Represents the truth value of the Riemann hypothesis, defined in
49704a .
Semantically,
RH ∈ { True , False } \operatorname{RH} \in \left\{\operatorname{True}, \operatorname{False}\right\} R H ∈ { T r u e , F a l s e } .
This symbol can be used in an assumption to express that a formula is valid conditionally on the truth of the Riemann hypothesis.
Definitions:
Fungrim symbol Notation Short description RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis
Source code for this entry:
Entry(ID("c03de4"),
SymbolDefinition(RiemannHypothesis, RiemannHypothesis, "Riemann hypothesis"),
Description("Represents the truth value of the Riemann hypothesis, defined in ", EntryReference("49704a"), "."),
Description("Semantically, ", Element(RiemannHypothesis, Set(True_, False_)), "."),
Description("This symbol can be used in an assumption to express that a formula is valid conditionally on the truth of the Riemann hypothesis.")) ( RH ) ⟺ ( Re ( s ) = 1 2 for all s ∈ C with 0 ≤ Re ( s ) ≤ 1 and ζ ( s ) = 0 ) \left(\operatorname{RH}\right) \iff \left(\operatorname{Re}(s) = \frac{1}{2} \;\text{ for all } s \in \mathbb{C} \text{ with } 0 \le \operatorname{Re}(s) \le 1 \;\mathbin{\operatorname{and}}\; \zeta\!\left(s\right) = 0\right) ( R H ) ⟺ ( R e ( s ) = 2 1 for all s ∈ C with 0 ≤ R e ( s ) ≤ 1 a n d ζ ( s ) = 0 ) TeX:
\left(\operatorname{RH}\right) \iff \left(\operatorname{Re}(s) = \frac{1}{2} \;\text{ for all } s \in \mathbb{C} \text{ with } 0 \le \operatorname{Re}(s) \le 1 \;\mathbin{\operatorname{and}}\; \zeta\!\left(s\right) = 0\right) Definitions:
Fungrim symbol Notation Short description RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis Re Re ( z ) \operatorname{Re}(z) R e ( z ) Real part CC C \mathbb{C} C Complex numbers RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function
Source code for this entry:
Entry(ID("9fa2a1"),
Formula(Equivalent(RiemannHypothesis, All(Equal(Re(s), Div(1, 2)), ForElement(s, CC), And(LessEqual(0, Re(s), 1), Equal(RiemannZeta(s), 0)))))) ( RH ) ⟺ ( Re ( ρ n ) = 1 2 for all n ∈ Z ≥ 1 ) \left(\operatorname{RH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\right) ( R H ) ⟺ ( R e ( ρ n ) = 2 1 for all n ∈ Z ≥ 1 ) TeX:
\left(\operatorname{RH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\right) Definitions:
Fungrim symbol Notation Short description RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis Re Re ( z ) \operatorname{Re}(z) R e ( z ) Real part RiemannZetaZero ρ n \rho_{n} ρ n Nontrivial zero of the Riemann zeta function ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("49704a"),
Formula(Equivalent(RiemannHypothesis, All(Equal(Re(RiemannZetaZero(n)), Div(1, 2)), ForElement(n, ZZGreaterEqual(1)))))) ( RH ) ⟺ ( ∣ π ( x ) − li ( x ) ∣ < x log ( x ) for all x ∈ [ 2 , ∞ ) ) \left(\operatorname{RH}\right) \iff \left(\left|\pi(x) - \operatorname{li}(x)\right| < \sqrt{x} \log(x) \;\text{ for all } x \in \left[2, \infty\right)\right) ( R H ) ⟺ ( ∣ π ( x ) − l i ( x ) ∣ < x log ( x ) for all x ∈ [ 2 , ∞ ) ) References:
https://mathoverflow.net/q/338066 TeX:
\left(\operatorname{RH}\right) \iff \left(\left|\pi(x) - \operatorname{li}(x)\right| < \sqrt{x} \log(x) \;\text{ for all } x \in \left[2, \infty\right)\right) Definitions:
