§25.16 Mathematical Applications

Keywords:: Riemann zeta function
Referenced by:: §25.1
Permalink:: http://dlmf.nist.gov/25.16

§25.16(i) Distribution of Primes
§25.16(ii) Euler Sums

§25.16(i) Distribution of Primes

Notes:: See Apostol (1976, pp. 75 and 79). For (25.16.2) see Apostol (2000). For (25.16.4) see Ingham (1932, p. 84).
Keywords:: Chebyshev $\psi$ -function, Riemann hypothesis, prime numbers, prime number theorem
Permalink:: http://dlmf.nist.gov/25.16.i

In studying the distribution of primes $p\leq x$ , Chebyshev (1851) introduced a function $\mathop{\psi\/}\nolimits\!\left(x\right)$ (not to be confused with the digamma function used elsewhere in this chapter), given by

25.16.1 $\mathop{\psi\/}\nolimits\!\left(x\right)=\sum _{{m=1}}^{\infty}\sum _{{p^{m}\leq x}}\mathop{\ln\/}\nolimits p,$

Defines:: $\mathop{\psi\/}\nolimits\!\left(x\right)$ : Chebyshev $\mathop{\psi\/}\nolimits$ -function
Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $m$ : nonnegative integer, $p$ : prime number and $x$ : real variable
Permalink:: http://dlmf.nist.gov/25.16.E1
Encodings:: TeX, pMML, png

which is related to the Riemann zeta function by

25.16.2 $\mathop{\psi\/}\nolimits\!\left(x\right)=x-\frac{{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(0\right)}{\mathop{\zeta\/}\nolimits\!\left(0\right)}-\sum _{\rho}\frac{x^{\rho}}{\rho}+\mathop{o\/}\nolimits\!\left(1\right),$ $x\to\infty$ ,

Symbols:: $\mathop{\psi\/}\nolimits\!\left(x\right)$ : Chebyshev $\mathop{\psi\/}\nolimits$ -function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than and $x$ : real variable
Referenced by:: §25.16(i)
Permalink:: http://dlmf.nist.gov/25.16.E2
Encodings:: TeX, pMML, png

where the sum is taken over the nontrivial zeros $\rho$ of $\mathop{\zeta\/}\nolimits\!\left(s\right)$ .

The prime number theorem (27.2.3) is equivalent to the statement

25.16.3 $\mathop{\psi\/}\nolimits\!\left(x\right)=x+\mathop{o\/}\nolimits\!\left(x\right),$ $x\to\infty$ .

Symbols:: $\mathop{\psi\/}\nolimits\!\left(x\right)$ : Chebyshev $\mathop{\psi\/}\nolimits$ -function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than and $x$ : real variable
Referenced by:: §25.10(i)
Permalink:: http://dlmf.nist.gov/25.16.E3
Encodings:: TeX, pMML, png

The Riemann hypothesis is equivalent to the statement

25.16.4 $\mathop{\psi\/}\nolimits\!\left(x\right)=x+\mathop{O\/}\nolimits\!\left(x^{{\frac{1}{2}+\epsilon}}\right),$ $x\to\infty$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\psi\/}\nolimits\!\left(x\right)$ : Chebyshev $\mathop{\psi\/}\nolimits$ -function and $x$ : real variable
Referenced by:: §25.16(i)
Permalink:: http://dlmf.nist.gov/25.16.E4
Encodings:: TeX, pMML, png

for every $\epsilon>0$ .

§25.16(ii) Euler Sums

Notes:: See Apostol and Vu (1984) and Basu and Apostol (2000).
Defines:: $\mathop{H\/}\nolimits\!\left(s,z\right)$ : generalized Euler sums and $\mathop{H\/}\nolimits\!\left(s\right)$ : Euler sums
Keywords:: Euler sums, Riemann zeta function
Permalink:: http://dlmf.nist.gov/25.16.ii

Euler sums have the form

25.16.5 $\mathop{H\/}\nolimits\!\left(s\right)=\sum _{{n=1}}^{\infty}\frac{h(n)}{n^{s}},$

Symbols:: $\mathop{H\/}\nolimits\!\left(s\right)$ : Euler sums, $n$ : nonnegative integer, $s$ : complex variable and $h(n)$ : sum
Permalink:: http://dlmf.nist.gov/25.16.E5
Encodings:: TeX, pMML, png

where $h(n)$ is given by (25.11.33).

$\mathop{H\/}\nolimits\!\left(s\right)$ is analytic for $\realpart{s}>1$ , and can be extended meromorphically into the half-plane $\realpart{s}>-2k$ for every positive integer $k$ by use of the relations

25.16.6 $\mathop{H\/}\nolimits\!\left(s\right)=-{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(s\right)+\EulerConstant\mathop{\zeta\/}\nolimits\!\left(s\right)+\frac{1}{2}\mathop{\zeta\/}\nolimits\!\left(s+1\right)+\sum _{{r=1}}^{k}\mathop{\zeta\/}\nolimits\!\left(1-2r\right)\mathop{\zeta\/}\nolimits\!\left(s+2r\right)+\sum _{{n=1}}^{\infty}\frac{1}{n^{s}}\int _{n}^{\infty}\frac{\mathop{\widetilde{B}_{{2k+1}}\/}\nolimits\!\left(x\right)}{x^{{2k+2}}}dx,$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{H\/}\nolimits\!\left(s\right)$ : Euler sums, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.16.E6
Encodings:: TeX, pMML, png

