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Multiple zeta function
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<h2 id="definition">Definition</h2> <p>Multiple zeta functions arise as special cases of the multiple polylogarithms </p> L i s 1 , … , s d ( μ 1 , … , μ d ) = ∑ k 1 > ⋯ > k d > 0 μ 1 k 1 ⋯ μ d k d k 1 s 1 ⋯ k d s d {\displaystyle \mathrm {Li} _{s_{1},\ldots ,s_{d}}(\mu _{1},\ldots ,\mu _{d})=\sum \limits _{k_{1}>\cdots >k_{d}>0}{\frac {\mu _{1}^{k_{1}}\cdots \mu _{d}^{k_{d}}}{k_{1}^{s_{1}}\cdots k_{d}^{s_{d}}}}} <p>which are generalizations of the <a href="/facts/Polylogarithm/CKrkJMKn">polylogarithm</a> functions. When all of the μ i {\displaystyle \mu _{i}} are <i>n</i>th <a href="/facts/Roots_of_unity/EqH0ho1Y">roots of unity</a> and the s i {\displaystyle s_{i}} are all nonnegative integers, the values of the multiple polylogarithm are called colored multiple zeta values of level n {\displaystyle n} . In particular, when n = 2 {\displaystyle n=2} , they are called Euler sums or alternating multiple zeta values, and when n = 1 {\displaystyle n=1} they are simply called multiple zeta values. Multiple zeta values are often written </p> ζ ( s 1 , … , s d ) = ∑ k 1 > ⋯ > k d > 0 1 k 1 s 1 ⋯ k d s d {\displaystyle \zeta (s_{1},\ldots ,s_{d})=\sum \limits _{k_{1}>\cdots >k_{d}>0}{\frac {1}{k_{1}^{s_{1}}\cdots k_{d}^{s_{d}}}}} <p>and Euler sums are written </p> ζ ( s 1 , … , s d ; ε 1 , … , ε d ) = ∑ k 1 > ⋯ > k d > 0 ε 1 k 1 ⋯ ε k d k 1 s 1 ⋯ k d s d {\displaystyle \zeta (s_{1},\ldots ,s_{d};\varepsilon _{1},\ldots ,\varepsilon _{d})=\sum \limits _{k_{1}>\cdots >k_{d}>0}{\frac {\varepsilon _{1}^{k_{1}}\cdots \varepsilon ^{k_{d}}}{k_{1}^{s_{1}}\cdots k_{d}^{s_{d}}}}} <p>where ε i = ± 1 {\displaystyle \varepsilon _{i}=\pm 1} . Sometimes, authors will write a bar over an s i {\displaystyle s_{i}} corresponding to an ε i {\displaystyle \varepsilon _{i}} equal to − 1 {\displaystyle -1} , so for example </p> ζ ( a ¯ , b ) = ζ ( a , b ; − 1 , 1 ) {\displaystyle \zeta ({\overline {a}},b)=\zeta (a,b;-1,1)} . <h2 id="integral-structure-and-identities">Integral structure and identities</h2> <p>It was noticed by Kontsevich that it is possible to express colored multiple zeta values (and thus their special cases) as certain multivariable <a href="/facts/Integral/V7lyV997">integrals</a>. This result is often stated with the use of a convention for iterated integrals, wherein </p> ∫ 0 x f 1 ( t ) d t ⋯ f d ( t ) d t = ∫ 0 x f 1 ( t 1 ) ( ∫ 0 t 1 f 2 ( t 2 ) ( ∫ 0 t 2 ⋯ ( ∫ 0 t d f d ( t d ) d t d ) ) d t 2 ) d t 1 {\displaystyle \int _{0}^{x}f_{1}(t)dt\cdots f_{d}(t)dt=\int _{0}^{x}f_{1}(t_{1})\left(\int _{0}^{t_{1}}f_{2}(t_{2})\left(\int _{0}^{t_{2}}\cdots \left(\int _{0}^{t_{d}}f_{d}(t_{d})dt_{d}\right)\right)dt_{2}\right)dt_{1}} <p>Using this convention, the result can be stated as follows:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> L i s 1 , … , s d ( μ 1 , … , μ d ) = ∫ 0 1 ( d t t ) s 1 − 1 d t a 1 − t ⋯ ( d t t ) s d − 1 d t a d − t {\displaystyle \mathrm {Li} _{s_{1},\ldots ,s_{d}}(\mu _{1},\ldots ,\mu _{d})=\int _{0}^{1}\left({\frac {dt}{t}}\right)^{s_{1}-1}{\frac {dt}{a_{1}-t}}\cdots \left({\frac {dt}{t}}\right)^{s_{d}-1}{\frac {dt}{a_{d}-t}}} where a j = ∏ i = 1 j μ i − 1 {\displaystyle a_{j}=\prod \limits _{i=1}^{j}\mu _{i}^{-1}} for j = 1 , 2 , … , d {\displaystyle j=1,2,\ldots ,d} . <p>This result is extremely useful due to a well-known result regarding products of iterated integrals, namely that </p> ( ∫ 0 x f 1 ( t ) d t ⋯ f n ( t ) d t ) ( ∫ 0 x f n + 1 ( t ) d t ⋯ f m ( t ) d t ) = ∑ σ ∈ S h n , m ∫ 0 x f σ ( 1 ) ( t ) ⋯ f σ ( m ) ( t ) {\displaystyle \left(\int _{0}^{x}f_{1}(t)dt\cdots f_{n}(t)dt\right)\!\left(\int _{0}^{x}f_{n+1}(t)dt\cdots f_{m}(t)dt\right)=\sum \limits _{\sigma \in {\mathfrak {Sh}}_{n,m}}\int _{0}^{x}f_{\sigma (1)}(t)\cdots f_{\sigma (m)}(t)} where S h n , m = { σ ∈ S m ∣ σ ( 1 ) < ⋯ < σ ( n ) , σ ( n + 1 ) < ⋯ < σ ( m ) } {\displaystyle {\mathfrak {Sh}}_{n,m}=\{\sigma \in S_{m}\mid \sigma (1)<\cdots <\sigma (n),\sigma (n+1)<\cdots <\sigma (m)\}} and S m {\displaystyle S_{m}} is the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> on m {\displaystyle m} symbols. <p>To utilize this in the context of multiple zeta values, define X = { a , b } {\displaystyle X=\{a,b\}} , X ∗ {\displaystyle X^{*}} to be the <a href="/facts/Free_monoid/h3nPVjqr">free monoid</a> generated by X {\displaystyle X} and A {\displaystyle {\mathfrak {A}}} to be the <a href="/facts/Free_module/o6rC6yoJ">free</a> Q {\displaystyle \mathbb {Q} } -<a href="/facts/Vector_space/M1pxMLx2">vector space</a> generated by X ∗ {\displaystyle X^{*}} . A {\displaystyle {\mathfrak {A}}} can be equipped with the <a href="/facts/Shuffle_algebra/AgsGQqgJ">shuffle product</a>, turning it into an <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra</a>. Then, the multiple zeta function can be viewed as an evaluation map, where we identify a = d t t {\displaystyle a={\frac {dt}{t}}} , b = d t 1 − t {\displaystyle b={\frac {dt}{1-t}}} , and define </p> ζ ( w ) = ∫ 0 1 w {\displaystyle \zeta (\mathbf {w} )=\int _{0}^{1}\mathbf {w} } for any w ∈ X ∗ {\displaystyle \mathbf {w} \in X^{*}} , <p>which, by the aforementioned integral <a href="/facts/Identity_(mathematics)/LR46j4uR">identity</a>, makes </p> ζ ( a s 1 − 1 b ⋯ a s d − 1 b ) = ζ ( s 1 , … , s d ) . {\displaystyle \zeta (a^{s_{1}-1}b\cdots a^{s_{d}-1}b)=\zeta (s_{1},\ldots ,s_{d}).