Multiple zeta function

In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by

n_2 > \\cdots > n_k > 0}\\ \\frac{1}{n_1^{s_1} \\cdots n_k^{s_k}} = \\sum_{n_1 > n_2 > \\cdots > n_k > 0}\\ \\prod_{i=1}^k \\frac{1}{n_i^{s_i}},\\!"}}'>ζ(s1,…,sk)=∑n1>n2>⋯>nk>0 1n1s1⋯nksk=∑n1>n2>⋯>nk>0 ∏i=1k1nisi,

and converge when Re(s1) + ... + Re(si) > i for all i. Like the Riemann zeta function, the multiple zeta functions can be analytically continued to be meromorphic functions (see, for example, Zhao (1999)). When s1, ..., sk are all positive integers (with s1 > 1) these sums are often called multiple zeta values (MZVs) or Euler sums. These values can also be regarded as special values of the multiple polylogarithms.

The k in the above definition is named the "depth" of a MZV, and the n = s1 + ... + sk is known as the "weight".

The standard shorthand for writing multiple zeta functions is to place repeating strings of the argument within braces and use a superscript to indicate the number of repetitions. For example,

Lis1,…,sd(μ1,…,μd)=∑k1>⋯>kd>0μ1k1⋯μdkdk1s1⋯kdsd

which are generalizations of the polylogarithm functions. When all of the n. In particular, when ζ(s1,…,sd)=∑k1>⋯>kd>01k1s1⋯kdsd

and Euler sums are written

\\cdots > k_d > 0} \\frac{\\varepsilon_1^{k_1}\\cdots \\varepsilon_d^{k_d}}{k_1^{s_1}\\cdots k_d^{s_d}}"}}'>ζ(s1,…,sd;ε1,…,εd)=∑k1>⋯>kd>0ε1k1⋯εdkdk1s1⋯kdsd

where εi equal to ∫0xf1(t)dt⋯fd(t)dt=∫0xf1(t1)(∫0t1f2(t2)(∫0t2⋯(∫0tdfd(td)dtd))dt2)dt1

Using this convention, the result can be stated as follows:

j=1,2,…,d.

This result is extremely useful due to a well-known result regarding products of iterated integrals, namely that

Sm is the symmetric group on X∗ to be the free monoid generated by ℚ-vector space generated by a=dtt, 𝐰∈X∗,

which, by the aforementioned integral identity, makes

ζ(s,t)=∑n>m≥1 1nsmt=∑n=2∞1ns∑m=1n−11mt=∑n=1∞1(n+1)s∑m=1n1mt

∑n=1∞Hn(n+1)2=ζ(2,1)=ζ(3)=∑n=1∞1n3,

where Hn are the harmonic numbers.

Special values of double zeta functions, with s > 0 and even, t > 1 and odd, but s+t = 2N+1 (taking if necessary ζ(0) = 0):

3ζ(2)ζ(3)−112ζ(5)A258983 420.08848338245436871429432783908692ζ(5)−2ζ(2)ζ(3)A258986 330.2137988682245925470995835745085ζ(3)ζ(5)−14724ζ(8)−52ζ(6,2)A258982 240.67452391403396814049156060825712((ζ(4))2−ζ(8))A258991

Note that if ζ(a) only.

Three parameters case

edit

In the particular case of only three parameters we have (with a > 1 and n, j, i integers):

j > i \\geq 1}\\ \\frac{1}{n^{a} j^{b} i^{c}} = \\sum_{n=1}^{\\infty} \\frac{1}{(n+2)^{a}} \\sum_{j=1}^n \\frac{1}{(j+1)^b} \\sum_{i=1}^{j} \\frac{1}{(i)^c} = \\sum_{n=1}^{\\infty} \\frac{1}{(n+2)^{a}} \\sum_{j=1}^n \\frac{H_{j,c}}{(j+1)^b}"}}'>ζ(a,b,c)=∑n>j>i≥1 1najbic=∑n=1∞1(n+2)a∑j=1n1(j+1)b∑i=1j1(i)c=∑n=1∞1(n+2)a∑j=1nHj,c(j+1)b

Euler reflection formula

edit

The above MZVs satisfy the Euler reflection formula:

ζ(a,b,c)+ζ(a,c,b)+ζ(b,a,c)+ζ(b,c,a)+ζ(c,a,b)+ζ(c,b,a)=ζ(a)ζ(b)ζ(c)+2ζ(a+b+c)−ζ(a)ζ(b+c)−ζ(b)ζ(a+c)−ζ(c)ζ(a+b) for 1"}}'>a,b,c>1

This function can be seen as a generalization of the reflection formulas.

Symmetric sums in terms of the zeta function

edit

Let {1,2,…,k}, let i={i1,...,ik} of exponents, define S are: i1,⋯,ik>1,, ∑σ∑n1≥n2≥⋯≥nk≥11ni1σ(1)ni2σ(2)⋯nikσ(k). Now thinking on the symmetric

group n=(n1,⋯,nk) has an isotropy group

(1,2,⋯,k): ni=nj, and ∑σ∈ΣkS(iσ(1),…,iσ(k))=∑partitions Π of {1,…,k}c(Π)ζ(i,Π) exactly Λ: letting ∑Π⪰Λ(Π) times. Thus, the conclusion will follow if Λ. To see this, note that Σk(n) has a unique cycle type specified by a partition that refines ∑σ∈Σ3S(iσ(1),iσ(2),iσ(3))=ζ(i1)ζ(i2)ζ(i3)+ζ(i1+i2)ζ(i3)+ζ(i1)ζ(i2+i3)+ζ(i1+i3)ζ(i2)+2ζ(i1+i2+i3) for 1"}}'>i1,i2,i3>1. This is the main result of.