Fungrim symbol Notation Short description RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value PrimePi π ( x ) \pi(x) π ( x ) Prime counting function LogIntegral li ( z ) \operatorname{li}(z) l i ( z ) Logarithmic integral Sqrt z \sqrt{z} z Principal square root Log log ( z ) \log(z) log ( z ) Natural logarithm ClosedOpenInterval [ a , b ) \left[a, b\right) [ a , b ) Closed-open interval Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("bfaeb5"),
Formula(Equivalent(RiemannHypothesis, All(Less(Abs(Sub(PrimePi(x), LogIntegral(x))), Mul(Sqrt(x), Log(x))), ForElement(x, ClosedOpenInterval(2, Infinity))))),
References("https://mathoverflow.net/q/338066")) ( RH ) ⟺ ( σ 1 ( n ) < e γ n log ( log ( n ) ) for all n ∈ Z ≥ 5041 ) \left(\operatorname{RH}\right) \iff \left(\sigma_{1}\!\left(n\right) < {e}^{\gamma} n \log\!\left(\log(n)\right) \;\text{ for all } n \in \mathbb{Z}_{\ge 5041}\right) ( R H ) ⟺ ( σ 1 ( n ) < e γ n log ( log ( n ) ) for all n ∈ Z ≥ 5 0 4 1 ) TeX:
\left(\operatorname{RH}\right) \iff \left(\sigma_{1}\!\left(n\right) < {e}^{\gamma} n \log\!\left(\log(n)\right) \;\text{ for all } n \in \mathbb{Z}_{\ge 5041}\right) Definitions:
Fungrim symbol Notation Short description RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis DivisorSigma σ k ( n ) \sigma_{k}\!\left(n\right) σ k ( n ) Sum of divisors function Exp e z {e}^{z} e z Exponential function ConstGamma γ \gamma γ The constant gamma (0.577...) Log log ( z ) \log(z) log ( z ) Natural logarithm ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("3142ec"),
Formula(Equivalent(RiemannHypothesis, All(Less(DivisorSigma(1, n), Mul(Mul(Exp(ConstGamma), n), Log(Log(n)))), ForElement(n, ZZGreaterEqual(5041)))))) ( RH ) ⟺ ( σ 1 ( n ) < H n + exp ( H n ) log ( H n ) for all n ∈ Z ≥ 2 ) \left(\operatorname{RH}\right) \iff \left(\sigma_{1}\!\left(n\right) < H_{n} + \exp\!\left(H_{n}\right) \log\!\left(H_{n}\right) \;\text{ for all } n \in \mathbb{Z}_{\ge 2}\right) ( R H ) ⟺ ( σ 1 ( n ) < H n + exp ( H n ) log ( H n ) for all n ∈ Z ≥ 2 ) References:
https://doi.org/10.2307/2695443 TeX:
\left(\operatorname{RH}\right) \iff \left(\sigma_{1}\!\left(n\right) < H_{n} + \exp\!\left(H_{n}\right) \log\!\left(H_{n}\right) \;\text{ for all } n \in \mathbb{Z}_{\ge 2}\right) Definitions:
Fungrim symbol Notation Short description RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis DivisorSigma σ k ( n ) \sigma_{k}\!\left(n\right) σ k ( n ) Sum of divisors function Exp e z {e}^{z} e z Exponential function Log log ( z ) \log(z) log ( z ) Natural logarithm ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("e4287f"),
Formula(Equivalent(RiemannHypothesis, All(Less(DivisorSigma(1, n), Add(HarmonicNumber(n), Mul(Exp(HarmonicNumber(n)), Log(HarmonicNumber(n))))), ForElement(n, ZZGreaterEqual(2))))),
References("https://doi.org/10.2307/2695443")) ( RH ) ⟺ ( λ n > 0 for all n ∈ Z ≥ 1 ) \left(\operatorname{RH}\right) \iff \left(\lambda_{n} > 0 \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\right) ( R H ) ⟺ ( λ n > 0 for all n ∈ Z ≥ 1 ) References:
https://doi.org/10.1006/jnth.1997.2137 TeX:
\left(\operatorname{RH}\right) \iff \left(\lambda_{n} > 0 \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\right) Definitions:
Fungrim symbol Notation Short description RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis KeiperLiLambda λ n \lambda_{n} λ n Keiper-Li coefficient ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("e68f82"),
