2013/12/16 by Oliver Knill, Knill, Oliver · 2 citations
Computer Science · Mathematics · #11M26 #11M99 #30D20 #68R10 #Advanced Operator Algebra Research #Complex Variables (math.CV) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph theory and applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #cs.DM #math.CV #math.SP #msc:11M26 #msc:11M99 #msc:30D20 #msc:68R10
paper · pdf · doi:10.48550/arxiv.1312.4239
24 pages and 7 figures
arxiv created 2013/12/16 · openalex publication_date 2013/12/16 · arxiv updated 2013/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the entire function zeta(n,s) which is the sum of l to the power -s, where l runs over the positive eigenvalues of the Laplacian of the circular graph C(n) with n vertices. We prove that the roots of zeta(n,s) converge for n to infinity to the line Re(s)=1/2 in the sense that for every compact subset K in the complement of this line, and large enough n, no root of the zeta function zeta(n,s) is in K. To prove this, we look at the Dirac zeta function, which uses the positive eigenvalues of the Dirac operator D=d+d^* of the circular graph, the square root of the Laplacian. We extend a Newton-Coates-Rolle type analysis for Riemann sums and use a derivative which has similarities with the Schwarzian derivative. As the zeta functions zeta(n,s) of the circular graphs are entire functions, the result does not say anything about the roots of the classical Riemann zeta function zeta(s), which is also the Dirac zeta function for the circle. Only for Re(s)>1, the values of zeta(n,s) converge suitably scaled to zeta(s). We also give a new solution to the discrete Basel problem which is giving expressions like zetan(2) = (n2-1)/12 or zetan(4) = (n2-1)(n2+11)/45 which allows to re-derive the values of the classical Basel problem zeta(2) = pi2/6 or zeta(4)=pi4/90 in the continuum limit.