CentralBinomialCoefficients Steven Finch. March 28, A(n +1)=(n +1)A(n), A(0) = 1

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1 CetralBiomialCoefficiets Steve Fich March The largest coefficiet of the polyomial ( + x) is [] It possesses recursio» + ad asymptotics A() = b/c = d/e ¼ A( +)=( +)A() A(0) = s A() π / as. Aother iterpretatio of A() is as the umber of sig choices + ad such that ± ± ± ± ±=0 if is eve ± ± ± ± ±= if is odd. {z } The latter is a especially attractive characterizatio of the th cetral biomial coefficiet. Cotrast this with the th cetral triomial coefficiet B() defied to be the largest coefficiet of the polyomial ( + x + x ). There is o simple closed-form expressio for B() []. It possesses recursio ( +)B( +)=( +)B()+B( ) B(0) = B() = ad asymptotics s B() 4π /. Here B() ca be iterpreted as the umber of solutios of ε + ε + ε + + ε =0 where each ε j { 0 }. Easy proofs of the asymptotics of A() adb() cabe based o such additive represetatios coupled with the Cetral Limit Theorem []. 0 Copyright c 007 by Steve R. Fich. All rights reserved..

2 Cetral Biomial Coefficiets 0.. Divisibility. Let ω( k) deote the umber of distict prime factors of ³. k Erdös[45]provedthat ω( ) l() l() as ad wodered what else could be said about the prime factors. Let the [6 7] c = lim N N NX = lim N f() = f() = N NX = X p ³ p6 k= p l(k) k = (f() c) =0. These two facts together express that f() c for almost all itegers hece ³ is almost always divisible by high powers of small primes. Let g() bethesmallestodd prime factor of ³. Whetherf() org() are bouded remais a ope questio. Sárközy [8] ad others [9 0 ] proved that ³ is ot square-free for ay >4. The largest for which ³ is ot divisible by p for ay odd prime p is =786. We tur attetio to ³ k the(k +) st elemet i the th row of Pascal s triagle. For each k the sequece of itegers such that ³ k is square-free has asymptotic desity c k where c = 6 π = c = Y p = (the latter is related to the Feller-Torier costat []). More geerally write k i base p: k = a 0 + a p + a p + + a`p` 0 a j <pfor all 0 j ` a`+ =0 ad defie c kp = Ỳ i=0 a i p + `X j=0 p a j (p a j+ ) if p k (p a j )(p a j+ ) k p if p>k.

3 Cetral Biomial Coefficiets The c k is equal to Q p c kp where the product is take over all primes p. We have c = c 4 = c 5 = c 6 = c 7 = ad 0 <c k =exp h (α + o()) k/ l(k) i as k where α = j= Z {x} j x / dx j (j +) j = ζ(j +/) j= j = j j X i>j i Itegrals ivolvig {x} = x bxc as such also appear i [ 4]. It follows that there are τ N square-free biomial coefficiets ³ k with 0 k< N where τ = k=0 c k =( ) = I words each row of Pascal s triagle possesses approximately 0 (o average). square-free etries 0.. Relevat Sums. Let ϕ deote the Golde mea ( + 5)/. We have [ ] ³ = ³ = = + π 7 X = = + π 7 X = ( ) + ³ ( ) + ³ = l(ϕ) 5 = l(ϕ) 5 = ³ = π 8 = ( ) + ³ = l(ϕ) 5 = ad more geerally [0] ³ = π 4 X = = ( ) + ³ = 5 8 5l(ϕ) 65 k X ( ) + k ³ = p k + q k π ³ = r k + s k 5l(ϕ) =

4 Cetral Biomial Coefficiets 4 for appropriate ratioals p k q k r k s k. Let L D deote the Dirichlet L-series with character (D/ ) adli k deote the k th polylogarithm fuctio. The followig are more difficult [ ]: = ³ = π 9 X = ( ) + ³ = 5l(ϕ) 5 ³ = = π 8 X = ( ) + ³ =l(ϕ) ³ = π = L () 4ζ() X ³ = = 4 = 7π4 40 ( ) + ³ = ζ () 5 = ( ) + ³ 4 = 8Li 4 ϕ + 7π l(ϕ) 5 +8l(ϕ)Li ϕ l(ϕ)4 6 Li 4 4ζ () l(ϕ) 5 ϕ 7π4 90 = ( ) + ³ 5 ³ = 5 = 9 π 8 = 5 Li 5 L (4) + π ζ() 9 ϕ +5l(ϕ)Li ϕ 4 9ζ(5) +4ζ() l(ϕ) 4π l(ϕ) + 4l(ϕ)5 ζ(5). 9 We woder whether the last two alteratig series possess expressios ivolvig L- series values rather tha polylogarithmic values. Let G = L 4 () deote Catala s costat. Other series iclude ³ =0 = ( π +) 9 ³ =0 ( ) ( = 4 5l(ϕ) +) 5 ³ =0 ( = 8G +) π l( + ) ³ =0 ( ) ( = π +) 6 l(ϕ)