} <p>Then, the integral identity on products gives<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> ζ ( w ) ζ ( v ) = ζ ( w ⧢ v ) . {\displaystyle \zeta (w)\zeta (v)=\zeta (w{\text{ ⧢ }}v).} <h2 id="two-parameters-case">Two parameters case</h2> <p>In the particular case of only two parameters we have (with <i>s</i> > 1 and <i>n</i>, <i>m</i> integers):<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> ζ ( s , t ) = ∑ n > m ≥ 1 1 n s m t = ∑ n = 2 ∞ 1 n s ∑ m = 1 n − 1 1 m t = ∑ n = 1 ∞ 1 ( n + 1 ) s ∑ m = 1 n 1 m t {\displaystyle \zeta (s,t)=\sum _{n>m\geq 1}\ {\frac {1}{n^{s}m^{t}}}=\sum _{n=2}^{\infty }{\frac {1}{n^{s}}}\sum _{m=1}^{n-1}{\frac {1}{m^{t}}}=\sum _{n=1}^{\infty }{\frac {1}{(n+1)^{s}}}\sum _{m=1}^{n}{\frac {1}{m^{t}}}} ζ ( s , t ) = ∑ n = 1 ∞ H n , t ( n + 1 ) s {\displaystyle \zeta (s,t)=\sum _{n=1}^{\infty }{\frac {H_{n,t}}{(n+1)^{s}}}} where H n , t {\displaystyle H_{n,t}} are the <a href="/facts/Harmonic_number/1b7v0hVn">generalized harmonic numbers</a>. <p>Multiple zeta functions are known to satisfy what is known as MZV duality, the simplest case of which is the famous identity of <a href="/facts/Leonhard_Euler/z2fsbjgj">Euler</a>: </p> ∑ n = 1 ∞ H n ( n + 1 ) 2 = ζ ( 2 , 1 ) = ζ ( 3 ) = ∑ n = 1 ∞ 1 n 3 , {\displaystyle \sum _{n=1}^{\infty }{\frac {H_{n}}{(n+1)^{2}}}=\zeta (2,1)=\zeta (3)=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}},\!} <p>where <i>H</i><i>n</i> are the <a href="/facts/Harmonic_number/1b7v0hVn">harmonic numbers</a>. </p><p>Special values of double zeta functions, with <i>s</i> > 0 and <a href="/facts/Parity_(mathematics)/7zq2UM4V">even</a>, <i>t</i> > 1 and <a href="/facts/Parity_(mathematics)/7zq2UM4V">odd</a>, but <i>s</i>+<i>t</i> = 2<i>N</i>+1 (taking if necessary <i>ζ</i>(0) = 0):<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> ζ ( s , t ) = ζ ( s ) ζ ( t ) + 1 2 [ ( s + t s ) − 1 ] ζ ( s + t ) − ∑ r = 1 N − 1 [ ( 2 r s − 1 ) + ( 2 r t − 1 ) ] ζ ( 2 r + 1 ) ζ ( s + t − 1 − 2 r ) {\displaystyle \zeta (s,t)=\zeta (s)\zeta (t)+{\tfrac {1}{2}}{\Big [}{\tbinom {s+t}{s}}-1{\Big ]}\zeta (s+t)-\sum _{r=1}^{N-1}{\Big [}{\tbinom {2r}{s-1}}+{\tbinom {2r}{t-1}}{\Big ]}\zeta (2r+1)\zeta (s+t-1-2r)} <table><tbody><tr><th><i>s</i></th><th><i>t</i></th><th>approximate value</th><th>explicit formulae</th><th><a href="/facts/OEIS/yow6cVsU">OEIS</a></th></tr><tr><td>2</td><td>2</td><td>0.811742425283353643637002772406</td><td> 3 4 ζ ( 4 ) {\displaystyle {\tfrac {3}{4}}\zeta (4)} </td><td>A197110</td></tr><tr><td>3</td><td>2</td><td>0.228810397603353759768746148942</td><td> 3 ζ ( 2 ) ζ ( 3 ) − 11 2 ζ ( 5 ) {\displaystyle 3\zeta (2)\zeta (3)-{\tfrac {11}{2}}\zeta (5)} </td><td>A258983</td></tr><tr><td>4</td><td>2</td><td>0.088483382454368714294327839086</td><td> ( ζ ( 3 ) ) 2 − 4 3 ζ ( 6 ) {\displaystyle \left(\zeta (3)\right)^{2}-{\tfrac {4}{3}}\zeta (6)} </td><td>A258984</td></tr><tr><td>5</td><td>2</td><td>0.038575124342753255505925464373</td><td> 5 ζ ( 2 ) ζ ( 5 ) + 2 ζ ( 3 ) ζ ( 4 ) − 11 ζ ( 7 ) {\displaystyle 5\zeta (2)\zeta (5)+2\zeta (3)\zeta (4)-11\zeta (7)} </td><td>A258985</td></tr><tr><td>6</td><td>2</td><td>0.017819740416835988362659530248</td><td></td><td>A258947</td></tr><tr><td>2</td><td>3</td><td>0.711566197550572432096973806086</td><td> 9 2 ζ ( 5 ) − 2 ζ ( 2 ) ζ ( 3 ) {\displaystyle {\tfrac {9}{2}}\zeta (5)-2\zeta (2)\zeta (3)} </td><td>A258986</td></tr><tr><td>3</td><td>3</td><td>0.213798868224592547099583574508</td><td> 1 2 ( ( ζ ( 3 ) ) 2 − ζ ( 6 ) ) {\displaystyle {\tfrac {1}{2}}\left(\left(\zeta (3)\right)^{2}-\zeta (6)\right)} </td><td>A258987</td></tr><tr><td>4</td><td>3</td><td>0.085159822534833651406806018872</td><td> 17 ζ ( 7 ) − 10 ζ ( 2 ) ζ ( 5 ) {\displaystyle 17\zeta (7)-10\zeta (2)\zeta (5)} </td><td>A258988</td></tr><tr><td>5</td><td>3</td><td>0.037707672984847544011304782294</td><td> 5 ζ ( 3 ) ζ ( 5 ) − 147 24 ζ ( 8 ) − 5 2 ζ ( 6 , 2 ) {\displaystyle 5\zeta (3)\zeta (5)-{\tfrac {147}{24}}\zeta (8)-{\tfrac {5}{2}}\zeta (6,2)} </td><td>A258982</td></tr><tr><td>2</td><td>4</td><td>0.674523914033968140491560608257</td><td> 25 12 ζ ( 6 ) − ( ζ ( 3 ) ) 2 {\displaystyle {\tfrac {25}{12}}\zeta (6)-\left(\zeta (3)\right)^{2}} </td><td>A258989</td></tr><tr><td>3</td><td>4</td><td>0.207505014615732095907807605495</td><td> 10 ζ ( 2 ) ζ ( 5 ) + ζ ( 3 ) ζ ( 4 ) − 18 ζ ( 7 ) {\displaystyle 10\zeta (2)\zeta (5)+\zeta (3)\zeta (4)-18\zeta (7)} </td><td>A258990</td></tr><tr><td>4</td><td>4</td><td>0.083673113016495361614890436542</td><td> 1 2 ( ( ζ ( 4 ) ) 2 − ζ ( 8 ) ) {\displaystyle {\tfrac {1}{2}}\left(\left(\zeta (4)\right)^{2}-\zeta (8)\right)} </td><td>A258991</td></tr></tbody></table> <p>Note that if s + t = 2 p + 2 {\displaystyle s+t=2p+2} we have p / 3 {\displaystyle p/3} irreducibles, i.e. these MZVs cannot be written as function of ζ ( a ) {\displaystyle \zeta (a)} only.