Formula(Equivalent(RiemannHypothesis, All(Greater(KeiperLiLambda(n), 0), ForElement(n, ZZGreaterEqual(1))))),
References("https://doi.org/10.1006/jnth.1997.2137")) ( RH ) ⟺ ( ∑ n = 1 ∞ ∣ λ n − a ( n ) ∣ 2 < ∞ where a ( n ) = log ( n ) 2 − log ( 2 π ) + 1 − γ 2 ) \left(\operatorname{RH}\right) \iff \left(\sum_{n=1}^{\infty} {\left|\lambda_{n} - a(n)\right|}^{2} < \infty\; \text{ where } a(n) = \frac{\log(n)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}\right) ( R H ) ⟺ ( n = 1 ∑ ∞ ∣ λ n − a ( n ) ∣ 2 < ∞ where a ( n ) = 2 log ( n ) − 2 log ( 2 π ) + 1 − γ ) References:
https://doi.org/10.7169/facm/1317045228 TeX:
\left(\operatorname{RH}\right) \iff \left(\sum_{n=1}^{\infty} {\left|\lambda_{n} - a(n)\right|}^{2} < \infty\; \text{ where } a(n) = \frac{\log(n)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}\right) Definitions:
Fungrim symbol Notation Short description RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis Sum ∑ n f ( n ) \sum_{n} f(n) ∑ n f ( n ) Sum Pow a b {a}^{b} a b Power Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value KeiperLiLambda λ n \lambda_{n} λ n Keiper-Li coefficient Infinity ∞ \infty ∞ Positive infinity Log log ( z ) \log(z) log ( z ) Natural logarithm Pi π \pi π The constant pi (3.14...) ConstGamma γ \gamma γ The constant gamma (0.577...)
Source code for this entry:
Entry(ID("a5d65f"),
Formula(Equivalent(RiemannHypothesis, Where(Less(Sum(Pow(Abs(Sub(KeiperLiLambda(n), a(n))), 2), For(n, 1, Infinity)), Infinity), Equal(a(n), Sub(Div(Log(n), 2), Div(Sub(Add(Log(Mul(2, Pi)), 1), ConstGamma), 2)))))),
References("https://doi.org/10.7169/facm/1317045228")) ( RH ) ⟺ ( log ( g ( n ) ) < f ( n ) for all n ∈ Z ≥ 1 where f ( y ) = solution* x ∈ ( 1 , ∞ ) [ li ( x ) = y ] ) \left(\operatorname{RH}\right) \iff \left(\log\!\left(g(n)\right) < \sqrt{f(n)} \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\; \text{ where } f(y) = \mathop{\operatorname{solution*}\,}\limits_{x \in \left(1, \infty\right)} \left[\operatorname{li}(x) = y\right]\right) ( R H ) ⟺ ( log ( g ( n ) ) < f ( n ) for all n ∈ Z ≥ 1 where f ( y ) = x ∈ ( 1 , ∞ ) s o l u t i o n * [ l i ( x ) = y ] ) References:
Marc Deleglise, Jean-Louis Nicolas, The Landau function and the Riemann Hypothesis, https://arxiv.org/abs/1907.07664 TeX:
\left(\operatorname{RH}\right) \iff \left(\log\!\left(g(n)\right) < \sqrt{f(n)} \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\; \text{ where } f(y) = \mathop{\operatorname{solution*}\,}\limits_{x \in \left(1, \infty\right)} \left[\operatorname{li}(x) = y\right]\right) Definitions:
Fungrim symbol Notation Short description RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis Log log ( z ) \log(z) log ( z ) Natural logarithm LandauG g ( n ) g(n) g ( n ) Landau's function Sqrt z \sqrt{z} z Principal square root ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n UniqueSolution solution* x ∈ S Q ( x ) \mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x) x ∈ S s o l u t i o n * Q ( x ) Unique solution LogIntegral li ( z ) \operatorname{li}(z) l i ( z ) Logarithmic integral OpenInterval ( a , b ) \left(a, b\right) ( a , b ) Open interval Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("65fa9f"),