5 Cetral Biomial Coefficiets 5 ad ³ =0 ³ =0 ( +) = π =0 X =0 ³ ( +) =L 8() ( +) =G but similar expressios for ³ =0 ( +) X ³ =0 X ³ =0 remai ope (as far as is kow). Batir [ ] recetly proved that 4 ³ = =8πG 4ζ() ( ) ³ = ( + l +) π 4 l( + ) ( ) ( = π +) 8 l( + ) ( ) ( +) ³ =0 4+ X ³ =0 ( +) ( ) ( +) =4ζ() 4πG ad also derived a complicated formula for P = / ³. We will barely metio cases for which ³ is i the umerator for example [5 7 ] ³ =0 =0 =0 ( +) = π l() ³ ( +) = π ³ =0 ( +) = (π l() + 4G) 8 4 ( +) = 4 L () ³ =0 ³ 4 ( +) = 7π 6 X Similar expressios for =0 =0 ³ ( +) X =0 ³ 4 ( +) = πζ() L (4) ³ 4 ( +) = 4G π. =0 ³ ( +) 4 X =0 ³ ( ) 4 ( +) remai ope (as far as is kow). Techiques i [4] might be helpful i evaluatig sums as these.

6 Cetral Biomial Coefficiets Addedum. Let χ k deote the k th Legedre chi fuctio: χ k (x) = j=0 x j+ (j +) k = (Li k(x) Li k ( x)) = i Ti k ( ix). Gosper [5] evaluated oe of the precedig ope sums: µq ( ) ( π arccosh (7) arccoth + µq 8χ = +) =0 ³ which is apparetly ew. The other sums all possess multiple Li k terms for example ³ =0 ( ) ( +) = 9 l(ϕ) l()l(ϕ) 4l(ϕ)Li µ ϕ µ 4l() l(ϕ)+ 7π l(ϕ) Li 0 ϕ 4 µ +4 Li ϕ Li +4Li ϕ ϕ ad further simplificatio seems to be impossible. + ζ () 0 Refereces [] N. J. A. Sloae O-Lie Ecyclopedia of Iteger Sequeces A00405 A0046 A6869. [] M. Petkovsek H. S. Wilf ad D. Zeilberger A = B A. K. Peters 996 pp ; available olie at wilf/aeqb.html; MR7980 (97j:0500). [] S. R. Fich Sigum equatios ad extremal coefficiets upublished ote (009). [4] P. Erdös Über die Azahl der Primfaktore vo ³ k Arch. Math. (Basel) 4 (97) 5 56; MR0988 (47 #84). [5] P. Goetgheluck O prime divisors of biomial coefficiets Math. Comp. 5 (988) 5 9; MR09459 (89f:0). [6] P. ³ Erdös R. L. Graham I. Z. Ruzsa ad E. G. Straus O the prime factors of Math. Comp. 9 (975) 8 9. MR06988 (5 #55).

7 Cetral Biomial Coefficiets 7 [7] J. W. Sader O primes ot dividig biomial coefficiets Math. Proc. Cambridge Philos. Soc. (99) 5 ; MR98408 (9m:099). [8] A. Sárközy O divisors of biomial coefficiets. I J. Number Theory 0 (985) 70 80; MR (86c:00). [9] G. Velammal Is the biomial coefficiet ³ squarefree? Hardy-Ramauja J. 8 (995) 45; MR0006 (95j:06). [0] A. Graville ad O. Ramaré Explicit bouds o expoetial sums ad the scarcity of squarefree biomial coefficiets Mathematika 4 (996) 7 07; available olie at adrew/aalytic.html; MR40709 (97m:0). [] P. A. Cutter Fidig prime pairs with particular gaps Math. Comp. 70 (00) ; MR869 (00c:74). [] S. R. Fich Arti s costat Mathematical Costats Cambridge Uiv. Press 00 pp [] S. R. Fich Euler-Mascheroi costat Mathematical Costats Cambridge Uiv. Press 00 pp [4] S. R. Fich Apéry s costat Mathematical Costats Cambridge Uiv. Press 00 pp [5] I. J. Zucker O the series P k= ³ k k k ad related sums J. Number Theory 0 (985) 9 0; MR (86e:08). [6] D. H. Lehmer Iterestig series ivolvig the cetral biomial coefficiet Amer. Math. Mothly 9 (985) ; MR0807 (87c:4000). [7] B. C. Berdt Ramauja s Notebooks. Part I Spriger-Verlag 985 pp ; MR0785 (86c:006). [8] J. M. Borwei D. J. Broadhurst ad J. Kamitzer Cetral biomial sums multiple Clause values ad zeta values Experimet. Math. 0 (00) 5 4; MR8569 (00k:05). [9] T. Sherma Summatio of Glaisher- ad Apéry-like series upublished mauscript (000) ura/participats.html. [0] J. M. Borwei ad R. Girgesoh Evaluatios of biomial series Aequatioes Math. 70 (005) 5 6; MR6798 (006e:05006).

8 Cetral Biomial Coefficiets 8 [] N. Batir Itegral represetatios of some series ivolvig ³ k k ad k some related series Appl. Math. Comput. 47 (004) ; MR0078 (004h:009). [] N. Batir O the series P ³ k k x k Proc. Idia Acad. Sci. Math. Sci. 5 k (005) 7 8; math.ac/050; MR8497 (006j:70). [] S. Ruscheweyh Eiige eue Darstelluge des Di- ud Trilogarithmus Math. Nachr. 57 (97) 7 44; MR060 (48 #458). [4] S. Weizierl Expasio aroud half-iteger values biomial sums ad iverse biomial sums J. Math. Phys. 45 (004) ; MR (005f:04). [5] R. W. Gosper Several cetral biomial sums upublished ote (008).