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h2 id="three-parameters-case">Three parameters case</h2> <p>In the particular case of only three parameters we have (with <i>a</i> > 1 and <i>n</i>, <i>j</i>, <i>i</i> integers): </p> ζ ( a , b , c ) = ∑ n > j > i ≥ 1 1 n a j b i c = ∑ n = 1 ∞ 1 ( n + 2 ) a ∑ j = 1 n 1 ( j + 1 ) b ∑ i = 1 j 1 ( i ) c = ∑ n = 1 ∞ 1 ( n + 2 ) a ∑ j = 1 n H j , c ( j + 1 ) b {\displaystyle \zeta (a,b,c)=\sum _{n>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}}}} <h2 id="euler-reflection-formula">Euler reflection formula</h2> <p>The above MZVs satisfy the Euler reflection formula: </p> ζ ( a , b ) + ζ ( b , a ) = ζ ( a ) ζ ( b ) − ζ ( a + b ) {\displaystyle \zeta (a,b)+\zeta (b,a)=\zeta (a)\zeta (b)-\zeta (a+b)} for a , b > 1 {\displaystyle a,b>1} <p>Using the shuffle relations, it is easy to <a href="/facts/Mathematical_proof/7ECDrU80">prove</a> that:<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> 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 ) + 2 ζ ( a + b + c ) − ζ ( a ) ζ ( b + c ) − ζ ( b ) ζ ( a + c ) − ζ ( c ) ζ ( a + b ) {\displaystyle \zeta (a,b,c)+\zeta (a,c,b)+\zeta (b,a,c)+\zeta (b,c,a)+\zeta (c,a,b)+\zeta (c,b,a)=\zeta (a)\zeta (b)\zeta (c)+2\zeta (a+b+c)-\zeta (a)\zeta (b+c)-\zeta (b)\zeta (a+c)-\zeta (c)\zeta (a+b)} for a , b , c > 1 {\displaystyle a,b,c>1} <p>This function can be seen as a generalization of the reflection formulas. </p> <h2 id="symmetric-sums-in-terms-of-the-zeta-function">Symmetric sums in terms of the zeta function</h2> <p>Let S ( i 1 , i 2 , ⋯ , i k ) = ∑ n 1 ≥ n 2 ≥ ⋯ n k ≥ 1 1 n 1 i 1 n 2 i 2 ⋯ n k i k {\displaystyle S(i_{1},i_{2},\cdots ,i_{k})=\sum _{n_{1}\geq n_{2}\geq \cdots n_{k}\geq 1}{\frac {1}{n_{1}^{i_{1}}n_{2}^{i_{2}}\cdots n_{k}^{i_{k}}}}} , and for a partition Π = { P 1 , P 2 , … , P l } {\displaystyle \Pi =\{P_{1},P_{2},\dots ,P_{l}\}} of the set { 1 , 2 , … , k } {\displaystyle \{1,2,\dots ,k\}} , let c ( Π ) = ( | P 1 | − 1 ) ! ( | P 2 | − 1 ) ! ⋯ ( | P l | − 1 ) ! {\displaystyle c(\Pi )=(\left|P_{1}\right|-1)!(\left|P_{2}\right|-1)!\cdots (\left|P_{l}\right|-1)!} . Also, given such a Π {\displaystyle \Pi } and a <i>k</i>-tuple i = { i 1 , . . . , i k } {\displaystyle i=\{i_{1},...,i_{k}\}} of exponents, define ∏ s = 1 l ζ ( ∑ j ∈ P s i j ) {\displaystyle \prod _{s=1}^{l}\zeta (\sum _{j\in P_{s}}i_{j})} . </p><p>The relations between the ζ {\displaystyle \zeta } and S {\displaystyle S} are: S ( i 1 , i 2 ) = ζ ( i 1 , i 2 ) + ζ ( i 1 + i 2 ) {\displaystyle S(i_{1},i_{2})=\zeta (i_{1},i_{2})+\zeta (i_{1}+i_{2})} and S ( i 1 , i 2 , i 3 ) = ζ ( i 1 , i 2 , i 3 ) + ζ ( i 1 + i 2 , i 3 ) + ζ ( i 1 , i 2 + i 3 ) + ζ ( i 1 + i 2 + i 3 ) . {\displaystyle S(i_{1},i_{2},i_{3})=\zeta (i_{1},i_{2},i_{3})+\zeta (i_{1}+i_{2},i_{3})+\zeta (i_{1},i_{2}+i_{3})+\zeta (i_{1}+i_{2}+i_{3}).} </p> <h3>Theorem 1 (Hoffman)</h3> <p>For any <a href="/facts/Real_number/R02gw5Pb">real</a> i 1 , ⋯ , i k > 1 , {\displaystyle i_{1},\cdots ,i_{k}>1,} , ∑ σ ∈ Σ k S ( i σ ( 1 ) , … , i σ ( k ) ) = ∑ partitions Π of { 1 , … , k } c ( Π ) ζ ( i , Π ) {\displaystyle \sum _{\sigma \in \Sigma _{k}}S(i_{\sigma (1)},\dots ,i_{\sigma (k)})=\sum _{{\text{partitions }}\Pi {\text{ of }}\{1,\dots ,k\}}c(\Pi )\zeta (i,\Pi )} . </p><p>Proof. Assume the i j {\displaystyle i_{j}} are all distinct. (There is no loss of generality, since we can take limits.) The left-hand side can be written as ∑ σ ∑ n 1 ≥ n 2 ≥ ⋯ ≥ n k ≥ 1 1 n i 1 σ ( 1 ) n i 2 σ ( 2 ) ⋯ n i k σ ( k ) {\displaystyle \sum _{\sigma }\sum _{n_{1}\geq n_{2}\geq \cdots \geq n_{k}\geq 1}{\frac {1}{{n^{i_{1}}}_{\sigma (1)}{n^{i_{2}}}_{\sigma (2)}\cdots {n^{i_{k}}}_{\sigma (k)}}}} . Now thinking on the symmetric </p><p>group Σ k {\displaystyle \Sigma _{k}} as acting on <i>k</i>-tuple n = ( 1 , ⋯ , k ) {\displaystyle n=(1,\cdots ,k)} of positive integers. A given <i>k</i>-tuple n = ( n 1 , ⋯ , n k ) {\displaystyle n=(n_{1},\cdots ,n_{k})} has an isotropy group </p><p> Σ k ( n ) {\displaystyle \Sigma _{k}(n)} and an associated partition Λ {\displaystyle \Lambda } of ( 1 , 2 , ⋯ , k ) {\displaystyle (1,2,\cdots ,k)} : Λ {\displaystyle \Lambda } is the set of <a href="/facts/Equivalence_class/1d2sh0TB">equivalence