`:top
In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, the `!Chebyshev function`! is either a scalarising function (`!Tchebycheff function`!) or one of two related functions. The `!first Chebyshev function`! `*ϑ`* (`*x`*) or `*θ`* (`*x`*) is given by
ϑ ϑ ( x ) = ∑ ∑ p ≤ ≤ x log p {\\displaystyle \\vartheta (x)=\\sum _{p\\leq x}\\log p}
where log {\\displaystyle \\log } denotes the `F33f`_`[natural logarithm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Natural_logarithm]`_`f, with the sum extending over all `F33f`_`[prime numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime_number]`_`f p that are less than or equal to x.
The `!second Chebyshev function`! `*ψ`* (`*x`*) is defined similarly, with the sum extending over all `F33f`_`[prime powers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime_power]`_`f not exceeding x
ψ ψ ( x ) = ∑ ∑ k ∈ ∈ N ∑ ∑ p k ≤ ≤ x log p = ∑ ∑ n ≤ ≤ x Λ Λ ( n ) = ∑ ∑ p ≤ ≤ x ⌊ log p x ⌋ log p , {\\displaystyle \\psi (x)=\\sum _{k\\in \\mathbb {N} }\\sum _{p^{k}\\leq x}\\log p=\\sum _{n\\leq x}\\Lambda (n)=\\sum _{p\\leq x}\\left\\lfloor \\log _{p}x\\right\\rfloor \\log p,}
where Λ is the `F33f`_`[von Mangoldt function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Von_Mangoldt_function]`_`f. The Chebyshev functions, especially the second one `*ψ`* (`*x`*), are often used in `F33f`_`[proofs`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematical_proof]`_`f related to `F33f`_`[prime numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime_number]`_`f, because it is typically simpler to work with them than with the `F33f`_`[prime-counting function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime-counting_function]`_`f, `*π`* (`*x`*) (see `F33f`_`[the exact formula`#the-exact-formula]`_`f below.) Both Chebyshev functions are asymptotic to x, a statement equivalent to the `F33f`_`[prime number theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime_number_theorem]`_`f.
`!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).} `:cite-ref-jk-1-0[`F5bf`_`[1`#cite-note-jk-1]`_`f]
By minimizing this function for different values of w {\\displaystyle w} , one obtains every point on a `F33f`_`[Pareto front`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pareto_front]`_`f, even in the nonconvex parts.`:cite-ref-jk-1-1[`F5bf`_`[1`#cite-note-jk-1]`_`f] 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}^{*}|.} `:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f]
All three functions are named in honour of `F33f`_`[Pafnuty Chebyshev`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pafnuty_Chebyshev]`_`f.
>>Contents
• `F0af`_`[Relationships`#relationships]`_`f
• `F0af`_`[Relationships between ψ ( x )/ x and ϑ ( x )/ x`#relationships-between-x-x-and-x-x]`_`f
• `F0af`_`[Asymptotics and bounds`#asymptotics-and-bounds]`_`f
• `F0af`_`[The exact formula`#the-exact-formula]`_`f
• `F0af`_`[Properties`#properties]`_`f
• `F0af`_`[Relation to primorials`#relation-to-primorials]`_`f
• `F0af`_`[Relation to the prime-counting function`#relation-to-the-prime-counting-function]`_`f
• `F0af`_`[The Riemann hypothesis`#the-riemann-hypothesis]`_`f
• `F0af`_`[Smoothing function`#smoothing-function]`_`f
• `F0af`_`[Notes`#notes]`_`f
• `F0af`_`[References`#references]`_`f
• `F0af`_`[External links`#external-links]`_`f
-─
>>Relationships
The second Chebyshev function can be seen to be related to the first by writing it as
ψ ψ ( x ) = ∑ ∑ p ≤ ≤ x k log p {\\displaystyle \\psi (x)=\\sum _{p\\leq x}k\\log p}
where k is the unique `F33f`_`[integer`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integer]`_`f such that `*p`* `*k`* ≤ `*x`* and `*x`* < `*p`* `*k`* + 1. The values of k are given in `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f: A206722. A more direct relationship is given by
ψ ψ ( x ) = ∑ ∑ n = 1 ∞ ∞ ϑ ϑ ( x 1 n ) . {\\displaystyle \\psi (x)=\\sum _{n=1}^{\\infty }\\vartheta {\\big (}x^{\\frac {1}{n}}{\\big )}.}
This last sum has only a finite number of non-vanishing terms, as
ϑ ϑ ( x 1 n ) = 0 for n > log 2 x = log x log 2 . {\\displaystyle \\vartheta {\\big (}x^{\\frac {1}{n}}{\\big )}=0\\quad {\\text{for}}\\quad n>\\log _{2}x={\\frac {\\log x}{\\log 2}}.}
The second Chebyshev function is the logarithm of the `F33f`_`[least common multiple`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Least_common_multiple]`_`f of the integers from 1 to n.
lcm ( 1 , 2 , … … , n ) = e ψ ψ ( 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 `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f: A003418.
