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EULER SUMS EVALUATABLE FROM INTEGRALS [PDF]
Communications of the Korean Mathematical Society, 2004 Summary: Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a proof of the classical Euler sum by following Lewin's method. We also consider some related formulas involving Euler sums, which are evaluatable from some known definite integrals.
June-Sang Choi
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Explicit Evaluation of Euler and Related Sums
Ramanujan Journal, 2005zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Junesang Choi, Hari Mohan Srivastava
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ON SUMS INVOLVING THE EULER TOTIENT FUNCTION
Bulletin of the Australian Mathematical Society, 2023AbstractLet $\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers $n_{1},\ldots ,n_{k}$ and let $\phi $ be the Euler totient function. For any real number $x>3$ and any integer $k\geq 2$ , we investigate the asymptotic behaviour of $\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $
Kiuchi, Isao, Tsuruta, Yuki
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A class of subsums of Euler’s sum
Analysis, 2016AbstractA class of sums of the typeis evaluated ...
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A variety of Euler's sum of powers conjecture
Czechoslovak Mathematical Journal, 2021zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Cai, Tianxin, Zhang, Yong
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EXPLICIT EVALUATION OF TRIPLE EULER SUMS
International Journal of Number Theory, 2008The triple Euler sum defined by [Formula: see text] with positive integers p, q, r, p ≥ 2, has not been fully evaluated yet except for the case of q = r = 1 and some particular cases such as p + q + r ≤ 10 or p = q = r. With the general theory developed for double Euler sums, we are able to produce identities among triple Euler sums with a variable or
Ong, Yao Lin +2 more
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A New Method for Investigating Euler Sums
The Ramanujan Journal, 2000zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Basu, Ankur, Apostol, Tom M.
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Generalization of the Euler Sum
2003In this chapter an investigation of a generalization of the identity (2.13) of Chapter 2 is undertaken. The investigation will also make use of the finite binomial type sums obtained in Chapter 4. A connection with renewal processes will be made. It will be proved that generated infinite sums may be represented in closed form that depend on k-dominant ...
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2003
A large literature exists relating Riemann’s zeta function $$\zeta (s) = \sum\limits_{{k = 1}}^{\infty } {\frac{1}{{{k^s}}}}, \quad \Re (s)\; > \;1, $$ and partial sums of the harmonic series, $$ h(n) = \sum\limits_{{k = 1}}^n {\frac{1}{k}.} $$ Much of the research originated from two striking formulas discovered by Euler, $$ \sum ...
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A large literature exists relating Riemann’s zeta function $$\zeta (s) = \sum\limits_{{k = 1}}^{\infty } {\frac{1}{{{k^s}}}}, \quad \Re (s)\; > \;1, $$ and partial sums of the harmonic series, $$ h(n) = \sum\limits_{{k = 1}}^n {\frac{1}{k}.} $$ Much of the research originated from two striking formulas discovered by Euler, $$ \sum ...
openaire +2 more sources