Having n_2>\\cdots n_k\\geq1}\\frac{1}{n_1^{i_1} n_2^{i_2}\\cdots n_k^{i_k}}"}}'>ζ(i1,i2,⋯,ik)=∑n1>n2>⋯nk≥11n1i1n2i2⋯nkik. To state the analog of Theorem 1 for the {1,2⋯,k}, let ∑σ∈Σkζ(iσ(1),…,iσ(k))=∑partitions Π of {1,…,k}c~(Π)ζ(i,Π).

Proof. We follow the same line of argument as in the preceding proof. The left-hand side is now n_2 > \\cdots > n_k \\geq1} \\frac{1}{{n^{i_1}}_{\\sigma(1)}{n^{i_2}}_{\\sigma(2)} \\cdots {n^{i_k}}_{\\sigma(k)} }"}}'>∑σ∑n1>n2>⋯>nk≥11ni1σ(1)ni2σ(2)⋯nikσ(k), and a term ∑Π⪰Λc~(Π)={1, if |Λ|=k0, otherwise . (1)

To prove this, note first that the sign of Σk(n). But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition ∑i1+⋯+ik=n,i1>1ζ(i1,⋯,ik)=ζ(n), where the sum is extended over k-tuples ∑i1+⋯+ik=n,i1>1S(i1,⋯,ik)=(n−1k−1)ζ(n). Second, in the case ζ′s and τ on the set Σ:ℑ→T be the function that sends a sequence in T whose last element is at most Cn on Cn(a1,…,al) = complement of τ is i1+⋯+ik=n.

For example, τ(i1,…,ik) are dual to each other, and refer to a sequence fixed by (i1,…,ik), then ζ(6,1)+ζ(5,2)+ζ(4,3)+ζ(3,4)+ζ(2,5)=ζ(7).

The Euler sum with alternations of sign appears in studies of the non-alternating Euler sum.

∑n=1∞H¯n(b)(n+1)a=ζ(a,b¯) with ∑n=1∞(−1)n(n+2)a∑n=1∞H¯n(c)(−1)(n+1)(n+1)b=ζ(a¯,b¯,c¯) with Hn(c)=+1+12c+13c+⋯

ϕ(s)=1−2(s−1)2(s−1)ζ(s) with 1"}}'>s>1

ζ(a,b¯)+ζ(b¯,a)=ζ(a)ϕ(b)−ϕ(a+b)

a=b we have a>1

ζ(a,b)+ζ(a¯,b¯)=∑s>0(a+b−s−1)![Za(a+b−s,s)(a−s)!(b−1)!+Zb(a+b−s,s)(b−s)!(a−1)!]

where s must be used for all value 1"}}'>>1 for which the argument of the factorials is ∑n=2∞ζ(n,k)=ζ(k+1) or more generally:

∑n=2∞ζ(n,a¯,b)=ζ(a+1‾,b)

limk→∞ζ(n,k)=ζ(n)−1

ζ(a,a,a)=16(ζ(a))3+13ζ(3a)−12ζ(a)ζ(2a)

The Mordell–Tornheim zeta function, introduced by Matsumoto (2003) who was motivated by the papers Mordell (1958) and Tornheim (1950), is defined by

0}\\frac{1}{ m_1^{s_1}\\cdots m_r^{s_r}(m_1+\\dots+m_r)^{s_{r+1}}}"}}'>ζMT,r(s1,…,sr;sr+1)=∑m1,…,mr>01m1s1⋯mrsr(m1+⋯+mr)sr+1

It is a special case of the Shintani zeta function.

• Tornheim, Leonard (1950). "Harmonic double series". American Journal of Mathematics. 72 (2): 303–314. doi:10.2307/2372034. ISSN 0002-9327. JSTOR 2372034. MR 0034860.
• Crandall, Richard E.; Buhler, Joe P. (1994). "On the evaluation of Euler Sums". Experimental Mathematics. 3 (4): 275. doi:10.1080/10586458.1994.10504297. MR 1341720.
• Zhao, Jianqiang (1999). "Analytic continuation of multiple zeta functions". Proceedings of the American Mathematical Society. 128 (5): 1275–1283. doi:10.1090/S0002-9939-99-05398-8. MR 1670846.
• Espinosa, Olivier; Moll, Victor Hugo (2010). "The evaluation of Tornheim double sums II". Ramanujan J. 22: 55–99. arXiv:0811.0557. doi:10.1007/s11139-009-9181-1. MR 2610609. S2CID 17055581.
• 1. ↑ Hoffman, Mike. "Multiple Zeta Values". Mike Hoffman's Home Page. U.S. Naval Academy. Retrieved June 8, 2012.
  2. 1 2 Hoffman, Michael (1992). "Multiple Harmonic Series". Pacific Journal of Mathematics. 152 (2): 276–278. doi:10.2140/pjm.1992.152.275. MR 1141796. Zbl 0763.11037.
  3. ↑ Euler, L. (1775). "Meditationes circa singulare serierum genus". Novi Comm. Acad. Sci. Petropol. 15 (20): 140–186.
  4. ↑ Hoffman, Michael (2012). "Multiple zeta values".