In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev functionϑ (x) or θ (x) is given by
Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalarizing function is used when one has several functions to be minimized and one wants to "scalarize" them to a single function:
By minimizing this function for different values of , one obtains every point on a Pareto front, even in the nonconvex parts.[1] Often the functions to be minimized are not but for some scalars . Then [2]
(The numerical value of ζ′(0)/ζ (0) is log(2π).) Here ρ runs over the nontrivial zeros of the zeta function, and ψ0 is the same as ψ, except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right:
From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of xω/ω over the trivial zeros of the zeta function, ω = −2, −4, −6, ..., i.e.
Similarly, the first term, x = x1/1, corresponds to the simple pole of the zeta function at 1. It being a pole rather than a zero accounts for the opposite sign of the term.
Properties
A theorem due to Erhard Schmidt states that, for some explicit positive constant K, there are infinitely many natural numbersx such that
The first Chebyshev function is the logarithm of the primorial of x, denoted x #:
This proves that the primorial x # is asymptotically equal to e(1 + o(1))x, where "o" is the little-o notation (see big O notation) and together with the prime number theorem establishes the asymptotic behavior of pn #.
Relation to the prime-counting function
The Chebyshev function can be related to the prime-counting function as follows. Define
Certainly π (x) ≤ x, so for the sake of approximation, this last relation can be recast in the form
The Riemann hypothesis
The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part1/2. In this case, |xρ| = √x, and it can be shown that
By the above, this implies
Good evidence that the hypothesis could be true comes from the fact proposed by Alain Connes and others, that if we differentiate the von Mangoldt formula with respect to x we get x = eu. Manipulating, we have the "Trace formula" for the exponential of the Hamiltonian operator satisfying
and
where the "trigonometric sum" can be considered to be the trace of the operator (statistical mechanics) eiuĤ, which is only true if ρ = 1/2 + iE(n).
Using the semiclassical approach the potential of H = T + V satisfies:
with Z (u) → 0 as u → ∞.
solution to this nonlinear integral equation can be obtained (among others) by
in order to obtain the inverse of the potential:
Smoothing function
The difference of the smoothed Chebyshev function and x 2/2 for x < 106
The smoothing function is defined as
Obviously
Variational formulation
The Chebyshev function evaluated at x = et minimizes the functional
so
Notes
^ abJoshua Knowles (2 May 2014). "Multiobjective Optimization Concepts, Algorithms and Performance Measures" (PDF). The University of Manchester. p. 34.
^Ho-Huu, V.; Hartjes, S.; Visser, H. G.; Curran, R. (2018). "An improved MOEA/D algorithm for bi-objective optimization problems with complex Pareto fronts and its application to structural optimization" (PDF). Delft University of Technology. Page 6 equation (2). doi:10.1016/j.eswa.2017.09.051.{{cite web}}: CS1 maint: multiple names: authors list (link)
^Apostol, Tom M. (2010). Introduction to Analytic Number Theory. Springer. pp. 75–76.
^Pierre Dusart, "Estimates of some functions over primes without R.H.". arXiv:1002.0442
^ Pierre Dusart, "Sharper bounds for ψ, θ, π, pk", Rapport de recherche no. 1998-06, Université de Limoges. An abbreviated version appeared as "The kth prime is greater than k(log k + log log k − 1) for k ≥ 2", Mathematics of Computation, Vol. 68, No. 225 (1999), pp. 411–415.
^ Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", Mathematische Annalen, 57 (1903), pp. 195–204.
^ G .H. Hardy and J. E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41 (1916) pp. 119–196.
Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN978-0-387-90163-3, MR 0434929, Zbl 0335.10001
Riemann's Explicit Formula, with images and movies
January 10, 2023
chebyshev, function, mathematics, either, scalarising, function, tchebycheff, function, related, functions, first, given, bythe, with, function, function, displaystyle, vartheta, where, displaystyle, denotes, natural, logarithm, with, extending, over, prime, n. In mathematics the Chebyshev function is either a scalarising function Tchebycheff function or one of two related functions The first Chebyshev function ϑ x or 8 x is given byThe Chebyshev function ps x with x lt 50 The function ps x x for x lt 104 The function ps x x for x lt 107 ϑ x p x ln p displaystyle vartheta x sum p leq x ln p where ln displaystyle ln denotes the natural logarithm with the sum extending over all prime numbers p that are less than or equal to x The second Chebyshev function ps x is defined similarly with the sum extending over all prime powers not exceeding x ps x k N p k x ln p n x L n p x log p x ln p displaystyle psi x sum k in mathbb N sum p k leq x ln p sum n leq x Lambda n sum p leq x left lfloor log p x right rfloor ln p where L is the von Mangoldt function The Chebyshev functions especially the second one ps x are often used in proofs related to prime numbers because it is typically simpler to work with them than with the prime counting function p x see the exact formula below Both Chebyshev functions are asymptotic to x a statement equivalent to the prime number theorem Tchebycheff function Chebyshev utility function or weighted Tchebycheff scalarizing function is used when one has several functions to be minimized and one wants to scalarize them to a single function f T c h b x w max i w i f i x displaystyle f Tchb x w max i w i f i x 1 By minimizing this function for different values of w displaystyle w one obtains every point on a Pareto front even in the nonconvex parts 1 Often the functions to be minimized are not f i displaystyle f i but f i z i displaystyle f i z i for some scalars z i displaystyle z i Then f T c h b x w max i w i f i x z i displaystyle f Tchb x w max i w i f i x z i 2 All three functions are named in honour of Pafnuty Chebyshev Contents 1 Relationships 2 Relationships between UNIQ postMath 0000000E QINU and UNIQ postMath 0000000F QINU 3 3 Asymptotics and bounds 4 The exact formula 5 Properties 6 Relation to primorials 7 Relation to the prime counting function 8 The Riemann hypothesis 9 Smoothing function 10 Variational formulation 11 Notes 12 References 13 External linksRelationships EditThe second Chebyshev function can be seen to be related to the first by writing it as ps x p x k log p displaystyle psi x sum p leq x k log p where k is the unique integer such that p k x and x lt p k 1 The values of k are given in OEIS A206722 A more direct relationship is given by ps x n 1 ϑ x 1 n displaystyle psi x sum n 1 infty vartheta big x frac 1 n big Note that this last sum has only a finite number of non vanishing terms as ϑ x 1 n 0 for n gt log 2 x log x log 2 displaystyle vartheta big x frac 1 n big 0 quad text for quad n gt log 2 x frac log x log 2 The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to n lcm 1 2 n e ps n displaystyle operatorname lcm 1 2 dots n e psi n Values of lcm 1 2 n for the integer variable n are given at OEIS A003418 Relationships between ps x x displaystyle psi x x and ϑ x x displaystyle vartheta x x 3 EditThe following theorem relates the two quotients ps x x displaystyle frac psi x x and ϑ x x displaystyle frac vartheta x x Theorem For x gt 0 displaystyle x gt 0 we have 0 ps x x ϑ x x log x 2 2 x log 2 displaystyle 0 leq frac psi x x frac vartheta x x leq frac log x 2 2 sqrt x log 2 Note This inequality implies that lim x ps x x ϑ x x 0 displaystyle lim x to infty left frac psi x x frac vartheta x x right 0 In other words if one of the ps x x displaystyle psi x x or ϑ x x displaystyle vartheta x x tends to a limit then so does the other and the two limits are equal Proof Since ps x n log 2 x ϑ x 1 n displaystyle psi x sum n leq log 2 x vartheta x 1 n we find that 0 ps x ϑ x 2 n log 2 x ϑ x 1 n displaystyle 0 leq psi x vartheta x sum 2 leq n leq log 2 x vartheta x 1 n But from the definition of ϑ x displaystyle vartheta x we have the trivial inequality ϑ x p x log x x log x displaystyle vartheta x leq sum p leq x log x leq x log x so 0 ps x ϑ x 2 n log 2 x x 1 n log