Solution of the Basel problem in the framework of distribution theory
FFebruary 21, 2021
Solution of the Basel problemin the framework of distribution theory
Andreas Aste
Department of Physics, University of Basel,Klingelbergstrasse 82, CH-4056 Basel, SwitzerlandE-Mail: [email protected]
Abstract
A simple proof of Euler’s formula which states that the sum of the reciprocals ofall natural numbers squared equals π / Mathematics Subject Classification MSC 2010:
Keywords:
Basel problem; Zeta function; distribution theory; generalized functions; testfunctions; summation of series a r X i v : . [ m a t h . G M ] F e b Introduction
The so-called
Basel problem to determine the sum ζ (2) = (cid:80) ∞ n =1 1 n = π / † ζ (2) (see [2] and referencestherein). A further simple method to derive Euler’s result using the the theory of distribu-tions and test functions which is based on elementary arguments like translational invarianceis presented in this letter.Distribution theory [1], which represents a mathematical discipline in its own right, isof fundamental significance for a rigourous treatment of quantum field theories in classicalspacetime [3, 4]. It is also hoped that the stunning exercise presented in this letter serves asan incentive for graduate students with some basic knowledge of distribution theory to studythe subject of generalized functions and their applications in theoretical physics in greaterdetail. ζ (2) We consider the distribution ∆ ∈ D (cid:48) ( R ) defined by the formal expression∆ ( x ) = ∞ (cid:88) n = −∞ e inx = . . . + e − ix + e − ix + e − x + 1 + e ix + e ix + e ix + . . . , (1)which acts on (smooth) test functions (with compact support) ϕ ∈ D ( R ) as a linear and, inthe sense of distributions, continuous functional according to∆ [ ϕ ] := ∞ (cid:88) n = −∞ ∞ (cid:90) −∞ e inx ϕ ( x ) dx = lim N →∞ N (cid:88) n = − N ∞ (cid:90) −∞ e inx ϕ ( x ) dx . (2)In fact, ∆ is well-defined by equation (2) as a distribution in D (cid:48) ( R ), the dual space of D ( R ),and equation (2) highlights the meaning of the formal definition (1) of ∆ as a generalized function [6]. Note that a more intuitive representation of ∆ as an alternative infinite sum ofDirac delta distributions is motivated in the appendix.By definition, ∆ is a periodic distribution invariant under a translation T π , i.e. formally( T π ∆ )( x ) = ∆ ( x + 2 π ) = ∞ (cid:88) n = −∞ e in ( x +2 π ) = ∞ (cid:88) n = −∞ e inx = ∆ ( x ) , (3)or in distributional notation( T π ∆ )[ ϕ ] = ∆ [ T π ϕ ] ∀ ϕ ∈ D ( R ) , where ( T π ϕ )( x ) = ϕ ( x − π ) , (4)and ∆ is symmetric ∆ ( x ) = ∆ ( − x ) . (5)Now since ∆ ( x ) is invariant with respect to a multiplication with e ix , i.e. e ix ∆ ( x ) = e ix ∞ (cid:88) n = −∞ e inx = ∞ (cid:88) n = −∞ e inx , (6)1 must vanish as a distribution on R \{ πn | n ∈ Z } , since only for x = 2 πn with n ∈ Z onehas a trivial factor e ix = 1; therefore the distributional support of ∆ must be contained in acorresponding discrete set supp ∆ ⊆ { πn | n ∈ Z } . (7)For a moment, the following considerations are restricted to the open interval I = (0 , π ).Calculating the first antisymmetric antiderivative ∆ of ∆ with x ∈ I ∆ ( x ) = lim (cid:15) (cid:38) x (cid:90) (cid:15) ∆ ( x (cid:48) ) dx (cid:48) = − i ∞ (cid:88) n = −∞ n (cid:54) =0 e inx n + x = 2 ∞ (cid:88) n =1 sin( nx ) n + x (8)with ∆ ( x ) = − ∆ ( − x ) , (9)∆ must be constant on I , since its derivative ∆ vanishes there. This also implies that theFourier sum in equation (8) represents a linear function on I . Calculating the mean value µ I, of ∆ on I according to µ I, = 12 π lim (cid:15) (cid:38) π − (cid:15) (cid:90) (cid:15) ∆ ( x ) dx , (10)the oscillatory terms ∼ e inx in equation (8) do not contribute to µ I, and one is left with µ I, = 12 π π (cid:90) xdx = π . (11)Finally turning to the antiderivative of ∆ on I ∆ ( x ) = x (cid:90) ∆ ( x (cid:48) ) dx (cid:48) = − (cid:88) n ∈ Z \ e inx n + 12 x = − ∞ (cid:88) n =1 cos( nx ) n + 12 x , (12)one arrives at an expression containing a series that converges absolutely to a continuousfunction on I . However, since the distributional derivative of ∆ is ∆ which is constant, i.e., π on I , ∆ must be of the form ∆ ( x ) = πx + γ , x ∈ I (13)with an integration constant γ . This constant can be calculated by considering the averagevalue of ∆ on I µ I, = 12 π π (cid:90) x dx π π π (cid:90) ( πx + γ ) dx = π + γ , (14)hence γ = − π /
