 1 [3 3]] [0 0 1]] pdfm Black 527 w(w)-36 b(e)526 b(w)-36 b(ould)527 b(\257nd)e(arbitrarily)i (large)g(in)-36 b(tegers)527 b Fr(x)f Ft(suc)-36 b(h)526 b(that)g(the)800 49914 y(left-hand)511 b(side)h(of)g(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[172 268 178 280] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(is)h(zero,)532 b(th)-36 b(us)511 b(w)-36 b(e)511 b(arriv)-36 b(e)513 b(at)e(a)h(con)-36 b(tradiction.)813 b(So)511 b(the)g(sequence)h(\()p Fr(a)48454 50113 y Fq(n)49080 49914 y Ft(\))49586 50113 y Fq(n)p Fp(\270)p Fo(1)51926 49914 y Ft(is)800 51464 y(not)433 b(automatic.)42586 b Fh(\244)2128 53848 y Ft(Cho)36 b(osing)658 b Fr(q)8608 54047 y Fq(i)9731 53848 y Ft(=)748 b Fr(p)12144 53366 y Fo(2)12144 54176 y Fq(i)12670 53848 y Ft(,)712 b(with)657 b Fr(p)17582 54047 y Fq(i)18613 53848 y Ft(the)f Fr(i)p Ft(-th)f(prime)h(n)-36 b(um)g(b)36 b(er,)711 b(w)-36 b(e)657 b(\257nd)e(that)h(the)g(sequence)g(\()p Fr(\271)p Ft(\()p Fr(n)p Ft(\))52275 53366 y Fo(2)800 55398 y Ft(\(mo)36 b(d)443 b Fr(p)p Ft(\)\))5907 55597 y Fq(n)p Fp(\270)p Fo(1)8168 55398 y Ft(is)434 b(not)f(automatic,)i(whic)-36 b(h)433 b(is)h(sligh)-36 b(tly)434 b(stronger)g(then)e(Corollary)p 0 1 0 0 TeXcolorcmyk 436 w(1)p (#Cor.1) [[447 218 453 230] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)2128 56947 y(Out)h(next)g(result)h(deals)g(with)f(the)g (automaticit)-36 b(y)435 b(of)f(m)-36 b(ultiplicativ)g(e)434 b(functions.)p Black 800 59351 a Fu(Corollary)535 b(3.)p Black 561 w Fj(L)-66 b(et)475 b Fr(f)533 b Ft(:)391 b Fi(N)g Fm(!)g Fi(Z)476 b Fj(b)-66 b(e)476 b(a)h(multiplic)-66 b(ative)474 b(function.)630 b(L)-66 b(et)476 b Fr(q)438 b Fm(\270)391 b Ft(2)477 b Fj(b)-66 b(e)476 b(an)g(inte)-66 b(ger.)630 b(Assume)800 60901 y(that)465 b(the)f(fol)66 b(lowing)465 b(c)-66 b(onditions)463 b(hold.)p Black 2779 62791 a Ft(\(1\))p Black 553 w Fj(Ther)-66 b(e)390 b(exist)f(in\257nitely)f(many)i(primes)g Fr(p)h Fj(such)g(that)f(ther) -66 b(e)390 b(exists)g(some)g Fr(h)41792 62990 y Fq(p)42690 62791 y Fm(\270)369 b Ft(1)391 b Fj(with)f Fr(q)48 b Fm(j)p Fr(f)142 b Ft(\()p Fr(p)50817 62309 y Fq(h)51359 62420 y Ff(p)51896 62791 y Ft(\))p Fj(.)p Black 2779 64516 a Ft(\(2\))p Black 553 w Fj(If)464 b Fr(b)6911 64715 y Fq(n)8002 64516 y Fj(denotes)h(the)f Fr(n)p Fj(-th)i(inte)-66 b(ger)462 b(with)j Fr(f)142 b Ft(\()p Fr(b)26611 64715 y Fq(n)27237 64516 y Ft(\))369 b Fm(6\264)g Ft(0)591 b(\(mo)36 b(d)442 b Fr(q)48 b Ft(\))p Fj(,)464 b(we)h(have)41238 63889 y Fq(b)41641 64012 y Ff(n)p Fg(+1)p 41238 64210 1997 54 v 41751 64974 a Fq(b)42154 65085 y Ff(n)43735 64516 y Fm(!)370 b