 About 839 results  books.google.com (e) Slow sum. £ (–1)" > m log(n) (f) Abel sum. >= 7 n log”(n) (g) Slow
trigonometric series. XL ": and, for 0 < x < T, XL #. • (h) Hardy–Littlewood sum. n=
1 n=1 2. SII] #" • (i) Euler sum. G(3) = G(2, 1) = 8 G(–2, 1). (j) EulerZagier sum [72
, Chapter ... 

 books.google.com It seems that the method of using the MellinBarnes integral is suitable to
consider upper bound estimates of multiple zetafunctions. The case of the Euler
Zagier sum §EZ,r(s1,.. .,s.) was studied by Ishikawa and Matsumoto [15];
especially, ... 

 books.google.com ... call EulerZagier sum, polyzeta, or MZV (Multiple Zeta Value) the following sum
, which appears as an extension of the Riemann zeta function to multiindices: C(
«). = C(«i,«a,...,«p). = £. ~^r4. *•. (2.1). ni>n2>>np>0 i 2 p This sum converges if ... 

 books.google.com 1998a, Flajolet 1998]. Let us start with the definition of EulerZagier sums ^ ' ' ' °fn
, (9.2) where Sj are positive integers and s\ > 1 and a, G {!,+!}. (9.3) We call n the
depth of the sum and N := s\ H  \ sn the weight. Not all of the above sums ... 

 books.google.com 3. Multivariable. EulerZagier. Sums. The rvariable generalization of the Euler
sum (2.1), defined by Cr(«l,...,»r) (3.1) f1 (mi + m2)~S2    (mi H  + mr)~8r , oo
oo oo is absolutely convergent in the region ..,sr)£Cr\K(srk+l+ + sr)>k (1 < fc ... 

 books.google.com 175(1), 949–976 (2012) The formula goes back to de Moivre, Bernoulli, Euler,
and later Binet, see: Beutelspacher, A., ... Cambridge University Press,
Cambridge (1997); Zagier, D.: Multiple Zeta Values. ... K.: The sum formula of
multiple zeta values and connection problem of the formal Knizhnik–
Zamolodchikov equation. 

 books.google.com Zsums are defined by fc is called the depth of the Zsum and w = mj + ... + mfc is
called the weight. If the sums go to ... xk — I the definition reduces to the Euler
Zagier sums [64,65]: Z(n;mi,...,mk;l,...,l) = Zmi,...,mfc(n). (6.19) For n = oo and zi = . 

 books.google.com To sum up all residues which lie inside the contour it is useful to know the
residues of the Gamma function:  (–1)" res (T(a + a), G = —a – n) ... + 8"Z11.1(n–
D]. where Zm.....m., (n) are EulerZagier sums defined by 1 1 Zmi,...my (n) = XD T .
. 

 books.google.com D. Borwein, J. M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums,
Proc. Edinburgh Math. ... J. M. Borwein, D. M. Bradley and D. J. Broadhurst,
Evaluations of kfold Euler/Zagier sums: A compendium of results for arbitrary k,
Electronic J. Combin. 4 (2) (1997) ... [169] D. M. Bradley, On the sum formula for
multiple. 

 books.google.com ik>0 k is called the depth of the Z sum and w : m1 +    + mk is called the weight.
If the sums go ... (4.171) For x1 :    : xk : 1 the definition reduces to the Euler—
Zagier sums [93—97]: Z(n;m1, . . . , rm; 1, . . . , 1) : Zm17___'mk(n). (4.172) For n ... 

 