classes</a> of the <a href="/facts/Equivalence_relation/1MQ5rtkW">relation</a> given by i ∼ j {\displaystyle i\sim j} iff n i = n j {\displaystyle n_{i}=n_{j}} , and Σ k ( n ) = { σ ∈ Σ k : σ ( i ) ∼ ∀ i } {\displaystyle \Sigma _{k}(n)=\{\sigma \in \Sigma _{k}:\sigma (i)\sim \forall i\}} . Now the term 1 n i 1 σ ( 1 ) n i 2 σ ( 2 ) ⋯ n i k σ ( k ) {\displaystyle {\frac {1}{{n^{i_{1}}}_{\sigma (1)}{n^{i_{2}}}_{\sigma (2)}\cdots {n^{i_{k}}}_{\sigma (k)}}}} occurs on the left-hand side of ∑ σ ∈ Σ k S ( i σ ( 1 ) , … , i σ ( k ) ) = ∑ partitions Π of { 1 , … , k } c ( Π ) ζ ( i , Π ) {\displaystyle \sum _{\sigma \in \Sigma _{k}}S(i_{\sigma (1)},\dots ,i_{\sigma (k)})=\sum _{{\text{partitions }}\Pi {\text{ of }}\{1,\dots ,k\}}c(\Pi )\zeta (i,\Pi )} exactly | Σ k ( n ) | {\displaystyle \left|\Sigma _{k}(n)\right|} times. It occurs on the right-hand side in those terms corresponding to partitions Π {\displaystyle \Pi } that are refinements of Λ {\displaystyle \Lambda } : letting ⪰ {\displaystyle \succeq } denote refinement, 1 n i 1 σ ( 1 ) n i 2 σ ( 2 ) ⋯ n i k σ ( k ) {\displaystyle {\frac {1}{{n^{i_{1}}}_{\sigma (1)}{n^{i_{2}}}_{\sigma (2)}\cdots {n^{i_{k}}}_{\sigma (k)}}}} occurs ∑ Π ⪰ Λ ( Π ) {\displaystyle \sum _{\Pi \succeq \Lambda }(\Pi )} times. Thus, the conclusion will follow if | Σ k ( n ) | = ∑ Π ⪰ Λ c ( Π ) {\displaystyle \left|\Sigma _{k}(n)\right|=\sum _{\Pi \succeq \Lambda }c(\Pi )} for any <i>k</i>-tuple n = { n 1 , ⋯ , n k } {\displaystyle n=\{n_{1},\cdots ,n_{k}\}} and associated partition Λ {\displaystyle \Lambda } . To see this, note that c ( Π ) {\displaystyle c(\Pi )} counts the permutations having <a href="/facts/Cycle_type/hAnfVIrf">cycle type</a> specified by Π {\displaystyle \Pi } : since any elements of Σ k ( n ) {\displaystyle \Sigma _{k}(n)} has a unique cycle type specified by a partition that refines Λ {\displaystyle \Lambda } , the result follows.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>For k = 3 {\displaystyle k=3} , the theorem says ∑ σ ∈ Σ 3 S ( i σ ( 1 ) , i σ ( 2 ) , i σ ( 3 ) ) = ζ ( i 1 ) ζ ( i 2 ) ζ ( i 3 ) + ζ ( i 1 + i 2 ) ζ ( i 3 ) + ζ ( i 1 ) ζ ( i 2 + i 3 ) + ζ ( i 1 + i 3 ) ζ ( i 2 ) + 2 ζ ( i 1 + i 2 + i 3 ) {\displaystyle \sum _{\sigma \in \Sigma _{3}}S(i_{\sigma (1)},i_{\sigma (2)},i_{\sigma (3)})=\zeta (i_{1})\zeta (i_{2})\zeta (i_{3})+\zeta (i_{1}+i_{2})\zeta (i_{3})+\zeta (i_{1})\zeta (i_{2}+i_{3})+\zeta (i_{1}+i_{3})\zeta (i_{2})+2\zeta (i_{1}+i_{2}+i_{3})} for i 1 , i 2 , i 3 > 1 {\displaystyle i_{1},i_{2},i_{3}>1} . This is the main result of.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p><p>Having ζ ( i 1 , i 2 , ⋯ , i k ) = ∑ n 1 > n 2 > ⋯ n k ≥ 1 1 n 1 i 1 n 2 i 2 ⋯ n k i k {\displaystyle \zeta (i_{1},i_{2},\cdots ,i_{k})=\sum _{n_{1}>n_{2}>\cdots n_{k}\geq 1}{\frac {1}{n_{1}^{i_{1}}n_{2}^{i_{2}}\cdots n_{k}^{i_{k}}}}} . To state the analog of Theorem 1 for the ζ ′ s {\displaystyle \zeta 's} , we require one bit of notation. For a partition </p><p> Π = { P 1 , ⋯ , P l } {\displaystyle \Pi =\{P_{1},\cdots ,P_{l}\}} of { 1 , 2 ⋯ , k } {\displaystyle \{1,2\cdots ,k\}} , let c ~ ( Π ) = ( − 1 ) k − l c ( Π ) {\displaystyle {\tilde {c}}(\Pi )=(-1)^{k-l}c(\Pi )} . </p> <h3>Theorem 2 (Hoffman)</h3> <p>For any real i 1 , ⋯ , i k > 1 {\displaystyle i_{1},\cdots ,i_{k}>1} , ∑ σ ∈ Σ k ζ ( i σ ( 1 ) , … , i σ ( k ) ) = ∑ partitions Π of { 1 , … , k } c ~ ( Π ) ζ ( i , Π ) {\displaystyle \sum _{\sigma \in \Sigma _{k}}\zeta (i_{\sigma (1)},\dots ,i_{\sigma (k)})=\sum _{{\text{partitions }}\Pi {\text{ of }}\{1,\dots ,k\}}{\tilde {c}}(\Pi )\zeta (i,\Pi )} . </p><p>Proof. We follow the same line of argument as in the preceding proof. The left-hand side is now ∑ σ ∑ n 1 > n 2 > ⋯ > n k ≥ 1 1 n i 1 σ ( 1 ) n i 2 σ ( 2 ) ⋯ n i k σ ( k ) {\displaystyle \sum _{\sigma }\sum _{n_{1}>n_{2}>\cdots >n_{k}\geq 1}{\frac {1}{{n^{i_{1}}}_{\sigma (1)}{n^{i_{2}}}_{\sigma (2)}\cdots {n^{i_{k}}}_{\sigma (k)}}}} , and a term 1 n 1 i 1 n 2 i 2 ⋯ n k i k {\displaystyle {\frac {1}{n_{1}^{i_{1}}n_{2}^{i_{2}}\cdots n_{k}^{i_{k}}}}} occurs on the left-hand since once if all the n i {\displaystyle n_{i}} are distinct, and not at all otherwise. Thus, it suffices to show ∑ Π ⪰ Λ c ~ ( Π ) = { 1 , if | Λ | = k 0 , otherwise . {\displaystyle \sum _{\Pi \succeq \Lambda }{\tilde {c}}(\Pi )={\begin{cases}1,{\text{ if }}\left|\Lambda \right|=k\\0,{\text{ otherwise }}.\end{cases}}} (1) </p><p>To prove this, note first that the sign of c ~ ( Π ) {\displaystyle {\tilde {c}}(\Pi )} is positive if the permutations of cycle type Π {\displaystyle \Pi } are <a href="/facts/Parity_of_a_permutation/ludhCMJz">even</a>, and negative if they are <a href="/facts/Parity_of_a_permutation/ludhCMJz">odd</a>: thus, the left-hand side of (1) is the signed sum of the number of even and odd permutations in the isotropy group Σ k ( n ) {\displaystyle \Sigma _{k}(n)} . But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition Λ {\displaystyle \Lambda } is { { 1 } , { 2 } , ⋯ , { k } } {\displaystyle \{\{1\},\{2\},\cdots ,\{k\}\}} .