x 1 n log 2 x x log x log x log 2 x 2 log x x log x 2 2 log 2 displaystyle begin aligned 0 leq psi x vartheta x amp leq sum 2 leq n leq log 2 x x 1 n log x 1 n amp leq log 2 x sqrt x log sqrt x amp frac log x log 2 frac sqrt x 2 log x amp frac sqrt x log x 2 2 log 2 end aligned Lastly divide by x displaystyle x to obtain the inequality in the theorem Asymptotics and bounds EditThe following bounds are known for the Chebyshev functions 1 2 in these formulas pk is the k th prime number p1 2 p2 3 etc ϑ p k k log k log log k 1 log log k 2 050735 log k for k 10 11 ϑ p k k log k log log k 1 log log k 2 log k for k 198 ϑ x x 0 006788 x log x for x 10 544 111 ps x x 0 006409 x log x for x e 22 0 9999 x lt ps x ϑ x lt 1 00007 x 1 78 x 3 for x 121 displaystyle begin aligned vartheta p k amp geq k left log k log log k 1 frac log log k 2 050735 log k right amp amp text for k geq 10 11 8px vartheta p k amp leq k left log k log log k 1 frac log log k 2 log k right amp amp text for k geq 198 8px vartheta x x amp leq 0 006788 frac x log x amp amp text for x geq 10 544 111 8px psi x x amp leq 0 006409 frac x log x amp amp text for x geq e 22 8px 0 9999 sqrt x amp lt psi x vartheta x lt 1 00007 sqrt x 1 78 sqrt 3 x amp amp text for x geq 121 end aligned Furthermore under the Riemann hypothesis ϑ x x O x 1 2 e ps x x O x 1 2 e displaystyle begin aligned vartheta x x amp O Big x frac 1 2 varepsilon Big psi x x amp O Big x frac 1 2 varepsilon Big end aligned for any e gt 0 Upper bounds exist for both ϑ x and ps x such that 4 3 ϑ x lt 1 000028 x ps x lt 1 03883 x displaystyle begin aligned vartheta x amp lt 1 000028x psi x amp lt 1 03883x end aligned for any x gt 0 An explanation of the constant 1 03883 is given at OEIS A206431 The exact formula EditIn 1895 Hans Carl Friedrich von Mangoldt proved 4 an explicit expression for ps x as a sum over the nontrivial zeros of the Riemann zeta function ps 0 x x r x r r z 0 z 0 1 2 log 1 x 2 displaystyle psi 0 x x sum rho frac x rho rho frac zeta 0 zeta 0 tfrac 1 2 log 1 x 2 The numerical value of z 0 z 0 is log 2p Here r runs over the nontrivial zeros of the zeta function and ps0 is the same as ps except that at its jump discontinuities the prime powers it takes the value halfway between the values to the left and the right ps 0 x 1 2 n x L n n lt x L n ps x 1 2 L x x 2 3 4 5 7 8 9 11 13 16 ps x otherwise displaystyle psi 0 x frac 1 2 left sum n leq x Lambda n sum n lt x Lambda n right begin cases psi x tfrac 1 2 Lambda x amp x 2 3 4 5 7 8 9 11 13 16 dots 5px psi x amp mbox otherwise end cases From the Taylor series for the logarithm the last term in the explicit formula can be understood as a summation of xw w over the trivial zeros of the zeta function w 2 4 6 i e k 1 x 2 k 2 k 1 2 log 1 x 2 displaystyle sum k 1 infty frac x 2k 2k tfrac 1 2 log left 1 x 2 right Similarly the first term x x1 1 corresponds to the simple pole of the zeta function at 1 It being a pole rather than a zero accounts for the opposite sign of the term Properties EditA theorem due to Erhard Schmidt states that for some explicit positive constant K there are infinitely many natural numbers x such that ps x x lt K x displaystyle psi x x lt K sqrt x and infinitely many natural numbers x such that ps x x gt K x displaystyle psi x x gt K sqrt x 5 6 In little o notation one may write the above as ps x x o x displaystyle psi x x neq o left sqrt x right Hardy and Littlewood 7 prove the stronger result that ps x x o x log log log x displaystyle psi x x neq o left sqrt x log log log x right Relation to primorials EditThe first Chebyshev function is the logarithm of the primorial of x denoted x ϑ x p x log p log p x p log x displaystyle vartheta x sum p leq x log p log prod p leq x p log left x right This proves that the primorial x is asymptotically equal to e 1 o 1 x where o is the little o notation see big O notation and together with the prime number theorem establishes the asymptotic behavior of pn Relation to the prime counting function EditThe Chebyshev function can be related to the prime counting function as follows Define P x n x L n log n displaystyle Pi x sum n leq x frac Lambda n log n Then P x n x L n n x d t t log 2 t 1 log x n x L n 2 x ps t d t t log 2 t ps x log x displaystyle Pi x sum n leq x Lambda n int n x frac dt t log 2 t frac 1 log x sum n leq x Lambda n int 2 x frac psi t dt t log 2 t frac psi x log x The transition from P to the prime counting function p is