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2012, University lecture series, ISBN 9780821853672, Volume 59, x, 190
Book
Journal of number theory, ISSN 0022-314X, 2019, Volume 202, pp. 278 - 297
We obtain reasonably tight upper and lower bounds on the sum ∑n⩽xφ(⌊x/n⌋), involving the Euler functions φ... 
Euler function | Exponent pair | Reciprocals | Integer part
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 11/2018, Volume 14, Issue 10, pp. 2699 - 2728
... of the gcd -sum function f ( gcd ( j , k ) ) and the function ∑ d | k , d s | j ( f ∗ μ ) ( d ) for any positive integers j and k , namely... 
gcd -sum functions | mean value formula | Euler totient function | Dedekind function | MATHEMATICS | GENERALIZED RAMANUJAN SUMS | THEOREM | EXTENSION | gcd-sum functions
Journal Article
1993, ISBN 9780387533452, xiii, 222
Book
Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 10/2018, Volume 466, Issue 1, pp. 1009 - 1042
In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers... 
Euler sums | Riemann zeta function | Multiple zeta (star) values | Stirling numbers | Harmonic numbers | Multiple harmonic (star) numbers | INTEGRALS | MATHEMATICS | MULTIPLE ZETA-VALUES | MATHEMATICS, APPLIED | SERIES
Journal Article
by Xu, Ce
Applied Mathematics and Computation, ISSN 0096-3003, 04/2019, Volume 346, pp. 594 - 611
In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series... 
Euler sum | Riemann zeta function | Multiple zeta (star) value | Multiple harmonic (star) sum | polylogarithm function | Harmonic number | INTEGRALS | MATHEMATICS, APPLIED | MULTIPLE ZETA VALUES
Journal Article
by Xu, C
BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, ISSN 0126-6705, 01/2020, Volume 43, Issue 1, pp. 847 - 877
...) and x is an element of [-1, 1), define the so-called Euler-type sums Sp(1) p(2)... p(m), p (x), which are the infinite sums whose general term is a product of harmonic numbers of index n, a power of n... 
MATHEMATICS | Polylogarithm function | IDENTITIES | Harmonic number | Multiple harmonic sum | Euler sum | Riemann zeta function | Multiple zeta value | EXPLICIT EVALUATION | MULTIPLE ZETA VALUES
Journal Article
MATHEMATICS, ISSN 2227-7390, 09/2019, Volume 7, Issue 9, p. 833
In this paper, we present some Euler-like sums involving partial sums of the harmonic and odd harmonic series... 
SUMMATION FORMULAS | MATHEMATICS | SERIES | closed form | ArcTan and ArcTanh functions | Catalan's constant | HARMONIC SUMS | Euler sums | Trigamma function | integral representation | partial fractions | FAMILY | Catalan’s constant
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 04/2017, Volume 13, Issue 3, pp. 655 - 672
In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers... 
Euler sum | Riemann zeta function | Polylogarithm function | harmonic number | INTEGRALS | MATHEMATICS | VALUES
Journal Article
Analysis, ISSN 0174-4747, 11/2016, Volume 36, Issue 4, pp. 231 - 243
A class of sums of the type is evaluated, where , and are positive integers with and are integers satisfying 
Euler’s formula | 11A07 | 11Y60 | Bernoulli numbers and polynomials | Euler’s sum | Kronecker symbol | 11B25 | 11A25 | 11B68 | subsums of Euler’s sum | 11F66 | Euler's sum | Euler's formula | subsums of Euler's sum
Journal Article
Journal of Difference Equations and Applications, ISSN 1023-6198, 07/2019, Volume 25, Issue 7, pp. 1007 - 1023
.... Using the decompositions, we discuss the evaluations of some Euler-type sums involving harmonic numbers and binomial coefficients, such as and... 
Riemann zeta function | Euler-type sums | harmonic numbers | binomial coefficients | MATHEMATICS, APPLIED | SERIES | IDENTITIES | DUALITY | EULER SUMS | Binomial coefficients | Decomposition | Sums
Journal Article
Applicable analysis and discrete mathematics, ISSN 1452-8630, 10/2017, Volume 11, Issue 2, pp. 369 - 385
In this paper we investigate a certain category of cotangent sums and more specifically the sum and associate the distribution of its values to a generalized totient function 𝜑(𝑛, 𝐴, 𝐵... 
Integers | Cotangent function | Mathematical functions | Cots | Euler totient function | Generalized totient function asymptotics | Fractional part | Cotangent sums | MATHEMATICS | MATHEMATICS, APPLIED | fractional part | generalized totient function asymptotics
Journal Article
by Xu, Ce
Journal of the Korean Mathematical Society, ISSN 0304-9914, 2018, Volume 55, Issue 5, pp. 1207 - 1220
.... Moreover, we prove that the Euler-type sums with hyperharmonic numbers: S(k, m; p) := Sigma(infinity)(n=1) h(n)((m)) (k)/n(p) (p >= m + 1, k = 1, 2, 3) can be expressed... 
Euler sums | Riemann zeta function | Generalized hyperharmonic numbers | Stirling numbers | Harmonic numbers | INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED | ZETA-FUNCTION | SERIES | generalized hyperharmonic numbers | VALUES | harmonic numbers
Journal Article
Journal of Integer Sequences, 2011, Volume 14, Issue 7
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 05/2014, Volume 10, Issue 3, pp. 737 - 762
...}.$$ We derive reciprocity theorems for the sums arising in these transformation formulas and investigate certain properties... 
Bernoulli polynomials | Hardy-Berndt sums | Euler-Maclaurin formula | Dedekind sums | MATHEMATICS | THETA-FUNCTIONS | ANALYTIC EISENSTEIN SERIES | Heterocyclic compounds | Mathematics - Number Theory
Journal Article
Discrete Mathematics, ISSN 0012-365X, 2009, Volume 309, Issue 10, pp. 3346 - 3363
.... Particularly, some of these identities are also related to the power sums and alternate power sums... 
Alternate power sums | Power sums | Bernoulli polynomials | Genocchi polynomials | Combinatorial identities | Euler polynomials | MATHEMATICS | SYMMETRY | SERIES | NUMBERS | RECURRENCE
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 08/2017, Volume 13, Issue 7, pp. 1695 - 1709
... ! , and then we produce the generating function and an integral representation for S n ( m ) . Using them we evaluate many interesting finite and infinite harmonic sums in closed form... 
Harmonic sums | Bell polynomials | Boole's formula | generalized harmonic numbers | Riemann zeta function | Apery constant | harmonic numbers | Stirling numbers | combinatorial identities | MATHEMATICS | HYPERGEOMETRIC-SERIES | EULER | Mathematics - Number Theory
Journal Article