Divisors, spin sums and the functional equation of the Zeta-Riemann function
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DIVISORS, SPIN SUMS AND THE FUNCTIONAL EQUATION OF THE ZETA-RIEMANN FUNCTION Michel Weber (Strasbourg) [Communicated by: Andr´ as S´ ark¨ ozy ]
Abstract Let Sn , n = 1, 2 . . . be the sequence of partial sums of independent spin random variables. We show that the distribution value of the divisors of Sn , is intimately related to the Zeta-Riemann function via the functional equation and Theta elliptic functions.
1. Main result The celebrated elliptic Theta function 2 Θ(u) = e−πn u ,
(1.1)
n∈Z
is linked to Gamma and Zeta-Riemann functions, via the functional equation valid for any complex s ∞ 1 1 1 1 1 − 12 s s ζ(s) = Θ(x) − 1 x 2 s−1 + x− 2 s− 2 dx − s(1 − s)−1 , (1.2) π Γ 2 2 1 which follows from the equation ∞ ∞ 2 1 1 − 12 s π Γ e−m πx x 2 s−1 dx. s ζ(s) = 2 m=1 0
(1.3)
We refer to [H], Chap. 11, Eq. (11.3) and (11.7) (see also [B] Part. 5, Chap. 3 p. 136). The purpose of this paper is to prove the existence of another functional equation of the Zeta-Riemann function, linking the left hand side of (1.2) to the distribution of the divisors of spin sums, in which the Theta function turns up to play a crucial role. Let ε = {εi , i ≥ 1} be a sequence of independent spin random Mathematics subject classification number: Primary 11M06; Secondary 11M99, 11A25, 60G50. Key words and phrases: independent spin sums, Theta function, Gamma function, Zeta-Riemann function, functional equation, divisors. 0031-5303/2005/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
120
m. weber
variables (P{εi = ±1} = 1/2) with basic probability space (Ω, A, P). Consider the sequence of partial sums Sn = ε1 + . . . + εn , n = 1, 2, . . .. Put for p, M even:
p υ(p, M ) = P p | SM − 1. (1.4) 2
When p is fixed, p2 P p | SM → 1, as M → ∞. Thus υ(p, M ) analyses the speed of convergence. When p and M simultaneously tend to infinity, this quantity in turn also appears, in the functional equation of the Zeta-Riemann function. We indeed prove the following result, which constitutes the main result of the paper. Theorem 1.1. There exist a sequence of pairs of even positive integers (pτ,Mτ), pτ ≤ Mτ such that for any complex s: 1 1 − 12 s +π s ζ(s) Γ s(1 − s) 2 1 − 12 s− 12 ∞ s−1 τ 2 τ 2 = lim 2 υ(pτ T, Mτ τ ) + . 2 2 T →∞ T T T 2 τ =T
It will result from the proof that the sequence of pairs (pτ , Mτ ) is intimately related to the diophantine approximation of the irrational number 2π. The paper is organized as follows: Section 2 is devoted to the study of the distribution of the divisors of Sn , and in Section 3, we prove the main result.
2. The distribution of divisors To begin, it will be necessary to recall some elementary, but useful facts. Since S2 takes values −2, 0, 2 only, it follows that P{S2 is even} = 1; and therefore by elementary considerations P{Sn is even} = 1 (resp. = 0)
⇔
n is even (resp. odd)
(2.1)
Now, let p, M be positive integers. It is possible to derive the exact value of P p | SM . By using the formula pδp|SM =
p−1
e2iπjSM /p ,
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