27 Functions of Number Theory Multiplicative Number Theory 27.10 Periodic Number-Theoretic Functions 27.12 Asymptotic Formulas: Primes

§27.11 Asymptotic Formulas: Partial Sums

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Keywords:: Dirichlet divisor problem, Dirichlet’s divisor problem, Dirichlet’s theorem, in arithmetic progressions, number theory, number-theoretic functions, prime number theorem, prime numbers, prime numbers in arithmetic progression
Notes:: See Apostol (1976, Chapters 3, 4). For (27.11.12), (27.11.14), and (27.11.15) see Prachar (1957, pp. 71–74).
Referenced by:: §25.15(ii), Erratum (V1.0.19) for Notation, Erratum (V1.1.8) for Section 27.11
Permalink:: http://dlmf.nist.gov/27.11
Change of Notation (effective with 1.1.8):: Immediately below (27.11.2), the bound $\theta_{0}$ for Dirichlet’s divisor problem (currently still unsolved) has been changed from $\frac{12}{37}$ Kolesnik (1969) to $\frac{131}{416}$ Huxley (2003).
See also:: Annotations for Ch.27

The behavior of a number-theoretic function for large is often difficult to determine because the function values can fluctuate considerably as increases. It is more fruitful to study partial sums and seek asymptotic formulas of the form

27.11.1 $\sum_{n\leq x}f(n)=F(x)+O\left(g(x)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, : positive integer and : real number
Permalink:: http://dlmf.nist.gov/27.11.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §27.11 and Ch.27

where is a known function of , and $O\left(g(x)\right)$ represents the error, a function of smaller order than for all in some prescribed range. For example, Dirichlet (1849) proves that for all $x\geq 1$ ,

27.11.2 $\sum_{n\leq x}d\left(n\right)=x\ln x+(2\gamma-1)x+O\left(\sqrt{x}\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\gamma$ : Euler’s constant, $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $\ln\NVar{z}$ : principal branch of logarithm function, : positive integer and : real number
A&S Ref:: 24.3.3 III
Referenced by:: §27.11, §27.11, Erratum (V1.1.8) for Section 27.11
Permalink:: http://dlmf.nist.gov/27.11.E2
Encodings:: TeX, pMML, png
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

where $\gamma$ is Euler’s constant (§5.2(ii)). Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number $\theta_{0}$ such that the error term in (27.11.2) is $O\left(x^{\theta}\right)$ for all $\theta>\theta_{0}$ . Huxley (2003) proves that $\theta_{0}\leq\frac{131}{416}$ .

Equations (27.11.3)–(27.11.11) list further asymptotic formulas related to some of the functions listed in §27.2. They are valid for all $x\geq 2$ . The error terms given here are not necessarily the best known.

27.11.3 $\sum_{n\leq x}\frac{d\left(n\right)}{n}=\frac{1}{2}(\ln x)^{2}+2\gamma\ln x+O% \left(1\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\gamma$ : Euler’s constant, $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $\ln\NVar{z}$ : principal branch of logarithm function, : positive integer and : real number
Referenced by:: §27.11
Permalink:: http://dlmf.nist.gov/27.11.E3
Encodings:: TeX, pMML, png
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

where $\gamma$ again is Euler’s constant.

27.11.4 $\sum_{n\leq x}\sigma_{1}\left(n\right)=\frac{{\pi}^{2}}{12}x^{2}+O\left(x\ln x% \right).$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$ : sum of powers of divisors of a number, $\ln\NVar{z}$ : principal branch of logarithm function, : positive integer and : real number
A&S Ref:: 24.3.3 III (in slightly different form)
Permalink:: http://dlmf.nist.gov/27.11.E4
Encodings:: TeX, pMML, png
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

27.11.5 $\sum_{n\leq x}\sigma_{\alpha}\left(n\right)=\frac{\zeta\left(\alpha+1\right)}{% \alpha+1}x^{\alpha+1}+O\left(x^{\beta}\right),$ $\alpha>0$ , $\alpha\neq 1$ , $\beta=\max(1,\alpha)$ .

