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{
"Title": "Multiple zeta values of length 5",
"Definition": "The multiple zeta value of length five CITE{Bachmann} is $\\zeta(s_1,s_2,s_3,s_4,s_5)=\\sum_{n_1>n_2>n_3>n_4>n_5>0}n_1^{-s_1}n_2^{-s_2}n_3^{-s_3}n_4^{-s_4}n_5^{-s_5}$, where $s_1\\geq2$ and $s_2,s_3,s_4,s_5\\geq1$ are integers.",
"Parameters": {
"s1": {
"display": "$s_1$",
"type": "Z",
"constraints": "$s_1\\geq2$"
},
"s2": {
"display": "$s_2$",
"type": "Z",
"constraints": "$s_2\\geq1$"
},
"s3": {
"display": "$s_3$",
"type": "Z",
"constraints": "$s_3\\geq1$"
},
"s4": {
"display": "$s_4$",
"type": "Z",
"constraints": "$s_4\\geq1$"
},
"s5": {
"display": "$s_5$",
"type": "Z",
"constraints": "$s_5\\geq1$"
}
},
"Tags": [
"period",
"special values",
"L-function",
"transcendental"
],
"Links": {
"Bachmann": {
"title": "Henrik Bachmann: Multiple Zeta Values, lecture notes, version 15 (2025)",
"url": "https://www.henrikbachmann.com/uploads/7/7/6/3/77634444/mzv_2025_lecture_notes_v15.pdf"
},
"Sage": {
"title": "SageMath reference: algebra of motivic multiple zeta values",
"url": "https://doc.sagemath.org/html/en/reference/modfrm/sage/modular/multiple_zeta.html"
}
},
"Data properties": {
"type": "R",
"complete": "no",
"complete-note": "it holds all 462 admissible length-five indices of weight $s_1+s_2+s_3+s_4+s_5\\leq12$; the weight grading represents each weight through 12 in full, including every term in its weight-sum identity",
"rigour": "heuristic (agreement-checked)",
"rigour details": "The entries are written to 100 significant digits. Sage's multiple-zeta cache CITE{Sage}, using PARI/GP `zetamultall`, is rebuilt separately at 150 and 200 decimal working digits, each with 16 additional guard bits. Every pair agrees to at least 155 significant digits. This is a heuristic accuracy check, not an error bound supplied by PARI.\n\nThe independent checker evaluates the iterated-integral representation CITE{Bachmann}, Section 2.1, by splitting at $1/2$ and summing convergent multiple-polylogarithm series in 550-bit ball arithmetic. It uses 650 terms with an explicit geometric bound for every omitted tail. Each resulting enclosure is finite and positive, has relative radius less than $10^{-125}$, and lies strictly inside the interval represented by the corresponding 100-digit decimal. Seven known-value controls pass, including ordinary zeta values and $\\zeta(2,1)=\\zeta(3)$. Deliberately shifting each stored decimal by ten units in its last place fails containment.\n\nThe weight-sum identity is checked for every weight from 6 through 12, and both displayed special values are checked against Arb's ordinary zeta and pi. All 462 values and these identities are checked again on the entries read back from the database. The attached generator, independent checker and numerical evidence reproduce these checks."
},
"Display properties": {
"number-header": "$\\zeta(s_1,s_2,s_3,s_4,s_5)$"
},
"Keywords": [
"multiple zeta values",
"depth five",
"MZV",
"multiple harmonic sums"
],
"Comments": {
"weight": "The weight is $w=s_1+s_2+s_3+s_4+s_5$, and the length is also called the depth.",
"sage-convention": "Sage uses the opposite argument order: the value indexed by $(s_1,s_2,s_3,s_4,s_5)$ is represented by `Multizeta(s5,s4,s3,s2,s1)` CITE{Sage}."
},
"Formulas": {
"sum-formula": "For every integer $w\\geq6$, $\\displaystyle\\sum_{\\substack{s_1+s_2+s_3+s_4+s_5=w\\\\s_1\\geq2,\\ s_2,s_3,s_4,s_5\\geq1}}\\zeta(s_1,s_2,s_3,s_4,s_5)=\\zeta(w)$, where $\\zeta(w)$ is a value of the HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}[Riemann zeta function] CITE{Bachmann}, Theorem 4.3.",
"special-values": "$\\zeta(2,1,1,1,1)=\\zeta(6)$ and $\\zeta(2,2,2,2,2)=\\pi^{10}/11!$ CITE{Bachmann}, Proposition 2.5 and Proposition 1.8."
},
"Programs": {
"sage": {
"language": "Sage",
"code": "import numberdb.sage as numberdb\nfrom sage.modular.multiple_zeta import Multizeta\n\ndef mzv5(s1, s2, s3, s4, s5, digits=100):\n return Multizeta(s5, s4, s3, s2, s1).n(digits=digits)\n\nmzv5(2, 1, 1, 1, 1)"
}
},
"Similar tables": [
{
"table": "HREF{Multiple_zeta_values_of_length_2}[Multiple zeta values of length 2]",
"relation": "The same summation convention at depth two."
},
{
"table": "HREF{Multiple_zeta_values_of_length_3}[Multiple zeta values of length 3]",
"relation": "The same summation convention at depth three."
},
{
"table": "HREF{Multiple_zeta_values_of_length_4}[Multiple zeta values of length 4]",
"relation": "The same summation convention at depth four."
},
{
"table": "HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}[Values of the Riemann zeta function at rational numbers]",
"relation": "The depth-one values appearing in the weight-sum identities."
}
]
}