On polylogarithms, Hurwitz zeta functions, and the Kubert identities (Q762203)

This very interesting and rich paper deals with classical analytic functions put in a new light, using modern techniques. It deals with the Hurwitz zeta-function \[ \zeta_{1- s}(x)=\sum^{\infty}_{n=0}(n+x)^{s-1} \] and the Fourier series (vid polylog) \(\ell_ s(x)=\sum e^{2\pi inx}n^{-s}\). Both functions are special cases of Lerch zeta-function. There is a classical functional equation connectin these functions. For integral values of s there is a relation with the Bernoulli polynomials. For positive integral values of s the functions \(\ell_ s(x)\) are connected with elemtnary functions, such as \(\log(2 \sin \pi x).\) Starting point is the Kubert identity. Let f be a function defined on \({\mathbb{Q}}/{\mathbb{Z}}\) or on \({\mathbb{R}}/{\mathbb{Z}}\), which satisfies \[ (*_ s)\quad f(x)=m^{s-1}\sum^{m-1}_{k=0}f((x+k)/m). \] We call a function f, defined on (0,1) even if \(f(x)=f(1-x)\) and odd if \(f(x)=-f(1- x)\). Examples of even functions, satisfying the Kubert identity are \(\zeta_ 3(x)+\zeta_ 3(1-x)\) for \(s=-2\) and \(\log(2 \sin\pi x)\) for \(s=1\). Examples of odd functions are cos \(\pi\) \(x\cdot \sin^{-3} \pi x\) for \(s=-2\) and the Bernoulli polynomials for positive integral values of s. It is proved that there are for each complex value of s exactly two linear independent continuous functions defined on (0,1) which satisfy \((*_ s)\). One generator can be chosen to be even, the other to be odd. Derivation with respect to x maps the space of Kubert functions with parameter s onto the space with parameter s-1, except when \(s=0.\) The author also deals with the generalization to functions defined on \({\mathbb{R}}/{\mathbb{Z}}\), instead of being defined on (0,1). In {\$} 5 of this paper the connection of these Kubert functions with algebraic number theory is found. Let A be a subgroup of \({\mathbb{Q}}/{\mathbb{Z}}\) and let \(U_ s({\mathbb{A}})\) be the additive group with one generator u(a) for each element of A, and defining relations \[ (*'_ s)\quad f(ma)=m^{s- 1}\sum^{m-1}_{k=0}f(a+k/m) \] (f is a map of A into a rational vector space V). Such a map is called universal if the induced mapping \(u(a)\to f(a)\) of \(U_ s(A)\otimes {\mathbb{Q}}\) to V is injective. The author now proves that the complex vector space splits, under the action of the automorphism group of A into a direct sum of 1-dimensional eigenspaces, with just one eigenspace corresponding to each continuous character \(\chi\) : Hom\((A,A)^ 0\to {\mathbb{C}}^ 0\). Furthermore, any inclusion \(A\subset A'\subset {\mathbb{Q}}/{\mathbb{Z}}\) gives rise to an embedding \(U_ s(A)\otimes {\mathbb{C}}\subset U_ s(A')\otimes {\mathbb{C}}\). Here \(Hom(A,A)^ 0\) is the automorphism group of A, identified with the group of invertible elements in the ring Hom(A,A) consisting of all homomorphisms from A to itself. The inclusion mentioned above has some Galois-like features. In the proof the author uses the following lemma. Let \(\chi:\Hom(A,A)^ 0\to {\mathbb{C}}^ 0\) be a continuous character. Then there is one and, up to a constant multiple, only one function \(f=f_{\chi}: A\to {\mathbb{C}}\) satisfying \((*'_ s)\) and satisfying \(f(ua)=\chi(u)f(a)\) for every \(u\in Hom(A,A)^ 0\) and every \(a\in A\). In the special case \(A={\mathbb{Q}}/{\mathbb{Z}}\), we have \(Hom(A,A)=\lim_{\leftarrow}{\mathbb{Z}}/m{\mathbb{Z}}={\hat {\mathbb{Z}}}\) and the connection with Dirichlet characters is clear. In this case \(f(1/n)=0\), if n is not in the conductor of \(\chi\) and if n is in the conductor, we have for \(n>0\) \(f(1/n)=n^{-s}\prod_{p| n}(1-{\bar \chi}(p)\cdot p^{s-1})(1-p^{-1})^{-1}.\) In the last paragraph on \({\mathbb{Q}}\)-linear relations there is a nice conjecture. Every \({\mathbb{Q}}\)-linear relation between the real numbers \(\zeta_ 2(x)\), \(x\in {\mathbb{Q}}\), \(0<x\leq 1\) is a consequence of the Kubert relations \((*_{-1})\). The author gives a proof of a theorem of Bass. Every \({\mathbb{Q}}\)-linear relation between the numbers \(\log(2 \sin\pi x)\) for rational \(x\in (0,1)\) is a consequence of the Kubert relations \((*_ 1)\) together with evenness. In an appendix the author describes the relation of the dilogarithm with the volume function and the Dehn invariant in hyperbolic three space.