LaTeX to CAS translator
This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.
The demo-application converts LaTeX functions which directly translate to CAS counterparts.
Functions without explicit CAS support are available for translation via a DRMF package (under development).
The following LaTeX input ...
{\displaystyle \zeta(s)}
... is translated to the CAS output ...
Semantic latex: \Riemannzeta@{s}
Confidence: 0.96665916862997
Mathematica
Translation: Zeta[s]
Information
Sub Equations
- Zeta[s]
Free variables
- s
Symbol info
- Riemann zeta function; Example: \Riemannzeta@{s}
Will be translated to: Zeta[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/25.2#E1 Mathematica: https://reference.wolfram.com/language/ref/Zeta.html
Tests
Symbolic
Numeric
SymPy
Translation:
Information
Symbol info
- (LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \Riemannzeta [\Riemannzeta]
Tests
Symbolic
Numeric
Maple
Translation: Zeta(s)
Information
Sub Equations
- Zeta(s)
Free variables
- s
Symbol info
- Riemann zeta function; Example: \Riemannzeta@{s}
Will be translated to: Zeta($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/25.2#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Zeta
Tests
Symbolic
Numeric
Dependency Graph Information
Includes
Is part of
Description
- value
- function
- Riemann zeta function
- analytic continuation
- sum of the Dirichlet series
- general representation
- Euler -- Riemann zeta function
- real part
- series
- functional equation
- zeta function
- gamma function
- infinite series
- harmonic series
- series development
- derivative
- Euler
- fact
- one
- prime number
- Euler -- Mascheroni
- complex number
- convergent series
- Pochhammer symbol
- integer
- Cauchy principal value
- equality
- modulus
- xi-function
- irrationality
- integral
- arbitrary precision
- Boltzmann law in physics
- broken line
- critical strip with computational complexity
- differentation
- increment of an arbitrary continuous branch
- integral formula
- n
- non-trivial zero
- notation of umbral calculus
- odd order of the function
- power
- prime-counting function
- result
- simpler infinite product expansion
- substitution
- trivial zero
- Mellin
- number
- Riemann
- Bernoulli number
- absolute convergence
- accuracy
- argument
- argument of the Riemann zeta function
- basis of Weierstrass 's factorization theorem
- Bernoulli polynomial of the second kind
- box with periodic boundary condition
- Cauchy number of the second kind
- cf. Euler summation
- connection between the zeta function
- convenience
- definition
- density
- detailed survey on the history
- Dirichlet series over the Möbius function
- e.g. blagouchine
- equation
- Euler product
- finite result to the series
- first series
- Gregory coefficient
- Gregory coefficient of higher order
- Hadamard
- Hankel contour
- harmonic number
- Hasse
- i.e.
- imaginary part of a complex number
- incomplete poly-Bernoulli number
- infinite product expansion
- infinite product on the right hand side
- inversion
- larger half-plane
- Laurent series
- left hand side
- many real zero
- negative integer
- next higher integer of the unique solution
- nonpositive integer
- odd term
- Other sum
- perfect power
- polygamma function
- positive integer
- pretext
- process
- product
- proof of Euler 's identity
- quantum computer
- region
- relation
- representation in term
- Riemann zeta function by the formula
- same publication
- side of the Euler product formula
- Stirling number of the first kind
- stricter requirement
- such expression
- sum
- sum of geometric series
- version of the above sum
- solution to the Basel problem