Ft(1)p Fj(.)800 66532 y(Then)464 b(the)h(se)-66 b(quenc)g(e)463 b Ft(\()p Fr(f)142 b Ft(\()p Fr(n)p Ft(\))590 b(\(mo)36 b(d)443 b Fr(q)48 b Ft(\)\))20372 66731 y Fq(n)p Fp(\270)p Fo(1)22664 66532 y Fj(is)465 b(not)g(automatic.)p Black 800 68936 a(Pr)-66 b(o)g(of.)p Black 552 w Ft(As)426 b(in)f(the)g(pro)36 b(of)426 b(of)h(Corollary)p 0 1 0 0 TeXcolorcmyk 427 w(2)p (#Cor.2) [[261 97 267 109] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)h(the)d(\257rst)g(condition)g(implies)h(that)f(for)i(ev)-36 b(ery)426 b Fr(k)470 b Ft(there)425 b(exist)800 70486 y(some)516 b Fr(n)f Ft(with)h Fr(f)142 b Ft(\()p Fr(n)p Ft(\))508 b Fm(\264)g Fr(f)142 b Ft(\()p Fr(n)351 b Ft(+)g(1\))508 b Fm(\264)h(\242)221 b(\242)g(\242)509 b(\264)f Fr(f)142 b Ft(\()p Fr(n)351 b Ft(+)f Fr(k)45 b Ft(\))508 b Fm(\264)h Ft(0)591 b(\(mo)36 b(d)442 b Fr(q)48 b Ft(\),)535 b(while)516 b(the)f(second)g(condition)800 72035 y(means)597 b(that)f(for)i(ev)-36 b(ery)598 b Fr(c)647 b(>)f Ft(0)598 b(there)e(are)h(only)h(\257nitely)f (man)-36 b(y)597 b(in)-36 b(tegers)597 b Fr(x)g Ft(suc)-36 b(h)596 b(that)g Fr(f)142 b Ft(\()p Fr(n)p Ft(\))647 b Fm(\264)h Ft(0)p Black Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black 800 886 a Fk(4)10493 b(A)444 b(CRITERION)d(F)-28 b(OR)444 b(NON-A)-28 b(UTOMA)-85 b(TICITY)440 b(OF)k(SEQUENCES)p Black 800 3542 a Ft(\(mo)36 b(d)443 b Fr(q)48 b Ft(\))554 b(holds)h(for)h(all)g Fr(n)576 b Fm(2)g Ft([)p Fr(x;)221 b Ft(\(1)379 b(+)f Fr(c)p Ft(\))p Fr(x)p Ft(].)943 b(T)-108 b(ogether)555 b(with)h(Theorem)p 0 1 0 0 TeXcolorcmyk 555 w(3)p (#Theo.3) [[420 685 426 697] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)586 b(w)-36 b(e)556 b(obtain)f(the)g(desired)800 5092 y(conclusion.)44746 b Fh(\244)2128 7400 y Ft(This)532 b(result)f(generalizes)i(a)f(theorem)f(of)h(S.)f(Y)-108 b(azdani)532 b([)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.Yaz) [[346 650 352 662] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)557 b(Theorem)531 b(2].)873 b(In)531 b(fact,)557 b(the)531 b(conditions)800 8950 y(on)521 b Fr(f)662 b Ft(are)521 b(relaxed)h(in)e(t)-36 b(w)g(o)522 b(asp)36 b(ects:)752 b(First,)543 b(the)520 b(in)-36 b(tegers)521 b Fr(h)31830 9149 y Fq(p)32879 8950 y Ft(are)g(allo)-36 b(w)g(ed)522 b(to)f(dep)36 b(end)520 b(on)g Fr(p)p Ft(.)840 b(More)800 10500 y(imp)36 b(ortan)-36 b(t,)585 b(the)555 b(lo)-36 b(w)g(er)555 b(b)36 b(ound)554 b(for)i(the)e(densit)-36 b(y)555 b(of)h(the)e(set)h(of)h(in)-36 b(tegers)555 b Fr(n)g Ft(satisfying)i Fr(q)623 b Fi(-)575 b Fr(f)142 b Ft(\()p Fr(n)p Ft(\))555 b(in)800 12049 y(the)501 b(second)h (condition)g(of)h(Corollary)p 