Some series representing the Riemann zeta function

2026/02/28 by Jean-François Burnol
Mathematics · #math.NT #msc:11Y60 #msc:11B83 #msc:33B15 #msc:41A60 #msc:05A16 #msc:11B68 #msc:11M41 #msc:30E15 #msc:60C05

paper · pdf

20 pages, 1 figure

arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

Given an integer b at least equal to 2, we obtain a representation of the Riemann zeta function in the complex plane as a finite linear combination of geometrically convergent series. The coefficients involve partial factorials and rational functions in b^s using the Bernoulli numbers. We obtain, for arbitrary b, and for s away from the poles, the asymptotic expansion of these rational functions to all orders in inverse powers of their index m. Each term of the development involves a periodic function in the base b logarithm of m, depending on s.

Citations

• Some series representing the zeta function for <inline-formula><tex-math id="M1"> \Re(s)&gt;1 </tex-math></inline-formula>
• Moments in the exact summation of the curious series of Kempner type
• Combinatorics on words and generating Dirichlet series of automatic sequences
• Philippe Flajolet's early work in combinatorics
• Singularity Analysis of Generating Functions
• Sums of Reciprocals of Integers Missing a Given Digit