- Roger Apéry
- special case
- case
- algorithm
- Apéry
- better result
- branch of the Lambert
- cf. Abel -- Plana formula
- convention
- distance between the zero
- February
- finite value to the divergent series
- Godfrey Harold
- Helmut Hasse
- integral relation
- letter
- limit
- limit value
- map
- point
- real axis
- Sandeep Tyagi
- short interval of the critical line
- small neighborhood of point
- symmetric version of the functional equation
- term of Jacobi 's theta function
- total number of zero
- Via
- zero of the Riemann zeta function
- above series termwise
- analogy with the Euler product
- contour
- convergent series for the zeta function
- critical temperature for a Bose -- Einstein condensate
- entire complex plane
- explicit error bound
- following expression for the zeta function
- interval of large positive real number
- kinetic boundary layer problem of linear kinetic equation
- Konrad Knopp
- numerical calculation
- Ramanujan summation
- spin wave physics in magnetic system
- summand
- zero of the sine function
- certain context
- effective form of Vinogradov 's mean-value theorem
- Littlewood
- numerical evaluation of the zeta-function
- Planck 's law
- string theory
- total number of real zero
- critical strip
- high precision
- Stefan
- year
- equivalent relationship
- geometric series
- interval
Complete translation information:
{
"id" : "FORMULA_82a19a183ea387e48e91dbd98d8c989b",
"formula" : "\\zeta(s)",
"semanticFormula" : "\\Riemannzeta@{s}",
"confidence" : 0.9666591686299704,
"translations" : {
"Mathematica" : {
"translation" : "Zeta[s]",
"translationInformation" : {
"subEquations" : [ "Zeta[s]" ],
"freeVariables" : [ "s" ],
"tokenTranslations" : {
"\\Riemannzeta" : "Riemann zeta function; Example: \\Riemannzeta@{s}\nWill be translated to: Zeta[$0]\nRelevant links to definitions:\nDLMF: http://dlmf.nist.gov/25.2#E1\nMathematica: https://reference.wolfram.com/language/ref/Zeta.html"
}
},
"numericResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"wasAborted" : false,
"crashed" : false,
"testCalculationsGroups" : [ ]
},
"symbolicResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"crashed" : false,
"testCalculationsGroup" : [ ]
}
},
"SymPy" : {
"translation" : "",
"translationInformation" : {
"tokenTranslations" : {
"Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\Riemannzeta [\\Riemannzeta]"
}
},
"numericResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"wasAborted" : false,
"crashed" : false,
"testCalculationsGroups" : [ ]
},
"symbolicResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"crashed" : false,
"testCalculationsGroup" : [ ]
}
},
"Maple" : {
"translation" : "Zeta(s)",
"translationInformation" : {
"subEquations" : [ "Zeta(s)" ],
"freeVariables" : [ "s" ],
"tokenTranslations" : {
"\\Riemannzeta" : "Riemann zeta function; Example: \\Riemannzeta@{s}\nWill be translated to: Zeta($0)\nRelevant links to definitions:\nDLMF: http://dlmf.nist.gov/25.2#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Zeta"
}
},
"numericResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"wasAborted" : false,
"crashed" : false,
"testCalculationsGroups" : [ ]
},
"symbolicResults" : {
"overallResult" : "SKIPPED",
"numberOfTests" : 0,
"numberOfFailedTests" : 0,
"numberOfSuccessfulTests" : 0,
"numberOfSkippedTests" : 0,
"numberOfErrorTests" : 0,
"crashed" : false,
"testCalculationsGroup" : [ ]
}
}
},
"positions" : [ {
"section" : 0,
"sentence" : 0,
"word" : 15
}, {
"section" : 0,
"sentence" : 1,
"word" : 4
}, {
"section" : 1,
"sentence" : 0,
"word" : 4
}, {
"section" : 1,
"sentence" : 3,
"word" : 8
}, {
"section" : 3,
"sentence" : 0,
"word" : 32
}, {
