Integer Linear Combinations of Dilog((a-1)/a) and Log((a-1)/a) that are VERY close to an INTEGER By Shalosh B. Ekhad Theorem: Fix a positive integer larger than 1, let's call it a. There exist\ EXPLICIT integer sequences A(n), B(n), C(n) (that depend on a, of course) such that abs(A(n)+B(n)*DiLog((a-1)/a)+C(n)*Log((a-1)/a))<= CONSTANT/max(A(n),B(n),C(n))^delta, where delta equals / / 1/3 2 \ \ / / 1/3 | |%1 6 (-16/9 a - 12 a) 8 a | | / | | %1 - |ln(|----- - ------------------- - --- - 1| a) + 2| / |ln(-|- ----- | | 6 1/3 3 | | / | | 12 \ \ %1 / / \ \ 2 / 1/3 2 \ 3 (-16/9 a - 12 a) 8 a 1/2 |%1 6 (-16/9 a - 12 a)| + ------------------- - --- - 1 + 1/2 I 3 |----- + -------------------| 1/3 3 | 6 1/3 | %1 \ %1 / \ \ | | | a) + 2| | | / / 3 2 %1 := 512 a - 8640 a - 2916 a 5 4 3 2 1/2 + 12 (-98304 a + 248832 a - 209952 a + 59049 a ) and in Maple format -(ln((1/6*(512*a^3-8640*a^2-2916*a+12*(-98304*a^5+248832*a^4-209952*a^3+59049*a ^2)^(1/2))^(1/3)-6*(-16/9*a^2-12*a)/(512*a^3-8640*a^2-2916*a+12*(-98304*a^5+ 248832*a^4-209952*a^3+59049*a^2)^(1/2))^(1/3)-8/3*a-1)*a)+2)/(ln(-(-1/12*(512*a ^3-8640*a^2-2916*a+12*(-98304*a^5+248832*a^4-209952*a^3+59049*a^2)^(1/2))^(1/3) +3*(-16/9*a^2-12*a)/(512*a^3-8640*a^2-2916*a+12*(-98304*a^5+248832*a^4-209952*a ^3+59049*a^2)^(1/2))^(1/3)-8/3*a-1+1/2*I*3^(1/2)*(1/6*(512*a^3-8640*a^2-2916*a+ 12*(-98304*a^5+248832*a^4-209952*a^3+59049*a^2)^(1/2))^(1/3)+6*(-16/9*a^2-12*a) /(512*a^3-8640*a^2-2916*a+12*(-98304*a^5+248832*a^4-209952*a^3+59049*a^2)^(1/2) )^(1/3)))*a)+2) Before going into the proof, for the sake of conreteness, let state the val\ ues for a from 2 to 20. We also state the smallest empirical delta between n=100 and n=200 for thes\ e sequences to be described shortly a= , 2, delta=, 0.11307349436299304360, Smallest Empirical: , 0.11975468723339055027 a= , 3, delta=, 0.10907900007828729022, Smallest Empirical: , 0.11448118390138794711 a= , 4, delta=, 0.10461893064197058074, Smallest Empirical: , 0.11334069289188594055 a= , 5, delta=, 0.10083377249937967161, Smallest Empirical: , 0.10464367334195500145 a= , 6, delta=, 0.097682084693486227942, Smallest Empirical: , 0.10744187713129081387 a= , 7, delta=, 0.095028808363002066670, Smallest Empirical: , 0.099789846021012024020 a= , 8, delta=, 0.092760875922467860907, Smallest Empirical: , 0.10160539651803793412 a= , 9, delta=, 0.090794142466844320753, Smallest Empirical: , 0.095688385805913902847 a= , 10, delta=, 0.089066826599657543137, Smallest Empirical: , 0.099340788525490997019 a= , 11, delta=, 0.087533087651406110975, Smallest Empirical: , 0.091615665553152487519 a= , 12, delta=, 0.086158340495084233399, Smallest Empirical: , 0.098636039279517327699 a= , 13, delta=, 0.084916029211226757099, Smallest Empirical: , 0.089251511838292892257 a= , 14, delta=, 0.083785418613566141285, Smallest Empirical: , 0.092243645384992278471 a= , 15, delta=, 0.082750067143890703254, Smallest Empirical: , 0.092090166075329284146 a= , 16, delta=, 0.081796754458751117187, Smallest Empirical: , 