made through the equation P x p x 1 2 p x 1 3 p x 3 displaystyle Pi x pi x tfrac 1 2 pi left sqrt x right tfrac 1 3 pi left sqrt 3 x right cdots Certainly p x x so for the sake of approximation this last relation can be recast in the form p x P x O x displaystyle pi x Pi x O left sqrt x right The Riemann hypothesis EditThe Riemann hypothesis states that all nontrivial zeros of the zeta function have real part 1 2 In this case x r x and it can be shown that r x r r O x log 2 x displaystyle sum rho frac x rho rho O left sqrt x log 2 x right By the above this implies p x li x O x log x displaystyle pi x operatorname li x O left sqrt x log x right Good evidence that the hypothesis could be true comes from the fact proposed by Alain Connes and others that if we differentiate the von Mangoldt formula with respect to x we get x e u Manipulating we have the Trace formula for the exponential of the Hamiltonian operator satisfying z 1 2 i H n z 1 2 i E n 0 displaystyle left zeta big tfrac 1 2 i hat H big right n geq zeta left tfrac 1 2 iE n right 0 and n e i u E n Z u e u 2 e u 2 d ps 0 d u e u 2 e 3 u e u Tr e i u H displaystyle sum n e iuE n Z u e frac u 2 e frac u 2 frac d psi 0 du frac e frac u 2 e 3u e u operatorname Tr big e iu hat H big where the trigonometric sum can be considered to be the trace of the operator statistical mechanics e iuĤ which is only true if r 1 2 iE n Using the semiclassical approach the potential of H T V satisfies Z u u 1 2 p e i u V x p 4 d x displaystyle frac Z u u frac 1 2 sqrt pi sim int infty infty e i left uV x frac pi 4 right dx with Z u 0 as u solution to this nonlinear integral equation can be obtained among others by V 1 x 4 p d 1 2 d x 1 2 N x displaystyle V 1 x approx sqrt 4 pi cdot frac d frac 1 2 dx frac 1 2 N x in order to obtain the inverse of the potential p N E Arg 3 1 2 i E displaystyle pi N E operatorname Arg xi left tfrac 1 2 iE right Smoothing function Edit The difference of the smoothed Chebyshev function and x 2 2 for x lt 106 The smoothing function is defined as ps 1 x 0 x ps t d t displaystyle psi 1 x int 0 x psi t dt Obviously ps 1 x x 2 2 displaystyle psi 1 x sim frac x 2 2 Variational formulation EditThe Chebyshev function evaluated at x e t minimizes the functional J f 0 f s z s c z s c s c d s 0 0 e s t f s f t d s d t displaystyle J f int 0 infty frac f s zeta s c zeta s c s c ds int 0 infty int 0 infty e st f s f t ds dt so f t ps e t e c t for c gt 0 displaystyle f t psi e t e ct quad text for c gt 0 Notes Edit a b Joshua Knowles 2 May 2014 Multiobjective Optimization Concepts Algorithms and Performance Measures PDF The University of Manchester p 34 Ho Huu V Hartjes S Visser H G Curran R 2018 An improved MOEA D algorithm for bi objective optimization problems with complex Pareto fronts and its application to structural optimization PDF Delft University of Technology Page 6 equation 2 doi 10 1016 j eswa 2017 09 051 a href Template Cite web html title Template Cite web cite web a CS1 maint multiple names authors list link Apostol Tom M 2010 Introduction to Analytic Number Theory Springer pp 75 76 Rosser J Barkley Schoenfeld Lowell 1962 Approximate formulas for some functions of prime numbers Illinois J Math 6 64 94 Pierre Dusart Estimates of some functions over primes without R H arXiv 1002 0442 Pierre Dusart Sharper bounds for ps 8 p pk Rapport de recherche no 1998 06 Universite de Limoges An abbreviated version appeared as The k th prime is greater than k log k log log k 1 for k 2 Mathematics of Computation Vol 68 No 225 1999 pp 411 415 Erhard Schmidt Uber die Anzahl der Primzahlen unter gegebener Grenze Mathematische Annalen 57 1903 pp 195 204 G H Hardy and J E Littlewood Contributions to the Theory of the Riemann Zeta Function and the Theory of the Distribution of Primes Acta Mathematica 41 1916 pp 119 196 Davenport Harold 2000 In Multiplicative Number Theory Springer p 104 ISBN 0 387 95097 4 Google Book Search References EditApostol Tom M 1976 Introduction to analytic number theory Undergraduate Texts in Mathematics New York Heidelberg Springer Verlag ISBN 978 0 387 90163 3 MR 0434929 Zbl 0335 10001External links EditWeisstein Eric W Chebyshev functions MathWorld Mangoldt summatory function PlanetMath Chebyshev functions PlanetMath Riemann s Explicit Formula with images and movies Retrieved from https en wikipedia org w index php title Chebyshev function amp oldid 1132414042, wikipedia, wiki, book, books, library,