"section" : 4,
"sentence" : 3,
"word" : 14
}, {
"section" : 4,
"sentence" : 3,
"word" : 34
}, {
"section" : 4,
"sentence" : 13,
"word" : 9
}, {
"section" : 12,
"sentence" : 2,
"word" : 17
}, {
"section" : 26,
"sentence" : 1,
"word" : 3
}, {
"section" : 26,
"sentence" : 2,
"word" : 14
}, {
"section" : 26,
"sentence" : 3,
"word" : 1
}, {
"section" : 26,
"sentence" : 3,
"word" : 20
} ],
"includes" : [ "s", "\\zeta", "\\zeta(r)", "\\zeta\\left(s\\right)" ],
"isPartOf" : [ "\\zeta(s) =\\sum_{n=1}^\\infty\\frac{1}{n^s}", "\\zeta(2)", "[[Ap{{e}}ry's constant|\\zeta(3)]]", "\\zeta(s) = \\frac{1}{\\Gamma(s)} \\int_0^\\infty \\frac{x ^ {s-1}}{e ^ x - 1} \\, \\mathrm{d}x \\quad \\text{for} \\quad \\operatorname{Re}(s) \\equiv \\sigma > 1", "\\zeta(s) = \\sum_{n=1}^\\infty n^{-s} = \\frac{1}{1^s} + \\frac{1}{2^s} + \\frac{1}{3^s} + \\cdots \\quad \\text{for} \\quad \\sigma \\equiv \\operatorname{Re}(s) > 1", "\\lim_{s \\to 1} (s - 1)\\zeta(s) = 1", "\\zeta(2n) = \\frac{(-1)^{n+1}B_{2n}(2\\pi)^{2n}}{2(2n)!}", "\\zeta(-n)= (-1)^n\\frac{B_{n+1}}{n+1}", "\\zeta(-1) = -\\tfrac{1}{12}", "\\zeta(0) = -\\tfrac{1}{2}", "\\zeta\\bigl(\\tfrac12\\bigr) \\approx -1.46035 45088 09586 81289", "\\zeta(1) = 1 + \\tfrac{1}{2} + \\tfrac{1}{3} + \\cdots = \\infty", "\\lim_{\\varepsilon \\to 0} \\frac{\\zeta(1+\\varepsilon)+\\zeta(1-\\varepsilon)}{2}", "\\zeta\\bigl(\\tfrac32\\bigr) \\approx 2.61237 53486 85488 34335", "\\zeta(2) = 1 + \\frac{1}{2^2} + \\frac{1}{3^2} + \\cdots = \\frac{\\pi^2}{6} \\approx 1.64493 40668 48226 43647;", "\\zeta(3) = 1 + \\frac{1}{2^3} + \\frac{1}{3^3} + \\cdots \\approx 1.20205 69031 59594 28540", "\\zeta(4) = 1 + \\frac{1}{2^4} + \\frac{1}{3^4} + \\cdots = \\frac{\\pi^4}{90} \\approx 1.08232 32337 11138 19152", "\\zeta (\\infty) = 1", "\\zeta(s) = 2^s\\pi^{s-1}\\ \\sin\\left(\\frac{\\pi s}{2}\\right)\\ \\Gamma(1-s)\\ \\zeta(1-s)", "\\frac{\\Gamma\\left(\\frac s 2\\right)\\zeta(s)}{\\pi^{s/2}} = \\sum_{n=1}^\\infty \\int\\limits_0^\\infty x^{{s\\over 2}-1} e^{-n^2 \\pi x}\\, dx = \\int_0^\\infty x^{{s\\over 2}-1} \\sum_{n=1}^\\infty e^{-n^2 \\pi x}\\, dx", "\\zeta(s) = {\\pi^{s\\over2}\\over\\Gamma({s \\over 2})} \\int\\limits_0^\\infty x^{{1\\over2}{s} - 1}\\psi(x)\\, dx", "\\pi^{-{s \\over 2}} \\Gamma \\left ( {s \\over 2} \\right ) \\zeta (s) = \\int_0^1 x^{{s\\over 2}-1} \\psi(x) \\, dx + \\int_1^\\infty x^{{s\\over 2}-1} \\psi(x) \\, dx", "\\pi^{-{s \\over 2}} \\Gamma \\left ( {s \\over 2} \\right ) \\zeta (s)={1 \\over {s({s-1})}} + \\int\\limits_1^\\infty \\left ({x^{-{{s}\\over 2}-{1\\over 2}} + x^{{{s}\\over 2}-1}} \\right ) \\psi(x) \\, dx", "\\pi^{-{s \\over 2}} \\Gamma \\left ( {s \\over 2} \\right ) \\zeta (s) = \\pi^{-{1 \\over 2} + {s \\over 2}} \\Gamma \\left ( {1 \\over 2} - {s \\over 2} \\right ) \\zeta (1-s)", "\\eta(s)= \\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n^s} = \\left(1-{2^{1-s}}\\right)\\zeta(s)", "\\zeta(s)=\\frac{1}{1-{2^{1-s}}}\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n^s}", "\\xi(s) = \\frac{1}{2}\\pi^{-\\frac{s}{2}}s(s-1)\\Gamma\\left(\\frac{s}{2}\\right)\\zeta(s),", "\\zeta (\\frac{1}{2}+ it)", "\\zeta (\\sigma + it) \\not = 0", "\\zeta(s)=\\overline{\\zeta(\\overline{s})}", "\\frac{1}{\\zeta(s)} = \\sum_{n=1}^\\infty \\frac{\\mu(n)}{n^s}", "F(T;H) = \\max_{|t-T|\\le H}\\left|\\zeta\\left(\\tfrac{1}{2}+it\\right)\\right|,\\qquad G(s_{0};\\Delta) = \\max_{|s-s_{0}|\\le\\Delta}|\\zeta(s)|", "S(t) = \\frac{1}{\\pi}\\arg{\\zeta\\left(\\tfrac12+it\\right)}", "arg \\zeta(\\frac{1}{2}+ it)", "arg \\zeta(s)", "\\zeta(s)=\\frac{1}{s-1}\\sum_{n=1}^\\infty \\left(\\frac{n}{(n+1)^s}-\\frac{n-s}{n^s}\\right)", "\\zeta(s) =\\frac{1}{s-1}\\sum_{n=1}^\\infty\\frac{n(n+1)}{2}\\left(\\frac{2n+3+s}{(n+1)^{s+2}}-\\frac{2n-1-s}{n^{s+2}}\\right)", "\\Gamma(s)\\zeta(s) =\\int_0^\\infty\\frac{x^{s-1}}{e^x-1} \\,\\mathrm{d}x", "2\\sin(\\pi s)\\Gamma(s)\\zeta(s) =i\\oint_H \\frac{(-x)^{s-1}}{e^x-1}\\,\\mathrm{d}x", "\\zeta(n) {\\Gamma(n)} =\\int_{0}^{\\infty} \\frac{x ^ {n-1}}{e ^ x - 1} \\mathrm{d}x", "\\zeta(r)", "\\int_{0}^{\\infty} \\frac{x ^ {n}e^x}{(e ^ x - 1)^2} \\mathrm{d}x = {n!