0.087114902933914477906 a= , 17, delta=, 0.080914715513406638934, Smallest Empirical: , 0.086636647435291299920 a= , 18, delta=, 0.080095084338138583765, Smallest Empirical: , 0.086472684653253896954 a= , 19, delta=, 0.079330483631504572348, Smallest Empirical: , 0.082966240864883326513 a= , 20, delta=, 0.078614717417040342692, Smallest Empirical: , 0.086771558498869510299 a= , 21, delta=, 0.077942537695280824896, Smallest Empirical: , 0.084474916650410247972 Proof of the Theorem Consider a generalization of Beukers's famous integral that established the \ improved Apery proof of the irrationality of Zeta(2). Let's call it, E(n, a) 1 1 / / n n n n | | x (1 - x ) y (1 - y) E(n, a) = | | ----------------------- dx dy | | / x y\(n + 1) | | |1 - ---| / / \ a / 0 0 It is readily seen that E(n,a) is a linear combination with RATIONAL number \ coefficients of 1, Dilog((a-1)/a)) and Log((a-1)/a) E(n,A)=A1(n)+B1(n)*DiLog((a-1)/a)+ C1(n)*Log((a-1)/a) This introduces three sequences of RATIONAL numbers, A1(n), B1(n), C1(n) (we\ supress the dependance on a for notational ease) Lemma 1: The absolute value of E(n,a) is asymptotic (up to a constant) to // 3 2 ||(512 a - 8640 a - 2916 a \\ 5 4 3 2 1/2 1/3 + 12 (-98304 a + 248832 a - 209952 a + 59049 a ) ) /6 - 6 2 / 3 2 (-16/9 a - 12 a) / (512 a - 8640 a - 2916 a / 5 4 3 2 1/2 1/3 8 a \ \n + 12 (-98304 a + 248832 a - 209952 a + 59049 a ) ) - --- - 1| a| 3 / / x (1 - x) y (1 - y) Proof: Maximize the function, -------------------, x y 1 - --- a in 0<=x,y<=1 using two-variable calculus Indeed taking partial derivatives with respect to x and y, and setting it eq\ ual to zero gives that the maximum is attained at the point 3 3 {x = RootOf(_Z - 2 _Z a + a), y = RootOf(_Z - 2 _Z a + a)} x (1 - x) y (1 - y) Now plug it into, ------------------- x y 1 - --- a Lemma 2: The three sequences of rational numbers A1(n), B1(n), C1(n) all are\ asymptotic, up to polynomial corrections to / / 1/3 2 | | %1 3 (-16/9 a - 12 a) 8 a |-|- ----- + ------------------- - --- - 1 | | 12 1/3 3 \ \ %1 / 1/3 2 \\ \n 1/2 |%1 6 (-16/9 a - 12 a)|| | + 1/2 I 3 |----- + -------------------|| a| | 6 1/3 || | \ %1 // / 3 2 %1 := 512 a - 8640 a - 2916 a 5 4 3 2 1/2 + 12 (-98304 a + 248832 a - 209952 a + 59049 a ) Proof: Using the amazing Multivariable extension of the Almkvist-Zeilberger \ algorithm, due to Moa Apagodu and Doron Zeilberger, it was discovered and proved that E(n,a) satisfies the following THIRD-ORDER\ linear recurrence equation with polynomial coefficients 3 2 a (n + 1) (a - 1) (32 a n + 76 a - 27 n - 66) E(n, a) 2 3 3 - ------------------------------------------------------- + a (512 a n 2 (32 a n + 44 a - 27 n - 39) (3 + n) 3 2 2 3 3 2 2 3 3 + 2752 a n - 1072 a n + 4800 a n - 5792 a n + 636 a n + 2736 a 2 2 3 2 2 - 10140 a n + 3456 a n - 81 n - 5796 a + 6068 a n - 441 n + 3472 a / 2 - 768 n - 432) E(n + 1, a) / ((32 a n + 44 a - 27 n - 39) (3 + n) ) + a / 2 3 2 2 3 2 2 3 2 (256 a n + 1632 a n - 120 a n + 3376 a n - 780 a n - 81 n + 2232 a 2 / - 