0 1 0 0 TeXcolorcmyk 504 w(3)p (#Cor.3) [[257 609 262 621] [1 1 1 [3 3]] [0 0 1]] pdfm Black 502 w(is)f(smaller)h(then)e(in)h([)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.Yaz) [[365 609 371 621] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)519 b(Theorem)502 b(2].)784 b(The)502 b(latter)g(theorem) 800 13599 y(requires)h Fr(q)533 b Fi(-)487 b Fr(f)142 b Ft(\()p Fr(p)p Ft(\))502 b(for)h(all)g(primes)f(in)h(a)g(residue)f (class,)521 b(whic)-36 b(h)502 b(b)-36 b(y)502 b(the)g(prime)g(n)-36 b(um)g(b)36 b(er)501 b(theorem)h(for)800 15149 y(arithmetic)434 b(progressions)f(implies)h(condition)g(\(2\))f(of)i(Corollary)p 0 1 0 0 TeXcolorcmyk 435 w(3)p (#Cor.3) [[377 581 383 593] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)2128 17318 y(No)-36 b(w)435 b(w)-36 b(e)433 b(return)g(to)g (the)g(pro)36 b(of)435 b(of)f(Theorem)p 0 1 0 0 TeXcolorcmyk 434 w(3)p (#Theo.3) [[299 561 304 573] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)2128 19487 y Fj(Pr)-66 b(o)g(of)477 b(of)f(The)-66 b(or)g(em)p 0 1 0 0 TeXcolorcmyk 475 w(3)p (#Theo.3) [[185 542 191 554] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)617 b Ft(Assume)446 b(that)g(\()p Fr(a)23108 19686 y Fq(n)23734 19487 y Ft(\))24240 19686 y Fq(n)p Fp(\270)p Fo(1)26515 19487 y Ft(is)h(a)g(sequence)f(satisfying)j(the)d (conditions)g(of)h(The-)800 21037 y(orem)p 0 1 0 0 TeXcolorcmyk 457 w(3)p (#Theo.3) [[109 528 115 540] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)463 b(and)456 b(it)h(is)f(generated)h(b)-36 b(y)456 b(a)h Fr(q)48 b Ft(-automaton.)647 b(F)-108 b(or)456 b(ev)-36 b(ery)457 b(in)-36 b(teger)456 b Fr(l)29 b Ft(,)464 b(let)457 b Fr(n)g Ft(b)36 b(e)456 b(an)g(in)-36 b(teger)457 b(suc)-36 b(h)800 22587 y(that)405 b Fr(a)4273 22786 y Fq(n)p Fo(+)p Fq(i)6321 22587 y Ft(=)368 b Fr(a)406 b Ft(for)g(all)h Fr(i)368 b Fm(\267)i Fr(q)15368 22105 y Fq(l)15714 22587 y Ft(.)569 b(W)-108 b(e)405 b(ma)-36 b(y)407 b(assume)e(that)h Fr(n)g Ft(is)g(divisible)h(b)-36 b(y)405 b Fr(q)38842 22105 y Fq(l)39188 22587 y Ft(.)569 b(Indeed,)411 b(b)-36 b(y)406 b(h)-36 b(yp)36 b(othesis,)800 24136 y(for)489 b(all)g(in)-36 b(tegers)488 b Fr(l)k Fm(\270)462 b Ft(1)489 b(there)e(exists)i(an)f(in)-36 b(teger)488 b Fr(m)g Ft(suc)-36 b(h)488 b(that)f Fr(a)34492 24335 y Fq(m)p Fo(+)p Fq(i)36894 24136 y Ft(=)462 b Fr(a)488 b Ft(for)g(0)463 b Fm(\267)f Fr(i)g(<)f Ft(2)p Fr(q)47842 23654 y Fq(l)48189 24136 y Ft(.)742 b(Let)488 b Fr(n)52453 24335 y Fq(l)800 25686 y Ft(b)36 b(e)507 b(the)h(least)g(in)-36 b(teger)507 b(suc)-36 b(h)507 b(that)g Fr(n)496 b Fm(\270)f Fr(m)507 b Ft(and)g