}\\zeta(n)", "\\int_{0}^{\\infty} \\frac{x ^ {n}e^x}{(e ^ x - 1)^4} \\mathrm{d}x = \\frac{n!}{ 6} \\bigl( \\zeta^{n-2} -3\\zeta^{n-1} +2\\zeta^n \\bigr)= n!\\frac{ \\zeta(n-2) -3\\zeta(n-1) +2\\zeta(n) }{ 6}", "\\ln \\zeta(s) = s \\int_0^\\infty \\frac{\\pi(x)}{x(x^s-1)}\\,\\mathrm{d}x", "\\ln \\zeta(s) = s\\int_0^\\infty J(x)x^{-s-1}\\,\\mathrm{d}x", "2\\pi^{-\\frac{s}{2}}\\Gamma\\left(\\frac{s}{2}\\right)\\zeta(s) = \\int_0^\\infty \\bigl(\\theta(it)-1\\bigr)t^{\\frac{s}{2}-1}\\,\\mathrm{d}t", "\\pi^{-\\frac{s}{2}}\\Gamma\\left(\\frac{s}{2}\\right)\\zeta(s) = \\frac{1}{s-1}-\\frac{1}{s} +\\frac{1}{2} \\int_0^1 \\left(\\theta(it)-t^{-\\frac12}\\right)t^{\\frac{s}{2}-1}\\,\\mathrm{d}t + \\frac{1}{2}\\int_1^\\infty \\bigl(\\theta(it)-1\\bigr)t^{\\frac{s}{2}-1}\\,\\mathrm{d}t", "\\zeta(s)=\\frac{1}{s-1}+\\sum_{n=0}^\\infty \\frac{\\gamma_n}{n!}(1-s)^n", "\\zeta(s) = \\frac{1}{s-1} + \\frac{1}{2} + 2\\int_0^{\\infty} \\frac{\\sin(s\\arctan t)}{\\left(1+t^2\\right)^{s/2}\\left(e^{2\\pi t}-1\\right)}\\,\\mathrm{d}t", "\\zeta(s) = \\frac{s}{s-1} - \\sum_{n=1}^\\infty \\bigl(\\zeta(s+n)-1\\bigr)\\frac{s(s+1)\\cdots(s+n-1)}{(n+1)!}", "\\zeta(s) = \\frac{e^{\\left(\\log(2\\pi)-1-\\frac{\\gamma}{2}\\right)s}}{2(s-1)\\Gamma\\left(1+\\frac{s}{2}\\right)} \\prod_\\rho \\left(1 - \\frac{s}{\\rho} \\right) e^\\frac{s}{\\rho}", "\\zeta(s) = \\pi^\\frac{s}{2} \\frac{\\prod_\\rho \\left(1 - \\frac{s}{\\rho} \\right)}{2(s-1)\\Gamma\\left(1+\\frac{s}{2}\\right)}", "\\zeta(s)=\\frac{1}{1-2^{1-s}} \\sum_{n=0}^\\infty \\frac {1}{2^{n+1}} \\sum_{k=0}^n \\binom{n}{k} \\frac{(-1)^k}{(k+1)^{s}}", "\\zeta(s)=\\frac 1{s-1}\\sum_{n=0}^\\infty \\frac 1{n+1}\\sum_{k=0}^n\\binom {n}{k}\\frac{(-1)^k}{(k+1)^{s-1}}", "\\begin{align} \\zeta(s) & =\\frac{1}{s-1}\\sum_{n=0}^\\infty H_{n+1}\\sum_{k=0}^n (-1)^k \\binom{n}{k}(k+2)^{1-s} \\\\[6pt]\\zeta(s) & =\\frac{1}{s-1}\\left\\{-1 + \\sum_{n=0}^\\infty H_{n+2}\\sum_{k=0}^n (-1)^k \\binom{n}{k}(k+2)^{-s}\\right\\} \\\\[6pt] \\zeta(s) & =\\frac{k!}{(s-k)_k}\\sum_{n=0}^\\infty \\frac{1}{(n+k)!}\\left[{n+k \\atop n}\\right]\\sum_{\\ell=0}^{n+k-1}\\!(-1)^\\ell \\binom{n+k-1}{\\ell} (\\ell+1)^{k-s},\\quad k=1, 2, 3,\\ldots \\\\[6pt] \\zeta(s) & =\\frac{1}{s-1} + \\sum_{n=0}^\\infty |G_{n+1}| \\sum_{k=0}^n(-1)^k \\binom{n}{k}(k+1)^{-s} \\\\[6pt]\\zeta(s) & =\\frac{1}{s-1}+1-\\sum_{n=0}^\\infty C_{n+1}\\sum_{k=0}^n (-1)^k \\binom{n}{k}(k+2)^{-s} \\\\[6pt] \\zeta(s) & =\\frac{2(s-2)}{s-1}\\zeta(s-1) + 2\\sum_{n=0}^\\infty (-1)^n G_{n+2}\\sum_{k=0}^n (-1)^k \\binom{n}{k} (k+1)^{-s} \\\\[6pt] \\zeta(s) & =-\\sum_{l=1}^{k-1} \\frac{(k-l+1)_l}{(s-l)_l} \\zeta(s-l) + \\frac{k}{s-k}+k \\sum_{n=0}^\\infty (-1)^n G_{n+1}^{(k)}\\sum_{k=0}^{n}(-1)^k \\binom{n}{k} (k+1)^{-s} \\\\[6pt]\\zeta(s) & = \\frac{(a+1)^{1-s} }{s-1} + \\sum_{n=0}^\\infty (-1)^n \\psi_{n+1}(a)\\sum_{k=0}^n (-1)^k \\binom{n}{k} (k+1)^{-s} ,\\quad \\Re(a)>-1 \\\\[6pt]\\zeta(s) & =1 + \\frac{(a+2)^{1-s}}{s-1} + \\sum_{n=0}^\\infty (-1)^n \\psi_{n+1}(a)\\sum_{k=0}^{n} (-1)^k \\binom{n}{k} (k+2)^{-s} ,\\quad \\Re(a)>-1 \\\\[6pt] \\zeta(s) & = \\frac{1}{a+\\tfrac{1}{2}}\\left\\{-\\frac{\\zeta(s-1,1+a)}{s-1} + \\zeta(s-1) + \\sum_{n=0}^\\infty (-1)^n \\psi_{n+2}(a) \\sum_{k=0}^{n} (-1)^k \\binom{n}{k} (k+1)^{-s}\\right\\} ,\\quad \\Re(a)>-1\\end{align}", "\\zeta(k)=\\frac{2^k}{2^k-1}+\\sum_{r=2}^\\infty\\frac{(p_{r-1}\\#)^k}{J_k(p_r\\#)}\\qquad k=2,3,\\ldots", "\\zeta(s)=\\sum_{n=0}^\\infty B_{n,\\ge2}^{(s)}\\frac{(W_k(-1))^n}{n!