<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> <h2 id="the-sum-and-duality-conjectures">The sum and duality conjectures</h2> <p>Source:<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p><p>We first state the sum conjecture, which is due to C. Moen.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p><p>Sum conjecture (Hoffman). For positive integers <i>k</i> and <i>n</i>, ∑ i 1 + ⋯ + i k = n , i 1 > 1 ζ ( i 1 , ⋯ , i k ) = ζ ( n ) {\displaystyle \sum _{i_{1}+\cdots +i_{k}=n,i_{1}>1}\zeta (i_{1},\cdots ,i_{k})=\zeta (n)} , where the sum is extended over <i>k</i>-tuples i 1 , ⋯ , i k {\displaystyle i_{1},\cdots ,i_{k}} of positive integers with i 1 > 1 {\displaystyle i_{1}>1} . </p><p>Three remarks concerning this <a href="/facts/Conjecture/oPWUb7SF">conjecture</a> are in order. First, it implies ∑ i 1 + ⋯ + i k = n , i 1 > 1 S ( i 1 , ⋯ , i k ) = ( n − 1 k − 1 ) ζ ( n ) {\displaystyle \sum _{i_{1}+\cdots +i_{k}=n,i_{1}>1}S(i_{1},\cdots ,i_{k})={n-1 \choose k-1}\zeta (n)} . Second, in the case k = 2 {\displaystyle k=2} it says that ζ ( n − 1 , 1 ) + ζ ( n − 2 , 2 ) + ⋯ + ζ ( 2 , n − 2 ) = ζ ( n ) {\displaystyle \zeta (n-1,1)+\zeta (n-2,2)+\cdots +\zeta (2,n-2)=\zeta (n)} , or using the relation between the ζ ′ s {\displaystyle \zeta 's} and S ′ s {\displaystyle S's} and Theorem 1, 2 S ( n − 1 , 1 ) = ( n + 1 ) ζ ( n ) − ∑ k = 2 n − 2 ζ ( k ) ζ ( n − k ) . {\displaystyle 2S(n-1,1)=(n+1)\zeta (n)-\sum _{k=2}^{n-2}\zeta (k)\zeta (n-k).} </p><p>This was proved by Euler<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> and has been rediscovered several times, in particular by Williams.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> Finally, C. Moen<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> has proved the same conjecture for <i>k</i>=3 by lengthy but elementary arguments. For the duality conjecture, we first define an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a> τ {\displaystyle \tau } on the set ℑ {\displaystyle \Im } of finite <a href="/facts/Sequence/gV2S4uqp">sequences</a> of positive integers whose first element is greater than 1. Let T {\displaystyle \mathrm {T} } be the set of <a href="/facts/Sequence/gV2S4uqp">strictly increasing</a> finite sequences of positive integers, and let Σ : ℑ → T {\displaystyle \Sigma :\Im \rightarrow \mathrm {T} } be the function that sends a sequence in ℑ {\displaystyle \Im } to its sequence of partial sums. If T n {\displaystyle \mathrm {T} _{n}} is the set of sequences in T {\displaystyle \mathrm {T} } whose last element is at most n {\displaystyle n} , we have two commuting involutions R n {\displaystyle R_{n}} and C n {\displaystyle C_{n}} on T n {\displaystyle \mathrm {T} _{n}} defined by R n ( a 1 , a 2 , … , a l ) = ( n + 1 − a l , n + 1 − a l − 1 , … , n + 1 − a 1 ) {\displaystyle R_{n}(a_{1},a_{2},\dots ,a_{l})=(n+1-a_{l},n+1-a_{l-1},\dots ,n+1-a_{1})} and C n ( a 1 , … , a l ) {\displaystyle C_{n}(a_{1},\dots ,a_{l})} = complement of { a 1 , … , a l } {\displaystyle \{a_{1},\dots ,a_{l}\}} in { 1 , 2 , … , n } {\displaystyle \{1,2,\dots ,n\}} arranged in increasing order. The our definition of τ {\displaystyle \tau } is τ ( I ) = Σ − 1 R n C n Σ ( I ) = Σ − 1 C n R n Σ ( I ) {\displaystyle \tau (I)=\Sigma ^{-1}R_{n}C_{n}\Sigma (I)=\Sigma ^{-1}C_{n}R_{n}\Sigma (I)} for I = ( i 1 , i 2 , … , i k ) ∈ ℑ {\displaystyle I=(i_{1},i_{2},\dots ,i_{k})\in \Im } with i 1 + ⋯ + i k = n {\displaystyle i_{1}+\cdots +i_{k}=n} . </p><p>For example, τ ( 3 , 4 , 1 ) = Σ − 1 C 8 R 8 ( 3 , 7 , 8 ) = Σ − 1 ( 3 , 4 , 5 , 7 , 8 ) = ( 3 , 1 , 1 , 2 , 1 ) . {\displaystyle \tau (3,4,1)=\Sigma ^{-1}C_{8}R_{8}(3,7,8)=\Sigma ^{-1}(3,4,5,7,8)=(3,1,1,2,1).} We shall say the sequences ( i 1 , … , i k ) {\displaystyle (i_{1},\dots ,i_{k})} and τ ( i 1 , … , i k ) {\displaystyle \tau (i_{1},\dots ,i_{k})} are dual to each other, and refer to a sequence fixed by τ {\displaystyle \tau } as self-dual.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p><p>Duality conjecture (Hoffman). If ( h 1 , … , h n − k ) {\displaystyle (h_{1},\dots ,h_{n-k})} is dual to ( i 1 , … , i k ) {\displaystyle (i_{1},\dots ,i_{k})} , then ζ ( h 1 , … , h n − k ) = ζ ( i 1 , … , i k ) {\displaystyle \zeta (h_{1},\dots ,h_{n-k})=\zeta (i_{1},\dots ,i_{k})} . </p><p>This sum conjecture is also known as <i>Sum Theorem</i>, and it may be expressed as follows: the Riemann zeta value of an integer <i>n</i> ≥ 2 is equal to the sum of all the valid (i.e. with <i>s</i>1 > 1) MZVs of the <a href="/facts/Partition_(number_theory)/LtyXWqwH">partitions</a> of length <i>k</i> and weight <i>n</i>, with 1 ≤ <i>k</i> ≤ <i>n</i> − 1. In formula:<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p> ∑ s 1 > 1 s 1 + ⋯ + s k = n ζ ( s 1 , … , s k ) = ζ ( n ) . {\displaystyle \sum _{\stackrel {s_{1}+\cdots +s_{k}=n}{s_{1}>1}}\zeta (s_{1},\ldots ,s_{k})=\zeta (n).