1670 a n - 522 n - 1170 a - 1086 n - 717) E(n + 2, a) / ( / 2 (32 a n + 44 a - 27 n - 39) (3 + n) ) + E(3 + n, a) = 0 and in Maple format -a^3*(n+1)^2*(a-1)*(32*a*n+76*a-27*n-66)/(32*a*n+44*a-27*n-39)/(3+n)^2*E(n,a)+a ^2*(512*a^3*n^3+2752*a^3*n^2-1072*a^2*n^3+4800*a^3*n-5792*a^2*n^2+636*a*n^3+ 2736*a^3-10140*a^2*n+3456*a*n^2-81*n^3-5796*a^2+6068*a*n-441*n^2+3472*a-768*n-\ 432)/(32*a*n+44*a-27*n-39)/(3+n)^2*E(n+1,a)+a*(256*a^2*n^3+1632*a^2*n^2-120*a*n ^3+3376*a^2*n-780*a*n^2-81*n^3+2232*a^2-1670*a*n-522*n^2-1170*a-1086*n-717)/(32 *a*n+44*a-27*n-39)/(3+n)^2*E(n+2,a)+E(3+n,a) = 0 It follows that the three sequences of rational numbers, A1(n), B1(n), C1(n)\ also satisy this recurrence, so let denote them all by X(n) 3 2 a (n + 1) (a - 1) (32 a n + 76 a - 27 n - 66) X(n) 2 3 3 - ---------------------------------------------------- + a (512 a n 2 (32 a n + 44 a - 27 n - 39) (3 + n) 3 2 2 3 3 2 2 3 3 + 2752 a n - 1072 a n + 4800 a n - 5792 a n + 636 a n + 2736 a 2 2 3 2 2 - 10140 a n + 3456 a n - 81 n - 5796 a + 6068 a n - 441 n + 3472 a / 2 - 768 n - 432) X(n + 1) / ((32 a n + 44 a - 27 n - 39) (3 + n) ) + a ( / 2 3 2 2 3 2 2 3 2 256 a n + 1632 a n - 120 a n + 3376 a n - 780 a n - 81 n + 2232 a 2 / - 1670 a n - 522 n - 1170 a - 1086 n - 717) X(n + 2) / ( / 2 (32 a n + 44 a - 27 n - 39) (3 + n) ) + X(3 + n) = 0 where X(n) stand for each of A1(n),B1(n), C1(n), that of course have differe\ nt initial conditions. Taking the leading term in n, gives us the constant-coefficient recurrence t\ hat approximates (up to polynomial corrections) The sequences A1(n), B1(n), C1(n) -a^3*(a-1)*X1(n)+a^2*(16*a^2-20*a+3)*X1(n+1)+a*(8*a+3)*X1(n+2)+X1(3+n) = 0 where X1(n) is a C-finite approximation with the same asymptotics (up to pol\ ynomial factors), thanks to the Poincare lemma. Solving the indicial equation -a^3*(a-1)+a^2*(16*a^2-20*a+3)*x+a*(8*a+3)*x^2+x^3 = 0 and taking the largest root, establishes the lemma. Lemma 3: Let d(n) be the least common multiple of the first n natural number\ s. Recall that d(n) is asymptotic to exp(n) A(n)=A1(n)*d(n)^2, B(n)=B1(n)*d(n)^2, C(n)=C1(n)*d(n)^2 are INTEGER sequencs\ Proof: Left to the reader We are now ready to prove the theorem: A(n), B(n), C(n) are asymptotic to / / 1/3 2 | | %1 3 (-16/9 a - 12 a) 8 a |-|- ----- + ------------------- - --- - 1 | | 12 1/3 3 \ \ %1 / 1/3 2 \\ \n 1/2 |%1 6 (-16/9 a - 12 a)|| | + 1/2 I 3 |----- + -------------------|| a| exp(2 n) | 6 1/3 || | \ %1 // / 3 2 %1 := 512 a - 8640 a - 2916 a 5 4 3 2 1/2 + 12 (-98304 a + 248832 a - 209952 a + 59049 a ) d(n)^2*E(n,a) is asymptotic to // 3 2 ||(512 a - 8640 a - 2916 a \\ 5 4 3 2 1/2 1/3 + 12 (-98304 a + 248832 a - 209952 a + 59049 a ) ) /6 - 6 2 / 3 2 (-16/9 a - 12 a) / (512 a - 8640 a - 2916 a / 5 4 3 2 1/2 1/3 8 a \ \n + 12 (-98304 a + 248832 a - 209952 a + 59049 a ) ) - --- - 1| a| 3 / / exp(2 n) Our delta is such that (Max(A1(n),B1(n),C1(n))*exp(2*n))^delta=exp(2*n)/E(n,a) Taking logarithms of both sides, and solving for delta, establishes the theo\ rem. QED.