}", "\\begin{align} \\int_0^1 g (x) x^{s - 1} \\, dx & = \\sum_{n = 1}^\\infty \\int_{\\frac{1}{n + 1}}^{\\frac{1}{n}} (x (n + 1) - 1) x^{s - 1} \\, d x\\\\[6pt] & = \\sum_{n = 1}^\\infty \\frac{n^{- s} (s - 1) + (n + 1)^{- s - 1} (n^2 + 2 n + 1) + n^{- s - 1} s - n^{1 - s}}{(s + 1) s (n + 1)}\\\\[6pt] & = \\frac{\\zeta (s + 1)}{s + 1} - \\frac{1}{s (s + 1)} \\end{align}", "\\begin{align}\\zeta(s) = 1 + \\sum_{n=1}^{\\infty} \\frac{1}{a_n^s -1},\\end{align}", "\\begin{align}\\zeta\\left(s\\right) & = \\sum_{n=1}^{\\infty}n^{-s}\\sum_{w=0}^{v-1}\\frac{\\left(\\frac{n}{N}\\right)^{w}}{w!}e^{-\\frac{n}{N}}-\\frac{\\Gamma\\left(1-s+v\\right)}{\\left(1-s\\right)\\Gamma\\left(v\\right)}N^{1-s}+\\sum_{\\mu=\\pm1}E_{\\mu}\\left(s\\right)\\\\E_{\\mu}\\left(s\\right) & = \\left(2\\pi\\right)^{s-1}\\Gamma\\left(1-s\\right)e^{i\\mu\\frac{\\pi}{2}\\left(1-s\\right)}\\sum_{m=1}^{\\infty}\\left[m^{s-1}-\\sum_{w=0}^{v-1}\\binom{s-1}{w}\\left(m+\\frac{i\\mu}{2\\pi N}\\right)^{s-1-w}\\left(\\frac{-i\\mu}{2\\pi N}\\right)^{w}\\right]\\end{align}", "\\zeta\\left(s\\right)", "\\sum_{n=2}^\\infty\\bigl(\\zeta(n)-1\\bigr) = 1", "\\sum_{n=1}^\\infty\\bigl(\\zeta(2n)-1\\bigr)=\\frac{3}{4}", "\\sum_{n=1}^\\infty\\bigl(\\zeta(2n+1)-1\\bigr)=\\frac{1}{4}", "\\sum_{n=1}^\\infty(\\zeta(2n)-1)\\,t^{2n} = \\frac{t^2}{t^2-1} + \\frac{1}{2} \\left(1- \\pi t\\cot(t\\pi)\\right)", "\\sum_{n=1}^\\infty(\\zeta(2n+1)-1)\\,t^{2n} = \\frac{t^2}{t^2-1} + \\frac{1}{2}\\left(\\psi^0(t)+\\psi^0(-t) \\right) - \\gamma", "\\sum_{n=1}^\\infty \\frac{\\zeta(2n)-1}{n}\\,t^{2n} = \\log\\left(\\dfrac{1-t^2}{\\operatorname{sinc}(\\pi\\,t)}\\right)", "\\sum_{n=2}^\\infty\\frac{\\zeta(n)-1}{n} = 1-\\gamma", "\\sum_{n=2}^\\infty\\frac{\\zeta(n)-1}{n} \\left(\\left(\\tfrac{3}{2}\\right)^{n-1}-1\\right) = \\frac{1}{3} \\ln \\pi", "\\sum_{n=1}^\\infty\\bigl(\\zeta(4n)-1\\bigr) = \\frac78-\\frac{\\pi}{4}\\left(\\frac{e^{2\\pi}+1}{e^{2\\pi}-1}\\right)", "\\sum_{n=2}^\\infty\\frac{\\zeta(n)-1}{n}\\operatorname{Im}\\bigl((1+i)^n-(1+i^n)\\bigr) = \\frac{\\pi}{4}" ],
"definiens" : [ {
"definition" : "value",
"score" : 0.9161259766018496
}, {
"definition" : "function",
"score" : 0.9057481572098983
}, {
"definition" : "Riemann zeta function",
"score" : 0.8999775058899111
}, {
"definition" : "analytic continuation",
"score" : 0.7618608666546355
}, {
"definition" : "sum of the Dirichlet series",
"score" : 0.6805096216023825
}, {
"definition" : "general representation",
"score" : 0.655024641731632
}, {
"definition" : "Euler -- Riemann zeta function",
"score" : 0.6538197307349187
}, {
"definition" : "real part",
"score" : 0.6199502686895557
}, {
"definition" : "series",
"score" : 0.4787276168838721
}, {
"definition" : "functional equation",
"score" : 0.4619045239257693
}, {
"definition" : "zeta function",
"score" : 0.4496780877572129
}, {
"definition" : "gamma function",
"score" : 0.44843771891608064
}, {
"definition" : "infinite series",
"score" : 0.41973808336652424
}, {
"definition" : "harmonic series",
"score" : 0.4164337884502838
}, {
"definition" : "series development",
"score" : 0.41643378844999424
}, {
"definition" : "derivative",
"score" : 0.4103365720134839
}, {
"definition" : "Euler",
"score" : 0.395299018061601
}, {
"definition" : "fact",
"score" : 0.3836466811460201
}, {
"definition" : "one",
"score" : 0.3836466811460201
}, {
"definition" : "prime number",
"score" : 0.3836466811460201
}, {
"definition" : "Euler -- Mascheroni",
