[Paper Review] Elementary evaluations of some Euler sums

Donal F. Connon|ArXiv.org|Oct 29, 2007

Advanced Mathematical Identities2 references3 citations

TL;DR

This paper presents elementary evaluations of specific Euler sums—series involving harmonic numbers and inverse powers—using direct analytical techniques. The author derives closed-form expressions for several classical Euler sums, offering accessible derivations without advanced special functions, thereby contributing explicit results for previously studied series in number theory and analysis.

ABSTRACT

This short note contains elementary evaluations of some Euler sums.

Motivation & Objective

• To provide accessible, elementary derivations of selected Euler sums without relying on advanced special functions.
• To evaluate specific instances of Euler sums involving harmonic numbers and zeta values.
• To contribute explicit closed-form expressions for classical Euler sums that are otherwise difficult to compute.
• To demonstrate that certain Euler sums can be resolved through basic calculus and series manipulation techniques.
• To offer a self-contained, elementary approach to results often derived using complex analysis or polylogarithmic identities.

Proposed method

• The author employs direct summation techniques and algebraic manipulation of series expansions.
• Elementary identities involving harmonic numbers and Riemann zeta functions are systematically applied.
• The method avoids complex analysis, focusing instead on real-variable techniques and known series transformations.
• Specific generating functions and summation by parts are used to simplify and evaluate the target Euler sums.
• The approach relies on known results from classical analysis, such as the Euler reflection formula and series identities.
• Each sum is evaluated step-by-step using only basic calculus and series algebra, ensuring accessibility.

Experimental results

Research questions

• RQ1 Can classical Euler sums be evaluated using only elementary methods rather than complex analysis?
• RQ2 What closed-form expressions can be derived for specific Euler sums involving harmonic numbers and inverse powers?
• RQ3 Are there systematic ways to evaluate Euler sums through real-variable techniques alone?
• RQ4 How do elementary derivations compare in simplicity and clarity to those using special functions?
• RQ5 Which known Euler sum identities can be re-derived with minimal advanced mathematical tools?

Key findings

• The paper successfully derives closed-form expressions for several Euler sums using only elementary techniques.
• Specific evaluations are provided for sums such as ∑(H_n / n^k) for small integer values of k, yielding exact rational and zeta-function combinations.
• The results confirm known values of Euler sums but are obtained through simpler, more transparent derivations.
• The method demonstrates that advanced tools like polylogarithms or contour integration are not necessary for evaluating certain Euler sums.
• The approach provides a pedagogically useful pathway to understanding Euler sums without requiring deep functional analysis.
• The work contributes explicit, verifiable results that can be used in further research on zeta functions and harmonic series.