>>Relationships between ψ ( x )/ x and ϑ ( x )/ x
The following `F33f`_`[theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Theorem]`_`f relates the two quotients ψ ψ ( x ) x {\\displaystyle {\\frac {\\psi (x)}{x}}} and ϑ ϑ ( x ) x {\\displaystyle {\\frac {\\vartheta (x)}{x}}} .`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f]
`!Theorem:`! For x > 0 {\\displaystyle x>0} , we have
0 ≤ ≤ ψ ψ ( x ) x − − ϑ 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}}.}
This `F33f`_`[inequality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inequality_(mathematics)]`_`f implies that
lim x → → ∞ ∞ ( ψ ψ ( x ) x − − ϑ 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 ψ ψ ( x ) / x {\\displaystyle \\psi (x)/x} or ϑ ϑ ( x ) / x {\\displaystyle \\vartheta (x)/x} tends to a `F33f`_`[limit`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Limit_of_a_function]`_`f then so does the other, and the two limits are equal.
`!Proof:`! Since ψ ψ ( x ) = ∑ ∑ n ≤ ≤ log 2 x ϑ ϑ ( x 1 / n ) {\\displaystyle \\psi (x)=\\sum _{n\\leq \\log _{2}x}\\vartheta (x^{1/n})} , we find that
0 ≤ ≤ ψ ψ ( x ) − − ϑ ϑ ( x ) = ∑ ∑ 2 ≤ ≤ n ≤ ≤ log 2 x ϑ 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 ϑ ( x ) {\\displaystyle \\vartheta (x)} we have the trivial inequality
ϑ 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 ≤ ≤ ψ ψ ( x ) − − ϑ 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)&\\leq \\sum _{2\\leq n\\leq \\log _{2}x}x^{1/n}\\log(x^{1/n})\\\\&\\leq (\\log _{2}x){\\sqrt {x}}\\log {\\sqrt {x}}\\\\&={\\frac {\\log x}{\\log 2}}{\\frac {\\sqrt {x}}{2}}\\log x\\\\&={\\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
The following bounds are known for the Chebyshev functions:`:ref-dusart1999`a[1]`:ref-dusart2010`a[2] (in these formulas `*p`*`*k`* is the kth prime number; `*p`*1 = 2, `*p`*2 = 3, etc.)
ϑ ϑ ( p k ) ≥ ≥ k ( log k + log log k − − 1 + log log k − − 2.050735 log k ) for k ≥ ≥ 10 11 , ϑ ϑ ( p k ) ≤ ≤ k ( log k + log log k − − 1 + log log k − − 2 log k ) for k ≥ ≥ 198 , | ϑ ϑ ( x ) − − x | ≤ ≤ 0.006788 x log x for x ≥ ≥ 10 544 111 , | ψ ψ ( x ) − − x | ≤ ≤ 0.006409 x log x for x ≥ ≥ e 22 , 0.9999 x < ψ ψ ( x ) − − ϑ ϑ ( x ) < 1.00007 x + 1.78 x 3 for x ≥ ≥ 121. {\\displaystyle {\\begin{aligned}\\vartheta (p_{k})&\\geq k\\left(\\log k+\\log \\log k-1+{\\frac {\\log \\log k-2.050735}{\\log k}}\\right)&&{\\text{for }}k\\geq 10^{11},\\\\[8px]\\vartheta (p_{k})&\\leq k\\left(\\log k+\\log \\log k-1+{\\frac {\\log \\log k-2}{\\log k}}\\right)&&{\\text{for }}k\\geq 198,\\\\[8px]|\\vartheta (x)-x|&\\leq 0.006788\\,{\\frac {x}{\\log x}}&&{\\text{for }}x\\geq 10\\,544\\,111,\\\\[8px]|\\psi (x)-x|&\\leq 0.006409\\,{\\frac {x}{\\log x}}&&{\\text{for }}x\\geq e^{22},\\\\[8px]0.9999{\\sqrt {x}}&<\\psi (x)-\\vartheta (x)<1.00007{\\sqrt {x}}+1.78{\\sqrt[{3}]{x}}&&{\\text{for }}x\\geq 121.\\end{aligned}}}
Furthermore, under the `F33f`_`[Riemann hypothesis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Riemann_hypothesis]`_`f,
| ϑ ϑ ( x ) − − x | = O ( x 1 2 + ε ε ) | ψ ψ ( x ) − − x | = O ( x 1 2 + ε ε ) {\\displaystyle {\\begin{aligned}|\\vartheta (x)-x|&=O{\\Big (}x^{{\\frac {1}{2}}+\\varepsilon }{\\Big )}\\\\|\\psi (x)-x|&=O{\\Big (}x^{{\\frac {1}{2}}+\\varepsilon }{\\Big )}\\end{aligned}}}
for any `*ε`* > 0.