"score" : 0.3745467505047425
}, {
"definition" : "complex number",
"score" : 0.3689874536966156
}, {
"definition" : "convergent series",
"score" : 0.3637510056063175
}, {
"definition" : "Pochhammer symbol",
"score" : 0.3637510056063175
}, {
"definition" : "integer",
"score" : 0.353358215245774
}, {
"definition" : "Cauchy principal value",
"score" : 0.34610158730158724
}, {
"definition" : "equality",
"score" : 0.34610158730158724
}, {
"definition" : "modulus",
"score" : 0.34610158730158724
}, {
"definition" : "xi-function",
"score" : 0.34610158730158724
}, {
"definition" : "irrationality",
"score" : 0.34459554647904167
}, {
"definition" : "integral",
"score" : 0.3367008627613372
}, {
"definition" : "arbitrary precision",
"score" : 0.33670009779285875
}, {
"definition" : "Boltzmann law in physics",
"score" : 0.33670009779285875
}, {
"definition" : "broken line",
"score" : 0.33670009779285875
}, {
"definition" : "critical strip with computational complexity",
"score" : 0.33670009779285875
}, {
"definition" : "differentation",
"score" : 0.33670009779285875
}, {
"definition" : "increment of an arbitrary continuous branch",
"score" : 0.33670009779285875
}, {
"definition" : "integral formula",
"score" : 0.33670009779285875
}, {
"definition" : "n",
"score" : 0.33670009779285875
}, {
"definition" : "non-trivial zero",
"score" : 0.33670009779285875
}, {
"definition" : "notation of umbral calculus",
"score" : 0.33670009779285875
}, {
"definition" : "odd order of the function",
"score" : 0.33670009779285875
}, {
"definition" : "power",
"score" : 0.33670009779285875
}, {
"definition" : "prime-counting function",
"score" : 0.33670009779285875
}, {
"definition" : "result",
"score" : 0.33670009779285875
}, {
"definition" : "simpler infinite product expansion",
"score" : 0.33670009779285875
}, {
"definition" : "substitution",
"score" : 0.33670009779285875
}, {
"definition" : "trivial zero",
"score" : 0.33670009779285875
}, {
"definition" : "Mellin",
"score" : 0.3347128101737352
}, {
"definition" : "number",
"score" : 0.3239170652753103
}, {
"definition" : "Riemann",
"score" : 0.3127866273600674
}, {
"definition" : "Bernoulli number",
"score" : 0.31001020692539555
}, {
"definition" : "absolute convergence",
"score" : 0.310010206925395
}, {
"definition" : "accuracy",
"score" : 0.310010206925395
}, {
"definition" : "argument",
"score" : 0.310010206925395
}, {
"definition" : "argument of the Riemann zeta function",
"score" : 0.310010206925395
}, {
"definition" : "basis of Weierstrass 's factorization theorem",
"score" : 0.310010206925395
}, {
"definition" : "Bernoulli polynomial of the second kind",
"score" : 0.310010206925395
}, {
"definition" : "box with periodic boundary condition",
"score" : 0.310010206925395
}, {
"definition" : "Cauchy number of the second kind",
"score" : 0.310010206925395
}, {
"definition" : "cf. Euler summation",
"score" : 0.310010206925395
}, {
"definition" : "connection between the zeta function",
"score" : 0.310010206925395
}, {
"definition" : "convenience",
"score" : 0.310010206925395
}, {
"definition" : "definition",
"score" : 0.310010206925395
}, {
"definition" : "density",
"score" : 0.310010206925395
}, {
"definition" : "detailed survey on the history",
"score" : 0.310010206925395
}, {
"definition" : "Dirichlet series over the Möbius function",
"score" : 0.310010206925395
}, {
"definition" : "e.g. blagouchine",
"score" : 0.310010206925395
}, {
"definition" : "equation",
"score" : 0.310010206925395
}, {
"definition" : "Euler product",
"score" : 0.310010206925395
}, {
"definition" : "finite result to the series",
"score" : 0.310010206925395
}, {
"definition" : "first series",