Formula(Equivalent(RiemannHypothesis, Where(All(Less(Log(LandauG(n)), Sqrt(f(n))), ForElement(n, ZZGreaterEqual(1))), Equal(f(y), UniqueSolution(Brackets(Equal(LogIntegral(x), y)), ForElement(x, OpenInterval(1, Infinity))))))),
References("Marc Deleglise, Jean-Louis Nicolas, The Landau function and the Riemann Hypothesis, https://arxiv.org/abs/1907.07664")) ( RH ) ⟺ ( 1 π ∫ 0 ∞ log ( ∣ ζ ( 1 2 + i t ) ζ ( 1 2 ) ∣ ) 1 t 2 d t = π 8 + γ 4 + log ( 8 π ) 4 − 2 ) \left(\operatorname{RH}\right) \iff \left(\frac{1}{\pi} \int_{0}^{\infty} \log\!\left(\left|\frac{\zeta\!\left(\frac{1}{2} + i t\right)}{\zeta\!\left(\frac{1}{2}\right)}\right|\right) \frac{1}{{t}^{2}} \, dt = \frac{\pi}{8} + \frac{\gamma}{4} + \frac{\log\!\left(8 \pi\right)}{4} - 2\right) ( R H ) ⟺ ( π 1 ∫ 0 ∞ log ( ∣ ∣ ∣ ∣ ∣ ζ ( 2 1 ) ζ ( 2 1 + i t ) ∣ ∣ ∣ ∣ ∣ ) t 2 1 d t = 8 π + 4 γ + 4 log ( 8 π ) − 2 ) References:
https://mathoverflow.net/q/279936 TeX:
\left(\operatorname{RH}\right) \iff \left(\frac{1}{\pi} \int_{0}^{\infty} \log\!\left(\left|\frac{\zeta\!\left(\frac{1}{2} + i t\right)}{\zeta\!\left(\frac{1}{2}\right)}\right|\right) \frac{1}{{t}^{2}} \, dt = \frac{\pi}{8} + \frac{\gamma}{4} + \frac{\log\!\left(8 \pi\right)}{4} - 2\right) Definitions:
Fungrim symbol Notation Short description RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis Pi π \pi π The constant pi (3.14...) Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x Integral Log log ( z ) \log(z) log ( z ) Natural logarithm Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function ConstI i i i Imaginary unit Pow a b {a}^{b} a b Power Infinity ∞ \infty ∞ Positive infinity ConstGamma γ \gamma γ The constant gamma (0.577...)
Source code for this entry:
Entry(ID("7783f9"),
Formula(Equivalent(RiemannHypothesis, Equal(Mul(Div(1, Pi), Integral(Mul(Log(Abs(Div(RiemannZeta(Add(Div(1, 2), Mul(ConstI, t))), RiemannZeta(Div(1, 2))))), Div(1, Pow(t, 2))), For(t, 0, Infinity))), Sub(Add(Add(Div(Pi, 8), Div(ConstGamma, 4)), Div(Log(Mul(8, Pi)), 4)), 2)))),
References("https://mathoverflow.net/q/279936")) ( RH ) ⟺ ( ∫ 0 ∞ 1 − 12 t 2 ( 1 + 4 t 2 ) 3 ∫ 1 / 2 ∞ log ( ∣ ζ ( σ + i t ) ∣ ) d σ d t = π ( 3 − γ ) 32 ) \left(\operatorname{RH}\right) \iff \left(\int_{0}^{\infty} \frac{1 - 12 {t}^{2}}{{\left(1 + 4 {t}^{2}\right)}^{3}} \int_{1 / 2}^{\infty} \log\!\left(\left|\zeta\!\left(\sigma + i t\right)\right|\right) \, d\sigma \, dt = \frac{\pi \left(3 - \gamma\right)}{32}\right) ( R H ) ⟺ ( ∫ 0 ∞ ( 1 + 4 t 2 ) 3 1 − 1 2 t 2 ∫ 1 / 2 ∞ log ( ∣ ζ ( σ + i t ) ∣ ) d σ d t = 3 2 π ( 3 − γ ) ) References:
https://doi.org/10.1007/BF01056314 TeX:
\left(\operatorname{RH}\right) \iff \left(\int_{0}^{\infty} \frac{1 - 12 {t}^{2}}{{\left(1 + 4 {t}^{2}\right)}^{3}} \int_{1 / 2}^{\infty} \log\!\left(\left|\zeta\!\left(\sigma + i t\right)\right|\right) \, d\sigma \, dt = \frac{\pi \left(3 - \gamma\right)}{32}\right) Definitions:
Fungrim symbol Notation Short description RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis Integral ∫ a b f ( x ) d x \int_{a}^{b} f(x) \, dx ∫ a b f ( x ) d x Integral Pow a b {a}^{b} a b Power Log log ( z ) \log(z) log ( z ) Natural logarithm Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function ConstI i i i Imaginary unit Infinity ∞ \infty ∞ Positive infinity Pi π \pi π The constant pi (3.14...) ConstGamma γ \gamma γ The constant gamma (0.577...)