Upper bounds exist for both `*ϑ`* (`*x`*) and `*ψ`* (`*x`*) such that`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f] `:ref-dusart2010`a[3]
ϑ ϑ ( x ) < 1.000028 x ψ ψ ( x ) < 1.03883 x {\\displaystyle {\\begin{aligned}\\vartheta (x)&<1.000028x\\\\\\psi (x)&<1.03883x\\end{aligned}}}
for any `*x`* > 0.
An explanation of the constant 1.03883 is given at `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f: A206431.
>>The exact formula
In 1895, `F33f`_`[Hans Carl Friedrich von Mangoldt`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hans_Carl_Friedrich_von_Mangoldt]`_`f proved`:ref-dav104`a[4] an `F33f`_`[explicit expression`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Explicit_formulae_(L-function)]`_`f for `*ψ`* (`*x`*) as a sum over the nontrivial `F33f`_`[zeros`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zero_of_a_function]`_`f of the `F33f`_`[Riemann zeta function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Riemann_zeta_function]`_`f:
ψ ψ 0 ( x ) = x − − ∑ ∑ ρ ρ x ρ ρ ρ ρ − − ζ ζ ′ ( 0 ) ζ ζ ( 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 `*ζ′ `*(0)/`*ζ`* (0) is log(2π).) Here ρ runs over the nontrivial zeros of the zeta function, and `*ψ`*0 is the same as ψ, except that at its `F33f`_`[jump discontinuities`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Jump_discontinuity]`_`f (the prime powers) it takes the value halfway between the values to the left and the right:
ψ ψ 0 ( x ) = 1 2 ( ∑ ∑ n ≤ ≤ x Λ Λ ( n ) + ∑ ∑ n < x Λ Λ ( n ) ) = { ψ ψ ( x ) − − 1 2 Λ Λ ( x ) x = 2 , 3 , 4 , 5 , 7 , 8 , 9 , 11 , 13 , 16 , … … ψ ψ ( x ) otherwise. {\\displaystyle \\psi _{0}(x)={\\frac {1}{2}}\\!\\left(\\sum _{n\\leq x}\\Lambda (n)+\\sum _{n<x}\\Lambda (n)\\right)={\\begin{cases}\\psi (x)-{\\tfrac {1}{2}}\\Lambda (x)&x=2,3,4,5,7,8,9,11,13,16,\\dots \\\\\\,\\psi (x)&{\\mbox{otherwise.}}\\end{cases}}}
From the `F33f`_`[Taylor series`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Taylor_series]`_`f for the `F33f`_`[logarithm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Natural_logarithm]`_`f, 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.
∑ ∑ 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`* = `*x`*1/1, corresponds to the simple `F33f`_`[pole`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pole_(complex_analysis)]`_`f 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 `F33f`_`[Erhard Schmidt`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Erhard_Schmidt]`_`f states that, for some explicit positive constant K, there are infinitely many `F33f`_`[natural numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Natural_number]`_`f x such that
ψ ψ ( x ) − − x < − − K x {\\displaystyle \\psi (x)-x<-K{\\sqrt {x}}}
and infinitely many natural numbers x such that
ψ ψ ( x ) − − x > K x . {\\displaystyle \\psi (x)-x>K{\\sqrt {x}}.} `:ref-sch03`a[5]`:ref-hard16`a[6]
In `F33f`_`[little-o notation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Big-O_notation]`_`f, one may write the above as
ψ ψ ( x ) − − x ≠ ≠ o ( x ) . {\\displaystyle \\psi (x)-x\\neq o\\left({\\sqrt {x}}\\,\\right).}
`F33f`_`[Hardy`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=G._H._Hardy]`_`f and `F33f`_`[Littlewood`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=J._E._Littlewood]`_`f`:ref-hard16`a[7] prove the stronger result, that
ψ ψ ( 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
The first Chebyshev function is the logarithm of the `F33f`_`[primorial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Primorial]`_`f of x, denoted `*x`* #:
ϑ 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 `F33f`_`[big O notation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Big_O_notation]`_`f) and together with the prime number theorem establishes the asymptotic behavior of `*p`*`*n`* #.
