The Chebyshev function , also known as Chebyshev function or similar, is one of two number-theoretic functions named after the Russian mathematician Pafnuti Lwowitsch Chebyshev . They gain in importance due to their connection with the prime number counting function and the prime number theorem, and thus the Riemann zeta function .
The first Chebyshev function , usually labeled or , is the sum of the logarithms of the prime numbers up to :
![\ theta \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/228647b7d4a18b6c8c0c390b439a61da8fafec76)
![\ vartheta](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00eaf197c35bbfa391b9477490a4af955416837)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![{\ displaystyle \ vartheta (x) = \ sum _ {p \ leq x \ atop p {\ text {prim}}} \ operatorname {log} (p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ccec04490b780f990e937ce9efe048e4b04515)
The second Chebyshev function is the summed function of the Mangoldt function :
![\ psi (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc)
![{\ displaystyle \ psi (x) = \ sum _ {n = 1} ^ {x} \ Lambda (n) = \ sum _ {p ^ {k} \ leq x} \ operatorname {log} (p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85b71cda0bf7fe2ef87656af545e32c8a934aaa)
where the Mangoldt function is defined as
![\ Lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733)
![{\ displaystyle \ Lambda (n) = {\ begin {cases} \ log (p) & {\ text {if}} n {\ text {can be represented as}} n = p ^ {k} {\ text { , where}} p {\ text {prim,}} k \ in \ mathbb {N} ^ {+} \\ 0 & {\ text {otherwise}} \ end {cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c590ddc903bff82df5cf8c1bc5a636b9c509d618)
Basic properties
The former Chebyshev function can also be represented as
![{\ displaystyle \ vartheta (x) = \ log (x _ {\ #}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2fb58f84a0f71d4fa8b0928d27b5d3042daf37)
where denotes the prime faculty .
![{\ displaystyle x _ {\ #}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1f04e88bf5e9857b5b0b1eb93b2fb983bef31a)
The second can also be written as the logarithm of the least common multiple from 1 to :
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle \ psi (x) = \ operatorname {log} (\ operatorname {kgV} (1,2,3, \ ldots, \ lfloor x \ rfloor))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0c2f3525b38025dd9129b963c54baac07b7b0db)
According to Erhard Schmidt, there are positive real values for every so that
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![{\ displaystyle \ psi (x) -x <-k {\ sqrt {x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2be99086d1f80f54bdcd44a51e3efbf773dcbe8b)
and
![{\ displaystyle \ psi (x) -x> k {\ sqrt {x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a776eb0c90174a15bc15626e45075478f5c8bfdc)
infinitely often.
Asymptotics
It applies
![{\ displaystyle \ lim _ {x \ to \ infty} {\ frac {x} {\ vartheta (x)}} = 1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0986027c0b54b8833f6363b3bd3ddcf113542173)
d. H.
![{\ displaystyle \ vartheta (n) \ sim n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c17c53f79c723427bfd8210469768c4de9ac2b0)
The same applies
![{\ displaystyle \ psi (n) \ sim n. \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29f90df817071017c7e1de7660d421a61d2f70ea)
Pierre Dusart found a number of barriers to the two functions:
![{\ displaystyle \ vartheta (p_ {k}) \ geq k \ left (\ ln k + \ ln \ ln k-1 + {\ frac {\ ln \ ln k-2 {,} 0553} {\ ln k}} \ right), \ qquad k \ geq \ exp (22)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eceebbd6c2714e812604378f0d5c70511c3078ad)
![{\ displaystyle \ vartheta (p_ {k}) \ leq k \ left (\ ln k + \ ln \ ln k-1 + {\ frac {\ ln \ ln k-2} {\ ln k}} \ right), \ qquad k \ geq 198}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24207a7f5a33b29406ea75cc4e22f1153c58f951)
![{\ displaystyle \ psi (p_ {k}) \ leq k \ left (\ ln k + \ ln \ ln k-1 + {\ frac {\ ln \ ln k-2} {\ ln k}} \ right) + 1 {,} 43 {\ sqrt {x}}, \ qquad k \ geq 198}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c68ff5c598a7185ea93fa6799a518a5447aed1b3)
![{\ displaystyle | \ vartheta (x) -x | \ leq 0 {,} 006788 \, {\ frac {x} {\ ln x}}, \ qquad x \ geq 10 {.} 544 {.} 111}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de574cc895b4defe2b2723a3ef5f014286a67e04)
![{\ displaystyle | \ psi (x) -x | \ leq 0 {,} 006409 \, {\ frac {x} {\ ln x}}, \ qquad x \ geq \ exp (22)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f44f5a427da989b1aa4cc4230e62b5b7165820fa)
![{\ displaystyle \ psi (x) - \ vartheta (x) <0 {,} 0000132 \, {\ frac {x} {\ ln x}}, \ qquad x \ geq \ exp (30).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c493a4dc57fc393df2951a59731747ed8d9cca9e)
Relationship between the two functions
It applies
![{\ displaystyle \ psi (x) = \ sum _ {p \ leq x} k \ log p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b13acdec2698b7719326d434d0f1411aacde5609)
where whole and then through and uniquely is determined.
![{\ displaystyle p ^ {k} \ leq x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea179ae590bada353c833bfccfabbdf31dbd100)
![{\ displaystyle p ^ {k + 1} \ geq x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db02b33574545682e2631632faf0e15c4faecd6d)
A more direct connection is created by
![{\ displaystyle \ psi (x) = \ sum _ {n = 1} ^ {\ infty} \ vartheta \ left (x ^ {\ frac {1} {n}} \ right) = \ sum _ {n = 1 } ^ {\ lfloor \ log _ {2} (x) \ rfloor} \ vartheta \ left (x ^ {\ frac {1} {n}} \ right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac792dae53d2d70d261dde422947cab53f2d98c)
Note that for![{\ displaystyle \ vartheta \ left (x ^ {\ frac {1} {n}} \ right) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/379e5606bac3e0103e9d9e5e7a755540c0e2b762)
The "exact formula"
In 1895 Hans Karl Friedrich von Mangoldt proved the following formula, which is also called "explicit formula" in English :
![{\ displaystyle \ psi (x) = x- \ sum _ {\ rho} {\ frac {x ^ {\ rho}} {\ rho}} - \ ln (2 \ pi) - {\ frac {1} { 2}} \ ln \ left (1-x ^ {- 2} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b366c63cc40b43e09bb9abe55dd627785e31af)
Here and is not prime or a prime power and the sum runs over all nontrivial zeros of the Riemann zeta function .
![x> 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/0549e1fb7ee2023519833093c6e3b60236e7d09f)
![\ zeta](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae)
credentials
-
^ Pierre Dusart: Sharper bounds for ψ, θ, π, p k . In: Rapport de recherche n ° 1998-06, Université de Limoges. PDF
-
↑ Eric W. Weisstein : Explicit Formula . In: MathWorld (English).
Web links