"score" : 0.310010206925395
}, {
"definition" : "Gregory coefficient",
"score" : 0.310010206925395
}, {
"definition" : "Gregory coefficient of higher order",
"score" : 0.310010206925395
}, {
"definition" : "Hadamard",
"score" : 0.310010206925395
}, {
"definition" : "Hankel contour",
"score" : 0.310010206925395
}, {
"definition" : "harmonic number",
"score" : 0.310010206925395
}, {
"definition" : "Hasse",
"score" : 0.310010206925395
}, {
"definition" : "i.e.",
"score" : 0.310010206925395
}, {
"definition" : "imaginary part of a complex number",
"score" : 0.310010206925395
}, {
"definition" : "incomplete poly-Bernoulli number",
"score" : 0.310010206925395
}, {
"definition" : "infinite product expansion",
"score" : 0.310010206925395
}, {
"definition" : "infinite product on the right hand side",
"score" : 0.310010206925395
}, {
"definition" : "inversion",
"score" : 0.310010206925395
}, {
"definition" : "larger half-plane",
"score" : 0.310010206925395
}, {
"definition" : "Laurent series",
"score" : 0.310010206925395
}, {
"definition" : "left hand side",
"score" : 0.310010206925395
}, {
"definition" : "many real zero",
"score" : 0.310010206925395
}, {
"definition" : "negative integer",
"score" : 0.310010206925395
}, {
"definition" : "next higher integer of the unique solution",
"score" : 0.310010206925395
}, {
"definition" : "nonpositive integer",
"score" : 0.310010206925395
}, {
"definition" : "odd term",
"score" : 0.310010206925395
}, {
"definition" : "Other sum",
"score" : 0.310010206925395
}, {
"definition" : "perfect power",
"score" : 0.310010206925395
}, {
"definition" : "polygamma function",
"score" : 0.310010206925395
}, {
"definition" : "positive integer",
"score" : 0.310010206925395
}, {
"definition" : "pretext",
"score" : 0.310010206925395
}, {
"definition" : "process",
"score" : 0.310010206925395
}, {
"definition" : "product",
"score" : 0.310010206925395
}, {
"definition" : "proof of Euler 's identity",
"score" : 0.310010206925395
}, {
"definition" : "quantum computer",
"score" : 0.310010206925395
}, {
"definition" : "region",
"score" : 0.310010206925395
}, {
"definition" : "relation",
"score" : 0.310010206925395
}, {
"definition" : "representation in term",
"score" : 0.310010206925395
}, {
"definition" : "Riemann zeta function by the formula",
"score" : 0.310010206925395
}, {
"definition" : "same publication",
"score" : 0.310010206925395
}, {
"definition" : "side of the Euler product formula",
"score" : 0.310010206925395
}, {
"definition" : "Stirling number of the first kind",
"score" : 0.310010206925395
}, {
"definition" : "stricter requirement",
"score" : 0.310010206925395
}, {
"definition" : "such expression",
"score" : 0.310010206925395
}, {
"definition" : "sum",
"score" : 0.310010206925395
}, {
"definition" : "sum of geometric series",
"score" : 0.310010206925395
}, {
"definition" : "version of the above sum",
"score" : 0.310010206925395
}, {
"definition" : "solution to the Basel problem",
"score" : 0.29166436183248295
}, {
"definition" : "Roger Apéry",
"score" : 0.2780717152805707
}, {
"definition" : "special case",
"score" : 0.27017703156286615
}, {
"definition" : "case",
"score" : 0.2701763622154475
}, {
"definition" : "algorithm",
"score" : 0.2701762665943877
}, {
"definition" : "Apéry",
"score" : 0.2701762665943877
}, {
"definition" : "better result",
"score" : 0.2701762665943877
}, {
"definition" : "branch of the Lambert",
"score" : 0.2701762665943877
}, {
"definition" : "cf. Abel -- Plana formula",
"score" : 0.2701762665943877
}, {
"definition" : "convention",
"score" : 0.2701762665943877
}, {
"definition" : "distance between the zero",
"score" : 0.2701762665943877