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$ : sum of powers of divisors of a number, : positive integer and : real number
Permalink:: http://dlmf.nist.gov/27.11.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §27.11 and Ch.27

27.11.6 $\sum_{n\leq x}\phi\left(n\right)=\frac{3}{{\pi}^{2}}x^{2}+O\left(x\ln x\right).$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, : positive integer and : real number
A&S Ref:: 24.3.2 III (in slightly different form)
Permalink:: http://dlmf.nist.gov/27.11.E6
Encodings:: TeX, pMML, png
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

27.11.7 $\sum_{n\leq x}\frac{\phi\left(n\right)}{n}=\frac{6}{{\pi}^{2}}x+O\left(\ln x% \right).$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, : positive integer and : real number
Permalink:: http://dlmf.nist.gov/27.11.E7
Encodings:: TeX, pMML, png
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

27.11.8 $\sum_{p\leq x}\frac{1}{p}=\ln\ln x+A+O\left(\frac{1}{\ln x}\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\ln\NVar{z}$ : principal branch of logarithm function, $p,p_{1},\ldots$ : prime numbers, : real number and : constant
Permalink:: http://dlmf.nist.gov/27.11.E8
Encodings:: TeX, pMML, png
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

where is a constant.

27.11.9 $\sum_{\begin{subarray}{c}p\leq x\\ p\equiv h\!\!\!\!\!\pmod{k}\end{subarray}}\frac{1}{p}=\frac{1}{\phi\left(k% \right)}\ln\ln x+B+O\left(\frac{1}{\ln x}\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\equiv$ : modular equivalence, $\ln\NVar{z}$ : principal branch of logarithm function, : positive integer, $p,p_{1},\ldots$ : prime numbers and : real number
Referenced by:: §27.11
Permalink:: http://dlmf.nist.gov/27.11.E9
Encodings:: TeX, pMML, png
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

where $\left(h,k\right)=1$ , , and is a constant depending on and .

27.11.10 $\sum_{p\leq x}\frac{\ln p}{p}=\ln x+O\left(1\right).$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\ln\NVar{z}$ : principal branch of logarithm function, $p,p_{1},\ldots$ : prime numbers and : real number
Permalink:: http://dlmf.nist.gov/27.11.E10
Encodings:: TeX, pMML, png
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

27.11.11 $\sum_{\begin{subarray}{c}p\leq x\\ p\equiv h\!\!\!\!\!\pmod{k}\end{subarray}}\frac{\ln p}{p}=\frac{1}{\phi\left(k% \right)}\ln x+O\left(1\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\equiv$ : modular equivalence, $\ln\NVar{z}$ : principal branch of logarithm function, : positive integer, $p,p_{1},\ldots$ : prime numbers and : real number
Referenced by:: §27.11, §27.11
Permalink:: http://dlmf.nist.gov/27.11.E11
Encodings:: TeX, pMML, png
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

where $\left(h,k\right)=1$ , .

Letting $x\to\infty$ in (27.11.9) or in (27.11.11) we see that there are infinitely many primes $p\equiv h\pmod{k}$ if are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions.

27.11.12 $\sum_{n\leq x}\mu\left(n\right)=O\left(xe^{-C\sqrt{\ln x}}\right),$ $x\to\infty$ ,

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\mu\left(\NVar{n}\right)$ : Möbius function, $\mathrm{e}$ : base of natural logarithm, $\ln\NVar{z}$ : principal branch of logarithm function, : positive integer, : real number and : constant
A&S Ref:: 24.3.1 III
Referenced by:: §27.11
Permalink:: http://dlmf.nist.gov/27.11.E12
Encodings:: TeX, pMML, png
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

for some positive constant ,

27.11.13 $\lim_{x\to\infty}\frac{1}{x}\sum_{n\leq x}\mu\left(n\right)=0,$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer and : real number
Referenced by:: §27.11
Permalink:: http://dlmf.nist.gov/27.11.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §27.11 and Ch.27

27.11.14 $\lim_{x\to\infty}\sum_{n\leq x}\frac{\mu\left(n\right)}{n}=0,$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer and : real number
A&S Ref:: 24.3.1 III
Referenced by:: §27.11
Permalink:: http://dlmf.nist.gov/27.11.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §27.11 and Ch.27

27.11.15 $\lim_{x\to\infty}\sum_{n\leq x}\frac{\mu\left(n\right)\ln n}{n}=-1.$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, $\ln\NVar{z}$ : principal branch of logarithm function, : positive integer and : real number
A&S Ref:: 24.3.1 III
Referenced by:: §27.11, §27.11
Permalink:: http://dlmf.nist.gov/27.11.E15
Encodings:: TeX, pMML, png
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3). The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if $\left(h,k\right)=1$ , then the number of primes $p\leq x$ with $p\equiv h\pmod{k}$ is asymptotic to $x/(\phi\left(k\right)\ln x)$ as $x\to\infty$ .

27.10 Periodic Number-Theoretic Functions 27.12 Asymptotic Formulas: Primes

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