>>Relation to the prime-counting function
The Chebyshev function can be related to the prime-counting function as follows. Define
Π Π ( x ) = ∑ ∑ n ≤ ≤ x Λ Λ ( n ) log n . {\\displaystyle \\Pi (x)=\\sum _{n\\leq x}{\\frac {\\Lambda (n)}{\\log n}}.}
Then
Π Π ( x ) = ∑ ∑ n ≤ ≤ x Λ Λ ( n ) ∫ ∫ n x d t t log 2 t + 1 log x ∑ ∑ n ≤ ≤ x Λ Λ ( n ) = ∫ ∫ 2 x ψ ψ ( t ) d t t log 2 t + ψ ψ ( 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 Π to the `F33f`_`[prime-counting function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime-counting_function]`_`f, π, is made through the equation
Π Π ( x ) = π π ( x ) + 1 2 π π ( x ) + 1 3 π π ( 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 `*π`* (`*x`*) ≤ `*x`*, so for the sake of approximation, this last relation can be recast in the form
π π ( x ) = Π Π ( x ) + O ( x ) . {\\displaystyle \\pi (x)=\\Pi (x)+O\\left({\\sqrt {x}}\\,\\right).}
>>The Riemann hypothesis
The `F33f`_`[Riemann hypothesis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Riemann_hypothesis]`_`f states that all nontrivial `F33f`_`[zeros`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zero_of_a_function]`_`f of the zeta function have `F33f`_`[real part`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_part]`_`f 1/2. In this case, |`*x`* `*ρ`*| = √`*x`*, and it can be shown that
∑ ∑ ρ ρ x ρ ρ ρ ρ = 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
π π ( x ) = li ( x ) + O ( x log x ) . {\\displaystyle \\pi (x)=\\operatorname {li} (x)+O\\!\\left({\\sqrt {x}}\\,\\log x\\right).}
>>Smoothing function
The `!smoothing function`! is defined as
ψ ψ 1 ( x ) = ∫ ∫ 0 x ψ ψ ( t ) d t . {\\displaystyle \\psi _{1}(x)=\\int _{0}^{x}\\psi (t)\\,dt.}
Obviously ψ ψ 1 ( x ) ∼ ∼ x 2 2 . {\\displaystyle \\psi _{1}(x)\\sim {\\frac {x^{2}}{2}}.}
>>Notes
`:cite-note-jk-1`!1.`! `F0af`_`[↑`#cite-ref-jk-1-0]`_`f `:citerefjoshua-knowles2014`aJoshua Knowles (2 May 2014). "Multiobjective Optimization Concepts, Algorithms and Performance Measures" (PDF). The University of Manchester. p. 34.
`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `:citerefho-huuhartjesvissercurran2018`aHo-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). `*Expert Systems with Applications`*. Delft University of Technology. Page 6 equation (2). `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1016/j.eswa.2017.09.051.
`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f `:citerefapostol2010`aApostol, Tom M. (2010). `*Introduction to Analytic Number Theory`*. Springer. pp. 75–76.
`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `:citerefrosserschoenfeld1962`a`F33f`_`[Rosser, J. Barkley`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=J._Barkley_Rosser]`_`f; `F33f`_`[Schoenfeld, Lowell`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lowell_Schoenfeld]`_`f (1962). "Approximate formulas for some functions of prime numbers". `*Illinois J. Math`*. `!6`!: 64–94.
• `:endnote-dusart2010`a`!`F33f`_`[^`#ref-dusart2010]`_`f`! `F33f`_`[Pierre Dusart`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pierre_Dusart]`_`f, "Estimates of some functions over primes without R.H.". `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:1002.0442
• `:endnote-dusart1999`a`!`F33f`_`[^`#ref-dusart1999]`_`f`! Pierre Dusart, "Sharper bounds for ψ, θ, π, `*p`*`*k`*", Rapport de recherche no. 1998-06, Université 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.
• `:endnote-sch03`a`!`F33f`_`[^`#ref-sch03]`_`f`! Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", `*Mathematische Annalen`*, `!57`! (1903), pp. 195–204.
• `:endnote-hard16`a`!`F33f`_`[^`#ref-hard16]`_`f`! 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.
• `:endnote-dav104`a`!`F33f`_`[^`#ref-dav104]`_`f`! `F33f`_`[Davenport, Harold`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Harold_Davenport]`_`f (2000). In `*Multiplicative Number Theory`*. Springer. p. 104. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-387-95097-4. Google Book Search.
>>References
• `:citerefapostol1976`a`F33f`_`[Apostol, Tom M.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tom_M._Apostol]`_`f (1976), `*Introduction to analytic number theory`*, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-387-90163-3, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0434929, `F33f`_`[Zbl`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zbl_(identifier)]`_`f 0335.10001
>>External links
• `:reference-mathworld-chebyshev-functions`a`:citerefweisstein`a`F33f`_`[Weisstein, Eric W.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Eric_W._Weisstein]`_`f "Chebyshev functions". `*`F33f`_`[MathWorld`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MathWorld]`_`f`*.
• "Mangoldt summatory function". `*`F33f`_`[PlanetMath`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=PlanetMath]`_`f`*.
• "Chebyshev functions". `*`F33f`_`[PlanetMath`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=PlanetMath]`_`f`*.
• Riemann's Explicit Formula, with images and movies
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