25.16.7 $\mathop{H\/}\nolimits\!\left(s\right)=\frac{1}{2}\mathop{\zeta\/}\nolimits\!\left(s+1\right)+\frac{\mathop{\zeta\/}\nolimits\!\left(s\right)}{s-1}-\sum _{{r=1}}^{k}\binom{s+2r-2}{2r-1}\mathop{\zeta\/}\nolimits\!\left(1-2r\right)\mathop{\zeta\/}\nolimits\!\left(s+2r\right)-\binom{s+2k}{2k+1}\sum _{{n=1}}^{\infty}\frac{1}{n}\int _{n}^{\infty}\frac{\mathop{\widetilde{B}_{{2k+1}}\/}\nolimits\!\left(x\right)}{x^{{s+2k+1}}}dx.$

Symbols:: $\mathop{H\/}\nolimits\!\left(s\right)$ : Euler sums, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\binom{m}{n}$ : binomial coefficient, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.16.E7
Encodings:: TeX, pMML, png

For integer $s$ ( $\geq 2$ ), $\mathop{H\/}\nolimits\!\left(s\right)$ can be evaluated in terms of the zeta function:

25.16.8

$\mathop{H\/}\nolimits\!\left(2\right)=2\mathop{\zeta\/}\nolimits\!\left(3\right),$

$\mathop{H\/}\nolimits\!\left(3\right)=\tfrac{5}{4}\mathop{\zeta\/}\nolimits\!\left(4\right),$

Symbols:: $\mathop{H\/}\nolimits\!\left(s\right)$ : Euler sums and $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function
Permalink:: http://dlmf.nist.gov/25.16.E8
Encodings:: TeX, TeX, pMML, pMML, png, png

25.16.9 $\mathop{H\/}\nolimits\!\left(a\right)=\frac{a+2}{2}\mathop{\zeta\/}\nolimits\!\left(a+1\right)-\frac{1}{2}\sum _{{r=1}}^{{a-2}}\mathop{\zeta\/}\nolimits\!\left(r+1\right)\mathop{\zeta\/}\nolimits\!\left(a-r\right),$ $a=2,3,4,\dots$ .

Symbols:: $\mathop{H\/}\nolimits\!\left(s\right)$ : Euler sums, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/25.16.E9
Encodings:: TeX, pMML, png

Also,

25.16.10 $\mathop{H\/}\nolimits\!\left(-2a\right)=\frac{1}{2}\mathop{\zeta\/}\nolimits\!\left(1-2a\right)=-\frac{\mathop{B_{{2a}}\/}\nolimits}{4a},$ $a=1,2,3,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{H\/}\nolimits\!\left(s\right)$ : Euler sums, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/25.16.E10
Encodings:: TeX, pMML, png

$\mathop{H\/}\nolimits\!\left(s\right)$ has a simple pole with residue $\mathop{\zeta\/}\nolimits\!\left(1-2r\right)$ ( $=-\mathop{B_{{2r}}\/}\nolimits/(2r)$ ) at each odd negative integer $s=1-2r$ , $r=1,2,3,\dots$ .

$\mathop{H\/}\nolimits\!\left(s\right)$ is the special case $\mathop{H\/}\nolimits\!\left(s,1\right)$ of the function

25.16.11 $\mathop{H\/}\nolimits\!\left(s,z\right)=\sum _{{n=1}}^{\infty}\frac{1}{n^{s}}\sum _{{m=1}}^{n}\frac{1}{m^{z}},$ $\realpart{(s+z)}>1$ ,

Symbols:: $\mathop{H\/}\nolimits\!\left(s,z\right)$ : generalized Euler sums, $\realpart{}$ : real part, $m$ : nonnegative integer, $n$ : nonnegative integer, $s$ : complex variable and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/25.16.E11
Encodings:: TeX, pMML, png

which satisfies the reciprocity law

25.16.12 $\mathop{H\/}\nolimits\!\left(s,z\right)+\mathop{H\/}\nolimits\!\left(z,s\right)=\mathop{\zeta\/}\nolimits\!\left(s\right)\mathop{\zeta\/}\nolimits\!\left(z\right)+\mathop{\zeta\/}\nolimits\!\left(s+z\right),$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{H\/}\nolimits\!\left(s,z\right)$ : generalized Euler sums, $s$ : complex variable and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/25.16.E12
Encodings:: TeX, pMML, png

when both $\mathop{H\/}\nolimits\!\left(s,z\right)$ and $\mathop{H\/}\nolimits\!\left(z,s\right)$ are finite.

For further properties of $\mathop{H\/}\nolimits\!\left(s,z\right)$ see Apostol and Vu (1984). Related results are:

25.16.13 $\sum _{{n=1}}^{\infty}\left(\frac{h(n)}{n}\right)^{2}=\frac{17}{4}\mathop{\zeta\/}\nolimits\!\left(4\right),$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $n$ : nonnegative integer and $h(n)$ : sum
Permalink:: http://dlmf.nist.gov/25.16.E13
Encodings:: TeX, pMML, png

25.16.14 $\sum _{{r=1}}^{\infty}\sum _{{k=1}}^{r}\frac{1}{rk(r+k)}=\frac{5}{4}\mathop{\zeta\/}\nolimits\!\left(3\right),$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.16.E14
Encodings:: TeX, pMML, png

25.16.15 $\sum _{{r=1}}^{\infty}\sum _{{k=1}}^{r}\frac{1}{r^{2}(r+k)}=\frac{3}{4}\mathop{\zeta\/}\nolimits\!\left(3\right).$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.16.E15
Encodings:: TeX, pMML, png

For further generalizations, see Flajolet and Salvy (1998).

§25.16 Mathematical Applications

Contents

§25.16(i) Distribution of Primes

§25.16(ii) Euler Sums