Source code for this entry:
Entry(ID("cf70ce"),
Formula(Equivalent(RiemannHypothesis, Equal(Integral(Mul(Div(Sub(1, Mul(12, Pow(t, 2))), Pow(Add(1, Mul(4, Pow(t, 2))), 3)), Integral(Log(Abs(RiemannZeta(Add(sigma, Mul(ConstI, t))))), For(sigma, Div(1, 2), Infinity))), For(t, 0, Infinity)), Div(Mul(Pi, Sub(3, ConstGamma)), 32)))),
References("https://doi.org/10.1007/BF01056314")) Definitions:
Source code for this entry:
Entry(ID("22ab47"),
SymbolDefinition(DeBruijnNewmanLambda, DeBruijnNewmanLambda, "De Bruijn-Newman constant")) ( RH ) ⟺ ( Λ = 0 ) \left(\operatorname{RH}\right) \iff \left(\Lambda = 0\right) ( R H ) ⟺ ( Λ = 0 ) References:
https://arxiv.org/abs/1801.05914 TeX:
\left(\operatorname{RH}\right) \iff \left(\Lambda = 0\right) Definitions:
Source code for this entry:
Entry(ID("a71ddd"),
Formula(Equivalent(RiemannHypothesis, Equal(DeBruijnNewmanLambda, 0))),
References("https://arxiv.org/abs/1801.05914")) Re ( ρ n ) = 1 2 \operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} R e ( ρ n ) = 2 1 Assumptions: n ∈ Z and n ≠ 0 and ( ∣ n ∣ < 103800788359 or RH ) n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0 \;\mathbin{\operatorname{and}}\; \left(\left|n\right| < 103800788359 \;\mathbin{\operatorname{or}}\; \operatorname{RH}\right) n ∈ Z a n d n = 0 a n d ( ∣ n ∣ < 1 0 3 8 0 0 7 8 8 3 5 9 o r R H ) References:
D. J. Platt (2016), Isolating some non-trivial zeros of zeta, Mathematics of Computation 86(307):1, DOI: 10.1090/mcom/3198 TeX:
\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2}
n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0 \;\mathbin{\operatorname{and}}\; \left(\left|n\right| < 103800788359 \;\mathbin{\operatorname{or}}\; \operatorname{RH}\right) Definitions:
Fungrim symbol Notation Short description Re Re ( z ) \operatorname{Re}(z) R e ( z ) Real part RiemannZetaZero ρ n \rho_{n} ρ n Nontrivial zero of the Riemann zeta function ZZ Z \mathbb{Z} Z Integers Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis
Source code for this entry:
Entry(ID("e6ff64"),
Formula(Equal(Re(RiemannZetaZero(n)), Div(1, 2))),
Variables(n),
Assumptions(And(Element(n, ZZ), NotEqual(n, 0), Or(Less(Abs(n), 103800788359), RiemannHypothesis))),
References("D. J. Platt (2016), Isolating some non-trivial zeros of zeta, Mathematics of Computation 86(307):1, DOI: 10.1090/mcom/3198")) ∣ π ( x ) − li ( x ) ∣ < x log ( x ) 8 π \left|\pi(x) - \operatorname{li}(x)\right| < \frac{\sqrt{x} \log(x)}{8 \pi} ∣ π ( x ) − l i ( x ) ∣ < 8 π x log ( x ) Assumptions: x ∈ R and x ≥ 2657 and RH x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \ge 2657 \;\mathbin{\operatorname{and}}\; \operatorname{RH} x ∈ R a n d x ≥ 2 6 5 7 a n d R H References:
L. Schoenfeld (1976). Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II. Mathematics of Computation. 30 (134): 337-360. DOI: 10.2307/2005976 TeX:
\left|\pi(x) - \operatorname{li}(x)\right| < \frac{\sqrt{x} \log(x)}{8 \pi}
