The Analytic and Arithmetic Mystery of Riemann s Zeta Function at Positive Integers

Transcription

1 The Analytic and Arithmetic Mystery of Riemann s Zeta Function at Positive Integers Hojoo Lee School of Mathematics, KIAS. Introduction The fundamental theorem of arithmetic (also known as the unique prime factorization theorem) in = {, 2, 3, } guarantees a simple but interesting equality that the infinite series is equal to the infinite product More generally, Riemann s zeta function is defined as for a complex number s with Re s >. Using Mobius arithmetric function, one also has the expression The purpose of this expository article is to offer a brief survey of known arithmetical results of the classical zeta function at positive integers and to sketch several interesting proofs of Euler s identity (2) = Is (5) IRRATIONAL? The study of the arithmetical nature of values of Riemann s zeta function (s) at positive integers s 2 is one of the undoubtedly fascinating topics in number theory. Euler established that, for any integer n, the positive 44 Korea Institute for Advanced Study

2 Science Café number (2n) is a rational multiple of 2n. More explicitly, he discovered the beautiful formula where the Bernoulli numbers B 0 =, B = -, B 2 =, B 3 = 0, B 4 = -, are a sequence of rational numbers defined by the combinatorial recursion relation 2 30 Euler s formula admits several interesting proofs, for instance, we refer to [, 5]. Since is a transcendental number, it follows from Euler s formula that (2) =, (4) =, () =, are transcendental. 2 Euler also tried, unsuccessfully, to determine the values of the zeta function at odd positive integers. Even up to 978 nothing had been established about the arithmetic properties of the values (3), (5), (7),. However, in 979, Apéry [2] announced an incredible proof of the irrationality of. Beukers offered a simplified proof [5] in the same year, and also presented a neat proof [] by revealing unexpected connections between modular forms and irrationality proofs. Later, a new proof was given by Nesterenko [0]. Though many tried to generalize Apéry s arguement in order to prove the irrationality of (5) = + + +, no interesting results were discovered for many years. Finally, in 2000, Rivoal [2] succeeded in obtaining the remarkable fact that there are infinitely many odd integers giving irrational values of the Riemann zeta function. See for example his work [4] with Ball. Furthemore, he [3] subsequently showed that at least one of the nine values (5), (7),, (9), (2) is irrational. This was immediately improved by Zudilin [], who proved that at least one of (5), (7), (9), () is irrational. Hata [9], Rhin and Viola [4] strengthened Apéry s result quantitatively to obtain a sharper measure of the irrationality of (3). We refer to Fischler s article [8] for a detailed survey of recent developments on irrationality of zeta function values Sketch of Apostol s proof [3] of Euler s identity The idea is to integrate the geometric series = +r+r 2 +, r (-, ). Letting we have the integral identity -r There are several ways to evaluate the double integral I defined over the unit square. We follow Apostol s approach. Introducing the 4 -rotation transformation the change of variables formula gives THE KIAS Newsletter

3 Using the integral identity we obtain Employing the subsitution we have A similar calculation gives the value It therefore follows that 4. Sketch of Choe s proof [7] of Euler s identity We begin with the well-known identity Taking x = sin t in this Maclaurin series yields Integrating this equality from 0 to 2, we obtain Now, combining this and the Wallis formula we have the equality In the view of the rearrangement equality we conclude 4 Korea Institute for Advanced Study

4 Science Café 5. Sketch of Papadimitriou s proof [] of Euler s identity The binomial theorem and De Moivre s formula give the identity Taking the imaginary part yields (5.) where we introduce the polynomial of degree m given by Taking into account the identity (5.), we immediately find that the polynomial P m of degree m admits m distinct real roots: Then, applying Vieta s formula to the polynomial P m, we get (5.2) Combining the inequality cot 2 x < < + cot 2 x, x and the equality (5.2), we have the estimation x 2 or Finally, by the sandwich theorem (or the squeeze theorem), letting m in the above estimation gives References [] E. De Amo, M. Díaz Carrillo, J. Fernández-Sánchez, Another proof of Euler s formula for (2k), Proc. Amer. Math. Soc. 39 (20), [2] R. Apéry, Irrationalité de (2) et (3), Astérisque, (979), -3. [3] T. M. Apostol, A proof that Euler missed: Evaluating (2) the easy way, Math. Intelligencer, 5 (983) [4] K.M. Ball, T. Rivoal, Irrationalité d une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math. 4 (200), [5] F. Beukers, A note on the irrationality of (2) and (3), Bull. London Math. Soc. (979), [] F. Beukers, Irrationality proofs using modular forms, Astérisque, 47 (987) THE KIAS Newsletter

5 [7] B. R. Choe, An elementary proof of Amer. Math. Monthly, 94 (987), 2-3. [8] S. Fischler, Irrationalité de valeurs de zêta (d après Apéry, Rivoal,... ), Astérisque, 294 (2004), [9] M. Hata, A new irrationality measure for (3), Acta Arith. 92 (2000), [0] Y. Nesterenko, A few remarks on (3). Math. Notes, 59 (99), [] I. Papadimitriou, A simple proof of the formula Amer. Math. Monthly, , [2] T. Rivoal, La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C. R. Acad. Sci. Paris Sér. I Math. 33 (2000), no. 4, [3] T. Rivoal, Irrationalitê d au moins un des neuf nombres (5), (7),, (2), C. R. Acad. Sci. Paris Sér. I Math. 33 (2000), no. 4, [4] G. Rhin, C. Viola, The group structure for (3), Acta Arith. 97 (200), [5] H. Tsumura, An elementary proof of Euler s formula for (2m), Amer. Math. Monthly, (5) (2004) [] W. Zudilin, One of the numbers (5), (7), (9), () is irrational, Uspekhi Mat. Nauk (Russian Math. Surveys) 5 (4) (200), no. 4, Hojoo Lee Hojoo Lee is a mathematics research fellow at KIAS. As an undergraduate in mathematics at Kwangwoon University, he was a big fan of lreland and Rosen s A Classical Introduction to Modern Number Theory. He is currently working in Geometric Analysis, which is at the interface of Differential Geometry and Partial Differential Equations. 48 Korea Institute for Advanced Study