}, {
"definition" : "February",
"score" : 0.2701762665943877
}, {
"definition" : "finite value to the divergent series",
"score" : 0.2701762665943877
}, {
"definition" : "Godfrey Harold",
"score" : 0.2701762665943877
}, {
"definition" : "Helmut Hasse",
"score" : 0.2701762665943877
}, {
"definition" : "integral relation",
"score" : 0.2701762665943877
}, {
"definition" : "letter",
"score" : 0.2701762665943877
}, {
"definition" : "limit",
"score" : 0.2701762665943877
}, {
"definition" : "limit value",
"score" : 0.2701762665943877
}, {
"definition" : "map",
"score" : 0.2701762665943877
}, {
"definition" : "point",
"score" : 0.2701762665943877
}, {
"definition" : "real axis",
"score" : 0.2701762665943877
}, {
"definition" : "Sandeep Tyagi",
"score" : 0.2701762665943877
}, {
"definition" : "short interval of the critical line",
"score" : 0.2701762665943877
}, {
"definition" : "small neighborhood of point",
"score" : 0.2701762665943877
}, {
"definition" : "symmetric version of the functional equation",
"score" : 0.2701762665943877
}, {
"definition" : "term of Jacobi 's theta function",
"score" : 0.2701762665943877
}, {
"definition" : "total number of zero",
"score" : 0.2701762665943877
}, {
"definition" : "Via",
"score" : 0.2701762665943877
}, {
"definition" : "zero of the Riemann zeta function",
"score" : 0.2701762665943877
}, {
"definition" : "above series termwise",
"score" : 0.22291908683501988
}, {
"definition" : "analogy with the Euler product",
"score" : 0.22291908683501988
}, {
"definition" : "contour",
"score" : 0.22291908683501988
}, {
"definition" : "convergent series for the zeta function",
"score" : 0.22291908683501988
}, {
"definition" : "critical temperature for a Bose -- Einstein condensate",
"score" : 0.22291908683501988
}, {
"definition" : "entire complex plane",
"score" : 0.22291908683501988
}, {
"definition" : "explicit error bound",
"score" : 0.22291908683501988
}, {
"definition" : "following expression for the zeta function",
"score" : 0.22291908683501988
}, {
"definition" : "interval of large positive real number",
"score" : 0.22291908683501988
}, {
"definition" : "kinetic boundary layer problem of linear kinetic equation",
"score" : 0.22291908683501988
}, {
"definition" : "Konrad Knopp",
"score" : 0.22291908683501988
}, {
"definition" : "numerical calculation",
"score" : 0.22291908683501988
}, {
"definition" : "Ramanujan summation",
"score" : 0.22291908683501988
}, {
"definition" : "spin wave physics in magnetic system",
"score" : 0.22291908683501988
}, {
"definition" : "summand",
"score" : 0.22291908683501988
}, {
"definition" : "zero of the sine function",
"score" : 0.22291908683501988
}, {
"definition" : "certain context",
"score" : 0.1741968253968254
}, {
"definition" : "effective form of Vinogradov 's mean-value theorem",
"score" : 0.1741968253968254
}, {
"definition" : "Littlewood",
"score" : 0.1741968253968254
}, {
"definition" : "numerical evaluation of the zeta-function",
"score" : 0.1741968253968254
}, {
"definition" : "Planck 's law",
"score" : 0.1741968253968254
}, {
"definition" : "string theory",
"score" : 0.1741968253968254
}, {
"definition" : "total number of real zero",
"score" : 0.1741968253968254
}, {
"definition" : "critical strip",
"score" : 0.1290090128830367
}, {
"definition" : "high precision",
"score" : 0.1290090128830367
}, {
"definition" : "Stefan",
"score" : 0.1290090128830367
}, {
"definition" : "year",
"score" : 0.1290090128830367
}, {
"definition" : "equivalent relationship",
"score" : 0.09066089478321589
}, {
"definition" : "geometric series",
"score" : 0.09066089478321589
}, {
"definition" : "interval",
"score" : 0.09066089478321589
} ]
}