x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \ge 2657 \;\mathbin{\operatorname{and}}\; \operatorname{RH} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value PrimePi π ( x ) \pi(x) π ( x ) Prime counting function LogIntegral li ( z ) \operatorname{li}(z) l i ( z ) Logarithmic integral Sqrt z \sqrt{z} z Principal square root Log log ( z ) \log(z) log ( z ) Natural logarithm Pi π \pi π The constant pi (3.14...) RR R \mathbb{R} R Real numbers RiemannHypothesis RH \operatorname{RH} R H Riemann hypothesis
Source code for this entry:
Entry(ID("375afe"),
Formula(Less(Abs(Sub(PrimePi(x), LogIntegral(x))), Div(Mul(Sqrt(x), Log(x)), Mul(8, Pi)))),
Variables(x),
Assumptions(And(Element(x, RR), GreaterEqual(x, 2657), RiemannHypothesis)),
References("L. Schoenfeld (1976). Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II. Mathematics of Computation. 30 (134): 337-360. DOI: 10.2307/2005976")) Represents the truth value of the generalized Riemann hypothesis for Dirichlet L-functions, defined in
e2a734 .
Semantically,
GRH ∈ { True , False } \operatorname{GRH} \in \left\{\operatorname{True}, \operatorname{False}\right\} G R H ∈ { T r u e , F a l s e } .
This symbol can be used in an assumption to express that a formula is valid conditionally on the truth of the generalized Riemann hypothesis.
Definitions:
Source code for this entry:
Entry(ID("dc593e"),
SymbolDefinition(GeneralizedRiemannHypothesis, GeneralizedRiemannHypothesis, "Generalized Riemann hypothesis"),
Description("Represents the truth value of the generalized Riemann hypothesis for Dirichlet L-functions, defined in ", EntryReference("e2a734"), "."),
Description("Semantically, ", Element(GeneralizedRiemannHypothesis, Set(True_, False_)), "."),
Description("This symbol can be used in an assumption to express that a formula is valid conditionally on the truth of the generalized Riemann hypothesis.")) ( GRH ) ⟺ ( Re ( ρ n , χ ) = 1 2 for all ( q , χ , n ) with q ∈ Z ≥ 1 and χ ∈ G q and n ∈ Z ∖ { 0 } ) \left(\operatorname{GRH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2} \;\text{ for all } \left(q, \chi, n\right) \text{ with } q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \setminus \left\{0\right\}\right) ( G R H ) ⟺ ( R e ( ρ n , χ ) = 2 1 for all ( q , χ , n ) with q ∈ Z ≥ 1 a n d χ ∈ G q a n d n ∈ Z ∖ { 0 } ) TeX:
\left(\operatorname{GRH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2} \;\text{ for all } \left(q, \chi, n\right) \text{ with } q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \setminus \left\{0\right\}\right) Definitions:
Fungrim symbol Notation Short description GeneralizedRiemannHypothesis GRH \operatorname{GRH} G R H Generalized Riemann hypothesis Re Re ( z ) \operatorname{Re}(z) R e ( z ) Real part DirichletLZero ρ n , χ \rho_{n,\chi} ρ n , χ Nontrivial zero of Dirichlet L-function ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n DirichletGroup G q G_{q} G q Dirichlet characters with given modulus ZZ Z \mathbb{Z} Z Integers
Source code for this entry:
Entry(ID("e2a734"),
Formula(Equivalent(GeneralizedRiemannHypothesis, All(Equal(Re(DirichletLZero(n, chi)), Div(1, 2)), For(Tuple(q, chi, n)), And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(n, SetMinus(ZZ, Set(0))))))))