Fr(q)26021 25204 y Fq(l)26367 25686 y Fm(j)p Fr(n)p Ft(.)801 b(Then)507 b Fr(n)496 b(<)f(m)345 b Ft(+)g Fr(q)38388 25204 y Fq(l)38734 25686 y Ft(,)527 b(and)507 b(therefore)g Fr(a)48437 25885 y Fq(n)p Fo(+)p Fq(i)50611 25686 y Ft(=)494 b Fr(a)800 27236 y Ft(for)525 b(0)f Fm(\267)f Fr(i)h(<)f(q)8736 26754 y Fq(l)9082 27236 y Ft(.)850 b(Let)524 b Fr(s)g Ft(b)36 b(e)524 b(the)g(state)g(of)i(the) d(automaton)i(reac)-36 b(hed)523 b(when)h(reading)h(all)g(digits)g(of)g Fr(n)800 28786 y Ft(except)549 b(the)f(last)i Fr(l)578 b Ft(digits.)924 b(Then)549 b(the)f(de\257nition)g(of)i Fr(s)e Ft(implies)h(that)g(all)h(states)e(accessible)i(from)f Fr(s)800 30335 y Ft(within)528 b(precisely)i Fr(l)558 b Ft(steps)528 b(return)f Fr(a)p Ft(.)863 b(T)-108 b(o)528 b(ev)-36 b(ery)530 b(state)e Fr(s)g Ft(de\257ne)g(a)g(set)h Fm(N)38843 30534 y Fq(s)39863 30335 y Fm(\265)i Fi(N)e Ft(suc)-36 b(h)528 b(that)g Fr(l)560 b Fm(2)530 b(N)52310 30534 y Fq(s)800 31885 y Ft(if)524 b(and)e(only)h(if)h(all)g(states)f (accessible)g(from)h Fr(s)e Ft(within)h(precisely)h Fr(l)552 b Ft(steps)522 b(return)g Fr(a)p Ft(.)846 b(Our)522 b(argumen)-36 b(t)800 33435 y(ab)36 b(o)-36 b(v)g(e)543 b(sho)-36 b(ws)543 b(that)f(for)h(ev)-36 b(ery)543 b Fr(l)584 b Fm(2)554 b Fi(N)543 b Ft(there)f(is)h(some)g(state)f Fr(s)g Ft(accessible)i (from)e(the)g(starting)h(state)800 34985 y(suc)-36 b(h)412 b(that)g Fr(l)399 b Fm(2)368 b(N)9640 35184 y Fq(s)10131 34985 y Ft(.)571 b(Hence,)417 b(since)c(there)f(are)h(only)g 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Ft(+)339 b Fr(q)9976 40702 y Fq(l)10662 41184 y Fm(\241)h Ft(1],)516 b(w)-36 b(e)499 b(ha)-36 b(v)g(e)499 b Fr(a)19650 41383 y Fq(n)20757 41184 y Ft(=)480 b Fr(a)p Ft(.)774 b(In)499 b(fact,)516 b(after)499 b(reading)g Fr(d)p Ft(,)516 b(the)498 b(automaton)h(is)g (in)g(state)800 42733 y Fr(s)1413 42932 y Fo(0)1939 42733 y Ft(,)417 b(then,)f(after)d(reading)f Fr(l)442 b Ft(arbitrary)413 b(digits,)k(it)c(is)g(in)f(some)h(state)g(returning)e Fr(a)p Ft(.)571 b(Hence,)417 b(our)412 b(theorem)800 44283 y(follo)-36 b(ws)436 b(with)d Fr(c)369 b Ft(=)g(\(1)295 b Fm(\241)g Fr(q)13747 43801 y Fp(\241)p Fo(1)15004 44283 y Ft(\))p Fr(d)16186 43801 y Fp(\241)p Fo(1)17444 44283 y Ft(.)34002 b Fe(2)22927 53833 y Fn(References)p Black 1354 55936 a Fd([1])p Black 554 w(J.-P)-92 b(.)419 b(Allouc)-31 b(he,)431 b(Note)419 b(on)e(the)h(transcendence)f(of)h(a)g(generating)h (function,)431 b(in)418 b Fc(New)440 b(T)-85 b(r)-57 b(ends)441 b(in)g(Pr)-57 b(ob)g(ability)442 b(and)3075 57264 y(Statistics,)397 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