} <p>For example, with length <i>k</i> = 2 and weight <i>n</i> = 7: </p> ζ ( 6 , 1 ) + ζ ( 5 , 2 ) + ζ ( 4 , 3 ) + ζ ( 3 , 4 ) + ζ ( 2 , 5 ) = ζ ( 7 ) . {\displaystyle \zeta (6,1)+\zeta (5,2)+\zeta (4,3)+\zeta (3,4)+\zeta (2,5)=\zeta (7).} <h2 id="euler-sum-with-all-possible-alternations-of-sign">Euler sum with all possible alternations of sign</h2> <p>The Euler sum with alternations of sign appears in studies of the non-alternating Euler sum.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p> <h3>Notation</h3> ∑ n = 1 ∞ H n ( b ) ( − 1 ) ( n + 1 ) ( n + 1 ) a = ζ ( a ¯ , b ) {\displaystyle \sum _{n=1}^{\infty }{\frac {H_{n}^{(b)}(-1)^{(n+1)}}{(n+1)^{a}}}=\zeta ({\bar {a}},b)} with H n ( b ) = + 1 + 1 2 b + 1 3 b + ⋯ {\displaystyle H_{n}^{(b)}=+1+{\frac {1}{2^{b}}}+{\frac {1}{3^{b}}}+\cdots } are the <a href="/facts/Harmonic_number/1b7v0hVn">generalized harmonic numbers</a>. ∑ n = 1 ∞ H ¯ n ( b ) ( n + 1 ) a = ζ ( a , b ¯ ) {\displaystyle \sum _{n=1}^{\infty }{\frac {{\bar {H}}_{n}^{(b)}}{(n+1)^{a}}}=\zeta (a,{\bar {b}})} with H ¯ n ( b ) = − 1 + 1 2 b − 1 3 b + ⋯ {\displaystyle {\bar {H}}_{n}^{(b)}=-1+{\frac {1}{2^{b}}}-{\frac {1}{3^{b}}}+\cdots } ∑ n = 1 ∞ H ¯ n ( b ) ( − 1 ) ( n + 1 ) ( n + 1 ) a = ζ ( a ¯ , b ¯ ) {\displaystyle \sum _{n=1}^{\infty }{\frac {{\bar {H}}_{n}^{(b)}(-1)^{(n+1)}}{(n+1)^{a}}}=\zeta ({\bar {a}},{\bar {b}})} ∑ n = 1 ∞ ( − 1 ) n ( n + 2 ) a ∑ n = 1 ∞ H ¯ n ( c ) ( − 1 ) ( n + 1 ) ( n + 1 ) b = ζ ( a ¯ , b ¯ , c ¯ ) {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{(n+2)^{a}}}\sum _{n=1}^{\infty }{\frac {{\bar {H}}_{n}^{(c)}(-1)^{(n+1)}}{(n+1)^{b}}}=\zeta ({\bar {a}},{\bar {b}},{\bar {c}})} with H ¯ n ( c ) = − 1 + 1 2 c − 1 3 c + ⋯ {\displaystyle {\bar {H}}_{n}^{(c)}=-1+{\frac {1}{2^{c}}}-{\frac {1}{3^{c}}}+\cdots } ∑ n = 1 ∞ ( − 1 ) n ( n + 2 ) a ∑ n = 1 ∞ H n ( c ) ( n + 1 ) b = ζ ( a ¯ , b , c ) {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{(n+2)^{a}}}\sum _{n=1}^{\infty }{\frac {H_{n}^{(c)}}{(n+1)^{b}}}=\zeta ({\bar {a}},b,c)} with H n ( c ) = + 1 + 1 2 c + 1 3 c + ⋯ {\displaystyle H_{n}^{(c)}=+1+{\frac {1}{2^{c}}}+{\frac {1}{3^{c}}}+\cdots } ∑ n = 1 ∞ 1 ( n + 2 ) a ∑ n = 1 ∞ H n ( c ) ( − 1 ) ( n + 1 ) ( n + 1 ) b = ζ ( a , b ¯ , c ) {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{(n+2)^{a}}}\sum _{n=1}^{\infty }{\frac {H_{n}^{(c)}(-1)^{(n+1)}}{(n+1)^{b}}}=\zeta (a,{\bar {b}},c)} ∑ n = 1 ∞ 1 ( n + 2 ) a ∑ n = 1 ∞ H ¯ n ( c ) ( n + 1 ) b = ζ ( a , b , c ¯ ) {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{(n+2)^{a}}}\sum _{n=1}^{\infty }{\frac {{\bar {H}}_{n}^{(c)}}{(n+1)^{b}}}=\zeta (a,b,{\bar {c}})} <p>As a variant of the <a href="/facts/Dirichlet_eta_function/32tUb6a7">Dirichlet eta function</a> we define </p> ϕ ( s ) = 1 − 2 ( s − 1 ) 2 ( s − 1 ) ζ ( s ) {\displaystyle \phi (s)={\frac {1-2^{(s-1)}}{2^{(s-1)}}}\zeta (s)} with s > 1 {\displaystyle s>1} ϕ ( 1 ) = − ln 2 {\displaystyle \phi (1)=-\ln 2} <h3>Reflection formula</h3> <p>The reflection formula ζ ( a , b ) + ζ ( b , a ) = ζ ( a ) ζ ( b ) − ζ ( a + b ) {\displaystyle \zeta (a,b)+\zeta (b,a)=\zeta (a)\zeta (b)-\zeta (a+b)} can be generalized as follows: </p> ζ ( a , b ¯ ) + ζ ( b ¯ , a ) = ζ ( a ) ϕ ( b ) − ϕ ( a + b ) {\displaystyle \zeta (a,{\bar {b}})+\zeta ({\bar {b}},a)=\zeta (a)\phi (b)-\phi (a+b)} ζ ( a ¯ , b ) + ζ ( b , a ¯ ) = ζ ( b ) ϕ ( a ) − ϕ ( a + b ) {\displaystyle \zeta ({\bar {a}},b)+\zeta (b,{\bar {a}})=\zeta (b)\phi (a)-\phi (a+b)} ζ ( a ¯ , b ¯ ) + ζ ( b ¯ , a ¯ ) = ϕ ( a ) ϕ ( b ) − ζ ( a + b ) {\displaystyle \zeta ({\bar {a}},{\bar {b}})+\zeta ({\bar {b}},{\bar {a}})=\phi (a)\phi (b)-\zeta (a+b)} <p>if a = b {\displaystyle a=b} we have ζ ( a ¯ , a ¯ ) = 1 2 [ ϕ 2 ( a ) − ζ ( 2 a ) ] {\displaystyle \zeta ({\bar {a}},{\bar {a}})={\tfrac {1}{2}}{\Big [}\phi ^{2}(a)-\zeta (2a){\Big ]}} </p> <h3>Other relations</h3> <p>Using the series definition it is easy to prove: </p> ζ ( a , b ) + ζ ( a , b ¯ ) + ζ ( a ¯ , b ) + ζ ( a ¯ , b ¯ ) = ζ ( a , b ) 2 ( a + b − 2 ) {\displaystyle \zeta (a,b)+\zeta (a,{\bar {b}})+\zeta ({\bar {a}},b)+\zeta ({\bar {a}},{\bar {b}})={\frac {\zeta (a,b)}{2^{(a+b-2)}}}} with a > 1 {\displaystyle a>1} ζ ( a , b , c ) + ζ ( a , b , c ¯ ) + ζ ( a , b ¯ , c ) + ζ ( a ¯ , b , c ) + ζ ( a , b ¯ , c ¯ ) + ζ ( a ¯ , b , c ¯ ) + ζ ( a ¯ , b ¯ , c ) + ζ ( a ¯ , b ¯ , c ¯ ) = ζ ( a , b , c ) 2 ( a + b + c − 3 ) {\displaystyle \zeta (a,b,c)+\zeta (a,b,{\bar {c}})+\zeta (a,{\bar {b}},c)+\zeta ({\bar {a}},b,c)+\zeta (a,{\bar {b}},{\bar {c}})+\zeta ({\bar {a}},b,{\bar {c}})+\zeta ({\bar {a}},{\bar {b}},c)+\zeta ({\bar {a}},{\bar {b}},{\bar {c}})={\frac {\zeta (a,b,c)}{2^{(a+b+c-3)}}}} with a > 1 {\displaystyle a>1} <p>A further useful relation is:<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> </p> ζ ( a , b ) + ζ ( a ¯ , b ¯ ) = ∑ s > 0 ( a + b − s − 1 ) ! [ Z a ( a + b − s , s ) ( a − s ) ! ( b − 1 ) ! + Z b ( a + b − s , s ) ( b − s ) ! ( a − 1 ) ! ] {\displaystyle \zeta (a,b)+\zeta ({\bar {a}},{\bar {b}})=\sum _{s>0}(a+b-s-1)!