A Note on the Irrationality of ζ(3) Work done as part of a summer programme in Chennai Mathematical Institute under Dr. Purusottam Rath.

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1 A Note on the Irrationality of ζ(3) MANJIL P. SAIKIA Work done as part of a summer programme in Chennai Mathematical Institute under Dr. Purusottam Rath. Abstract. We give an account of F. Beukers [2] proof of the irrationality of ζ(3). Key Words: Zeta functions, Apéry s constant.. Introduction At the Journees Arithmetiques held at Marseille Luminy in June 978, Roger Apéry gave an elmentary proof of the irrationality of ζ(3) However due to the complexity of the proof there was a general disagreement amongst the mathematicians present there as to the validity of the proof presented. Two months later a complete exposition of the proof was presented at the International Congress of Mathematicians in Helsinki in August 978 by H. Cohen. This proof was based on the lecture by Apéry but also contained some ideas of Cohen and Don Zaiger. Then in 979 F. Beukers gave a very simple proof of the result in [2]. In this note we shall give an account of the proof given by Beukers. Beukers proof uses some elementary calculus involving few double and triple integrals. 2. Preliminaries We prove in this section few non-trivial lemmas which shall be required in our proof of the irrationality of ζ(3). Lemma 2.. The LCM of, 2, 3..., n is denoted by d n, then d n p n p n p log n log p n π(n). p log n log p < Proof. If p, p 2,..., p π(n) are all the prime numbers less than or equal to n. If p is a prime bigger than n then p does not divide any of, 2, 3,..., n, so p doesn t divide d n. Thus d n p a p a 2 2, p a π(n) π(n) for some a i s (i, 2, 3,..., π(n)). We have that p log p n n and p log p n + > n hold for any prime less or equal than d n. So p log p n d n. Thus a i log pi n. Hence, d n p n p log p n logn p n p logp < p n p logn logp p n plog p n p n n nπ(n). Lemma 2.2. n π (n) < 3 n. Proof. Accoring to the prime number theorem we have π(n) < large n. So, n π (n) < 3 n. (log 3)n log n for sufficiently Currently a Summer Fellow at Chennai Mathematical Institute, Siruseri, Chennai 63, India.

2 2 Manjil P. Saikia Lemma 2.3. Let r and s be non-negative integers. If r > s then, x r y s dxdy is a rational number whose denominator is a divisor of xy d2 r. xr y s dxdy is a rational number whose denominator is a divisor of d 3 r. If r s, then xy x r y r dxdy ζ(2)... xy xy xr y r dxdy 2{ζ(3)... } r 3 When r we let the sums r 2 and r 3 r 2. vanish. Proof. Let σ be any non-negative integer. We consider the integral, x r+σ y s+σ dxdy (2.) xy It is very easy to see that when the denominator of the I n is expressed as a Geometric Series we get the following, ( k (xy) k x r+σ y s+σ dxdy k x k+r+σ dx)( y k+s+σ dy) ( k + r + σ +. k + s σ + ) ( k + s + σ + k + r + σ + ) (2.2) k k+ The above is true since r > s. So, finally we have, r s ( s + + σ r + σ ) (2.3) If we put σ now then the first part of our lemma follows immediately. Differentiating with respect to σ and putting σ, I n changes to, And the summation becomes r s ( (s + ) r 2 This proves the second part of the lemma. log xy xy xr y s dxdy (2.4)

3 We now assume r s, then it is easy to see that, A Note on the Irrationality of ζ(3) 3 x r+σ y r+σ xy dxdy k (k + r + σ + ) 2 By putting σ as before we get the third part of the lemma as follows, This equals, (r + ) + 2 (r + 2) Thus, ( r 2 + (r + ) 2 + (r + 2) ) ( r 2 ) ζ(2) r 2 Again, we differentiate with respect to σ and put σ. Then, The above equals to log xy xy xr y r dxdy k 2 (k + r + ) 3 which in turn equals to 2( (r + ) + 3 (r + 2) +...) 3 Thus, 2(( r 3 + (r + ) ) ( r 3 )) 2(ζ(3) r 3 ) which proves the final part of the lemma. Lemma 2.4. dz. ( xy)z xy Proof. We substitute ( xy)z u in the integral which gives ( xy)dz du. Changing the limits we get,

4 4 Manjil P. Saikia xy ( xy)z dz ( u)( xy) du xy xy o u du log( u) xy xy xy Lemma 2.5. If F (u, v, w) u( u)v( v)w( w) ( uv)w and u, v, w (o, ) then Max F (u, v, w) 27. Proof. We have So, ( uv)w w + uvw 2( ( w) uvw) F (u, v, w) 2 ( w)w u( u) v( v) Again, Max t t( t 2 ) , t 3 Max t t( t) 2. 2, t 2 So, Max [F (u, v, w)] Theorem 3.. ζ(3) is irrational. Proof. We consider the integral, 3. Irrationality of ζ(3) xy P n(x)p n (y)dxdy, where P n (r) ( d dr )n r n ( r) n. n!. It is clear from Lemma 2.3 that for some A n, B n Z we have,

5 A Note on the Irrationality of ζ(3) 5 Using Lemma 2.4 I n changes into, (A n + B n ζ(3))d 3 n (3.) P n (x)p n (y) ( xy)z dxdydz After partially integrating the above integral with respect to x n-times the interal changes into, We now use the substitution w (xyz) n ( x) n P n (y) dxdydz (3.2) ( ( xy)z) n+ z ( xy)z to get, ( x) n ( w) n P n (y) ( xy)w dxdydw [x( x)y( y)w( w)] n [ ( xy)w] n+ dxdydw (3.3) where the last equality again follows from an n-fold integration by parts with respect to w like earlier. From Lemma 2.5 and Lemma 2.4 it follows that the integral is bounded above by ( 27 )n ( xy)w dxdydw ( 27 )n The above is equal to 2( 27 )n ζ(3) by Lemma 2.3. Since integral (3.3) is not zero, so we have xy dxdy < A n + B n ζ(3) d 3 n < 2ζ(3)( 27 )n and hence by Lemma 2. and Lemma 2.2 < A n + B n ζ(3) < 2ζ(3)d 3 n( 27 )n < 2ζ(3) < ( 4 5 )n (3.4) for sufficiently large n, which proves the irrationality of ζ(3). 4. Acknowledgements The author wishes to thank Dr. P. Rath for his immense help and guidance and Prof. F. Beukers for [3].

6 6 Manjil P. Saikia References [] M. Aigner, G. M. Ziegler, Proofs from THE BOOK, Springer India, 2. [2] F. Beukers, A note on the Irrationality of ζ(2) and ζ(3), Bull. Lon. Math. Soc. (979), [3] F. Beukers, Elementary Number Theory WISB32, Lecture Notes, 29. [4] J. M. Borwein, P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley Intescience, 998. Department of Mathematical Sciences, Tezpur University, Sonitpur, Pin address: manjil msi9@agnee.tezu.ernet.in, manjil.saikia@gmail.com