3, an finally Euler’s famous result∆ (0) = − ∞ (cid:88) n =1 n = γ = − π → ∞ (cid:88) n =1 n = π like ∆ , ∆ et cetera , further values of the Euler-Riemann zeta function like ζ (4) = ∞ (cid:88) n =1 n = π , ζ (6) = ∞ (cid:88) n =1 n = π , . . . (16)follow directly from strategy outlined above. 2 Explicit representation of ∆ as an infinite sum of Diracdelta distributions We consider the following sequence { δ N } N ∈ N ⊂ D (cid:48) ( R ) of distributions [6] represented by thefunctions δ N ( x ) = Θ( π − x ) N (cid:88) n = − N e inx = N (cid:80) n = − N e inx | x | < π | x | ≥ π (17)with supp δ N = [ − π, π ], where Θ( x ) = (cid:40) x > x ≤ δ N ( x ) = e − iNx + e − i ( N − x + . . . + e − ix + 1 + e ix + . . . + e iNx and e ix δ N ( x ) = δ N ( x ) + e i ( N +1) x − e − iNx (19)for x ∈ ( − π, π ) one immediately obtains the compact representation δ N ( x ) = e i ( N +1) x − e − iNx e ix − e i ( N +1 / x − e − i ( N +1 / x e ix/ − e − ix/ = sin(( N + 1 / x )sin( x/ , x ∈ ( − π, π ) \{ } (20)and from the definition (17) one has δ N (0) = 2 N + 1 which removes the singularity appearingat x = 0 in the representation (20). Only the term e i x = 1 in definition (17) contributes tothe integral ∞ (cid:90) −∞ δ N ( x ) dx = π (cid:90) − π δ N ( x ) dx = 2 π . (21)For illustrative purposes, the graph of δ is depicted in Fig. 1. In fact, { δ N } N ∈ N ⊂ D (cid:48) ( R )is a δ -sequence converging to 2 π times the Dirac delta distribution δ for N → ∞ . Applying δ N on a (smooth) test function ϕ ∈ D ( R ) (with compact support) leads to δ N [ ϕ ] = ∞ (cid:90) −∞ δ N ( x ) ϕ ( x ) dx = π (cid:90) − π sin(( N + 1 / x )sin( x/ ϕ ( x ) dx = 2 π (cid:90) − π sin(( N + 1 / x ) x x/ x/ ϕ ( x ) dx = 2 π (cid:90) − π sin(( N + 1 / x ) x β ( x ) x/ x/ ϕ ( x ) dx (22)where a smooth bump function β ∈ D ( R ) with the properties β ( x ) = 1 for | x | ≤ π and β ( x ) = 0 for | x | ≥ π/ σ ( x ) = (cid:40) x/ = x/ x/ | x | ∈ (0 , π/ x = 0 , σ ∈ C ∞ ([ − π/ , π/ , (23)i.e. since σ is a smooth function on the interval [ − π/ , π/ ϕ ( x ) = β ( x ) σ ( x ) ϕ ( x ) issmooth and has compact support: ˜ ϕ ∈ D ( R ). Furthermore, ˜ ϕ (0) = ϕ (0) holds.Now, equation (22) becomes, with x (cid:48) = ( N + 1 / x in the limit N → ∞ in the sense ofdistributions δ N [ ϕ ] = ∞ (cid:90) −∞ δ N ( x ) ϕ ( x ) dx = 2 π (cid:90) − π sin(( N + 1 / x ) x ˜ ϕ ( x ) dx = 2 ( N +1 / π (cid:90) − ( N +1 / π sin( x (cid:48) ) x (cid:48) ˜ ϕ ( x (cid:48) / ( N +1 / dx (cid:48) x -20020406080100 ( x ) Figure 1: The graph of δ defined by equation (17). N →∞ −−−−→ ∞ (cid:90) −∞ sin( x (cid:48) ) x (cid:48) ˜ ϕ (0) dx (cid:48) = 2 π ˜ ϕ (0) = 2 πϕ (0) = 2 πδ [ ϕ ] . (24)The normalization of the δ -distribution follows from equation (21), i.e., as a byproduct of thederivation presented above the integrallim N →∞ ( N +1 / π (cid:90) − ( N +1 / π sin( x ) x dx = ∞ (cid:90) −∞ sin( x ) x dx = π (25)is obtained. Neglecting the cutoff in definition (17) leads to the periodic distributional identity∆ ( x ) = ∞ (cid:88) n = −∞ e inx = 2 π ∞ (cid:88) n = −∞ δ ( x − πn ) or ∆ [ ϕ ] = 2 π ∞ (cid:88) n = −∞ ϕ (2 πn ) . (26)One readily expresses the antisymmetric antiderivative of ∆ by the help of the floorfunction (cid:98)·(cid:99) and the ceiling function (cid:100)·(cid:101) ∆ ( x ) = π (cid:18)(cid:22) x π (cid:23) + (cid:24) x π (cid:25)(cid:19) , (27)which simplifies to ∆ ( x ) = π sign( x ) (28)on the open interval ( − π, π ), and the symmetric antiderivative of ∆ is represented by thecontinuous function∆ ( x ) = πx (cid:18)(cid:22) x π (cid:23) + (cid:24) x π (cid:25)(cid:19) − π (cid:18)(cid:24) x π (cid:25)(cid:22) x π (cid:23)(cid:19) − π − (cid:88) n ∈ Z \{ } e inx n + x . (29)4 eferences [1] Schwartz, L.: G´en´eralisation de la notion de fonction, de d´erivation, de transformationde Fourier et applications math´ematiques et physiques.
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