{\Big [}{\frac {Z_{a}(a+b-s,s)}{(a-s)!(b-1)!}}+{\frac {Z_{b}(a+b-s,s)}{(b-s)!(a-1)!}}{\Big ]}} <p>where Z a ( s , t ) = ζ ( s , t ) + ζ ( s ¯ , t ) − [ ζ ( s , t ) + ζ ( s + t ) ] 2 ( s − 1 ) {\displaystyle Z_{a}(s,t)=\zeta (s,t)+\zeta ({\bar {s}},t)-{\frac {{\Big [}\zeta (s,t)+\zeta (s+t){\Big ]}}{2^{(s-1)}}}} and Z b ( s , t ) = ζ ( s , t ) 2 ( s − 1 ) {\displaystyle Z_{b}(s,t)={\frac {\zeta (s,t)}{2^{(s-1)}}}} </p><p>Note that s {\displaystyle s} must be used for all value > 1 {\displaystyle >1} for which the argument of the factorials is ⩾ 0 {\displaystyle \geqslant 0} </p> <h2 id="other-results">Other results</h2> <p>For all positive integers a , b , … , k {\displaystyle a,b,\dots ,k} : </p> ∑ n = 2 ∞ ζ ( n , k ) = ζ ( k + 1 ) {\displaystyle \sum _{n=2}^{\infty }\zeta (n,k)=\zeta (k+1)} or more generally: ∑ n = 2 ∞ ζ ( n , a , b , … , k ) = ζ ( a + 1 , b , … , k ) {\displaystyle \sum _{n=2}^{\infty }\zeta (n,a,b,\dots ,k)=\zeta (a+1,b,\dots ,k)} ∑ n = 2 ∞ ζ ( n , k ¯ ) = − ϕ ( k + 1 ) {\displaystyle \sum _{n=2}^{\infty }\zeta (n,{\bar {k}})=-\phi (k+1)} ∑ n = 2 ∞ ζ ( n , a ¯ , b ) = ζ ( a + 1 ¯ , b ) {\displaystyle \sum _{n=2}^{\infty }\zeta (n,{\bar {a}},b)=\zeta ({\overline {a+1}},b)} ∑ n = 2 ∞ ζ ( n , a , b ¯ ) = ζ ( a + 1 , b ¯ ) {\displaystyle \sum _{n=2}^{\infty }\zeta (n,a,{\bar {b}})=\zeta (a+1,{\bar {b}})} ∑ n = 2 ∞ ζ ( n , a ¯ , b ¯ ) = ζ ( a + 1 ¯ , b ¯ ) {\displaystyle \sum _{n=2}^{\infty }\zeta (n,{\bar {a}},{\bar {b}})=\zeta ({\overline {a+1}},{\bar {b}})} lim k → ∞ ζ ( n , k ) = ζ ( n ) − 1 {\displaystyle \lim _{k\to \infty }\zeta (n,k)=\zeta (n)-1} 1 − ζ ( 2 ) + ζ ( 3 ) − ζ ( 4 ) + ⋯ = | 1 2 | {\displaystyle 1-\zeta (2)+\zeta (3)-\zeta (4)+\cdots =|{\frac {1}{2}}|} ζ ( a , a ) = 1 2 [ ( ζ ( a ) ) 2 − ζ ( 2 a ) ] {\displaystyle \zeta (a,a)={\tfrac {1}{2}}{\Big [}(\zeta (a))^{2}-\zeta (2a){\Big ]}} ζ ( a , a , a ) = 1 6 ( ζ ( a ) ) 3 + 1 3 ζ ( 3 a ) − 1 2 ζ ( a ) ζ ( 2 a ) {\displaystyle \zeta (a,a,a)={\tfrac {1}{6}}(\zeta (a))^{3}+{\tfrac {1}{3}}\zeta (3a)-{\tfrac {1}{2}}\zeta (a)\zeta (2a)} <h2 id="mordelltornheim-zeta-values">Mordell–Tornheim zeta values</h2> <p>The Mordell–Tornheim zeta function, introduced by Matsumoto (2003) who was motivated by the papers Mordell (1958) and Tornheim (1950), is defined by </p> ζ M T , r ( s 1 , … , s r ; s r + 1 ) = ∑ m 1 , … , m r > 0 1 m 1 s 1 ⋯ m r s r ( m 1 + ⋯ + m r ) s r + 1 {\displaystyle \zeta _{MT,r}(s_{1},\dots ,s_{r};s_{r+1})=\sum _{m_{1},\dots ,m_{r}>0}{\frac {1}{m_{1}^{s_{1}}\cdots m_{r}^{s_{r}}(m_{1}+\dots +m_{r})^{s_{r+1}}}}} <p>It is a special case of the <a href="/facts/Shintani_zeta_function/QCFWDa7p">Shintani zeta function</a>. </p> <ul><li>Tornheim, Leonard (1950). 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Number Theory</i>. 6 (3): 501–514. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.157.9158">10.1.1.157.9158</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2FS1793042110003058">10.1142/S1793042110003058</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2652893">2652893</a>.</li> <li>Basu, Ankur (2011). "On the evaluation of Tornheim sums and allied double sums". <i>Ramanujan J</i>. 26 (2): 193–207. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs11139-011-9302-5">10.1007/s11139-011-9302-5</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2853480">2853480</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120229489">120229489</a>.</li></ul> <h2 id="notes">Notes</h2> <h2 id="external-links">External links</h2> <ul><li>Borwein, Jonathan; Zudilin, Wadim. <a href="http://carma.newcastle.edu.au/MZVs/">"Lecture notes on the Multiple Zeta Function"</a>.</li> <li>Hoffman, Michael (2012). <a href="http://www.usna.edu/Users/math/meh/mult.html">"Multiple zeta values"</a>.</li> <li>Zhao, Jianqiang (2016). <i>Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values</i>. Series on Number Theory and its Applications. Vol. 12. World Scientific Publishing. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2F9634">10.1142/9634</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-4689-39-7.</li> <li>Burgos Gil, José Ignacio; Fresán, Javier. <a href="http://javier.fresan.perso.math.cnrs.fr/mzv.pdf">"Multiple zeta values: from numbers to motives"</a> (PDF).</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Zhao, Jianqiang (2010). "Standard relations of multiple polylogarithm values at roots of unity". Documenta Mathematica. 15: 1–34. arXiv:0707.1459. doi:10.4171/dm/291. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Zhao, Jianqiang (2016). Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values. Series on Number Theory and its Applications. Vol. 12. World Scientific Publishing. doi:10.1142/9634. ISBN 978-981-4689-39-7. <a href="978-981-4689-39-7" target="_blank">978-981-4689-39-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Hoffman, Mike. "Multiple Zeta Values". Mike Hoffman's Home Page. U.S. Naval Academy. Retrieved June 8, 2012. <a href="http://www.usna.edu/Users/math/meh/mult.html" target="_blank">http://www.usna.edu/Users/math/meh/mult.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Zhao, Jianqiang (2016). Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values. Series on Number Theory and its Applications. Vol. 12. World Scientific Publishing. doi:10.1142/9634. ISBN 978-981-4689-39-7. <a href="978-981-4689-39-7" target="_blank">978-981-4689-39-7</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Zhao, Jianqiang (2016). Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values. Series on Number Theory and its Applications. Vol. 12. World Scientific Publishing. doi:10.1142/9634. ISBN 978-981-4689-39-7. <a href="978-981-4689-39-7" target="_blank">978-981-4689-39-7</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Borwein, David; Borwein, Jonathan; Bradley, David (September 23, 2004). "Parametric Euler Sum Identities" (PDF). CARMA, AMSI Honours Course. The University of Newcastle. Retrieved June 3, 2012. <a href="http://carma.newcastle.edu.au/MZVs/parasums.pdf" target="_blank">http://carma.newcastle.edu.au/MZVs/parasums.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Borwein, David; Borwein, Jonathan; Bradley, David (September 23, 2004). "Parametric Euler Sum Identities" (PDF). CARMA, AMSI Honours Course. The University of Newcastle. Retrieved June 3, 2012. <a href="http://carma.newcastle.edu.au/MZVs/parasums.pdf" target="_blank">http://carma.newcastle.edu.au/MZVs/parasums.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Broadhurst, D. J. (1996). "On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory". arXiv:hep-th/9604128. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Broadhurst, D. J. (1996). "On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory". arXiv:hep-th/9604128. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>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. <a href="http://projecteuclid.org/euclid.pjm/1102636166" target="_blank">http://projecteuclid.org/euclid.pjm/1102636166</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Ramachandra Rao, R. Sita; M. V. Subbarao (1984). "Transformation formulae for multiple series". Pacific Journal of Mathematics. 113 (2): 417–479. doi:10.2140/pjm.1984.113.471. <a href="https://doi.org/10.2140%2Fpjm.1984.113.471" target="_blank">https://doi.org/10.2140%2Fpjm.1984.113.471</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>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. <a href="http://projecteuclid.org/euclid.pjm/1102636166" target="_blank">http://projecteuclid.org/euclid.pjm/1102636166</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>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. <a href="http://projecteuclid.org/euclid.pjm/1102636166" target="_blank">http://projecteuclid.org/euclid.pjm/1102636166</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Moen, C. "Sums of Simple Series". Preprint. <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Euler, L. (1775). "Meditationes circa singulare serierum genus". Novi Comm. Acad. Sci. Petropol. 15 (20): 140–186. <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Williams, G. T. (1958). "On the evaluation of some multiple series". Journal of the London Mathematical Society. 33 (3): 368–371. doi:10.1112/jlms/s1-33.3.368. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Moen, C. "Sums of Simple Series". Preprint. <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>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. <a href="http://projecteuclid.org/euclid.pjm/1102636166" target="_blank">http://projecteuclid.org/euclid.pjm/1102636166</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Hoffman, Mike. "Multiple Zeta Values". Mike Hoffman's Home Page. U.S. Naval Academy. Retrieved June 8, 2012. <a href="http://www.usna.edu/Users/math/meh/mult.html" target="_blank">http://www.usna.edu/Users/math/meh/mult.html</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Broadhurst, D. J. (1996). "On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory". arXiv:hep-th/9604128. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Broadhurst, D. J. (1996). "On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory". arXiv:hep-th/9604128. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> </ol>