# Hurwitz's zeta function

The Hurwitz zeta function (after Adolf Hurwitz ) is one of the many known zeta functions that plays an important role in analytic number theory , a branch of mathematics .

The formal definition of complex is ${\ displaystyle s, q}$

${\ displaystyle \ zeta (s, q) = \ sum _ {n = 0} ^ {\ infty} {\ frac {1} {(q + n) ^ {s}}} \ qquad \ quad \ mathrm {Re } (s)> 1 {\ text {and Re}} (q)> 0}$

The series converges absolutely and can be expanded to a meromorphic function for all${\ displaystyle s \ not = 1.}$

The Riemann zeta function is then${\ displaystyle \ zeta (s, 1).}$

## Analytical continuation

The Hurwitz zeta function may be a function meromorphic continued to be so for all complex is defined. At is a simple pole with residual 1. ${\ displaystyle s \ not = 1}$${\ displaystyle s = 1}$

It then applies

${\ displaystyle \ lim _ {s \ to 1} \ left [\ zeta (s, q) - {\ frac {1} {s-1}} \ right] = {\ frac {- \ Gamma '(q) } {\ Gamma (q)}} = - \ psi (q)}$

## Series representations

Helmut Hasse found the series representation in 1930

${\ displaystyle \ zeta (s, q) = {\ frac {1} {s-1}} \ sum _ {n = 0} ^ {\ infty} {\ frac {1} {n + 1}} \ sum _ {k = 0} ^ {n} (- 1) ^ {k} {n \ choose k} (q + k) ^ {1-s}}$

for and . ${\ displaystyle q> -1}$${\ displaystyle s \ in \ mathbb {C} \ setminus \ {1 \}}$

### Laurent development

The Laurent development around is: ${\ displaystyle s = 1}$

${\ displaystyle \ zeta (s, q) = {\ frac {1} {s-1}} + \ sum _ {n = 0} ^ {\ infty} {\ frac {(-1) ^ {n} \ gamma _ {n} (q)} {n!}} (s-1) ^ {n}}$

with . are the generalized Stieltjes constants : ${\ displaystyle 0 ${\ displaystyle \ gamma _ {n} (q)}$

${\ displaystyle \ gamma _ {n} (q): = \ lim _ {N \ to \ infty} \ left (\ sum _ {k = 0} ^ {N} {\ frac {\ log ^ {n} ( k + q)} {k + q}} - {\ frac {\ log ^ {n + 1} (N + q)} {n + 1}} \ right)}$

For ${\ displaystyle n = 0,1,2, \ dots}$

### Fourier series

${\ displaystyle \ zeta (s, a) = 2 (2 \ pi) ^ {s-1} \ Gamma (1-s) \ left (\ sin \ left ({\ frac {\ pi s} {2}} \ right) \ sum _ {k = 1} ^ {\ infty} {\ frac {\ cos (2 \ pi ak)} {k ^ {1-s}}} + \ cos \ left ({\ frac {\ pi s} {2}} \ right) \ sum _ {k = 1} ^ {\ infty} {\ frac {\ sin (2 \ pi ak)} {k ^ {1-s}}} \ right)}$

with . ${\ displaystyle \ mathrm {Re} (s) <1 {\ text {and}} 0

## Integral representation

The integral representation is

${\ displaystyle \ zeta (s, q) = {\ frac {1} {\ Gamma (s)}} \ int \ limits _ {0} ^ {\ infty} {\ frac {t ^ {s-1} e ^ {- qt}} {1-e ^ {- t}}} \ mathrm {d} t}$

where and${\ displaystyle \ mathrm {Re} (s)> 1}$${\ displaystyle \ mathrm {Re} (q)> 0}$

## Hurwitz formula

Hurwitz's formula is a representation of the function for and you reads: ${\ displaystyle 0 \ leq x \ leq 1}$${\ displaystyle s> 1.}$

${\ displaystyle \ zeta (1-s, x) = {\ frac {1} {2s}} \ left [e ^ {- \ mathrm {i} \ pi s / 2} \ beta (x; s) + e ^ {\ mathrm {i} \ pi s / 2} \ beta (1-x; s) \ right]}$

in which

${\ displaystyle \ beta (x; s) = 2 \ Gamma (s + 1) \ sum _ {n = 1} ^ {\ infty} {\ frac {\ exp (2 \ pi \ mathrm {i} nx)} {(2 \ pi n) ^ {s}}} = {\ frac {2 \ Gamma (s + 1)} {(2 \ pi) ^ {s}}} {\ mbox {Li}} _ {s} (e ^ {2 \ pi \ mathrm {i} x})}$

It denotes the polylogarithm . ${\ displaystyle {\ mbox {Li}} _ {s} (z)}$

## Functional equation

For everyone and applies ${\ displaystyle s}$${\ displaystyle 1 \ leq m \ leq n}$

${\ displaystyle \ zeta \ left (1-s, {\ frac {m} {n}} \ right) = {\ frac {2 \ Gamma (s)} {(2 \ pi n) ^ {s}}} \ sum _ {k = 1} ^ {n} \ cos \ left ({\ frac {\ pi s} {2}} - {\ frac {2 \ pi km} {n}} \ right) \; \ zeta \ left (s, {\ frac {k} {n}} \ right).}$

## values

### zeropoint

Since for and results in the Riemann zeta function or this multiplied by a simple function of , this leads to the complicated zero point calculation of the Riemann zeta function with the Riemann conjecture . ${\ displaystyle q = 1}$${\ displaystyle q = {\ tfrac {1} {2}}}$${\ displaystyle s}$

For these , the Hurwitz zeta function has no zeros with a real part greater than or equal to 1. ${\ displaystyle q}$

For and , on the other hand, there are zeros for every stiffener with a positive-real . This was proved for rational and non-algebraic-irrational ones by Davenport and Heilbronn ; for algebraic irrational by Cassels . ${\ displaystyle 0 ${\ displaystyle q \ not = {\ tfrac {1} {2}}}$${\ displaystyle 1 <\ mathrm {Re} (s) <1+ \ epsilon}$${\ displaystyle \ epsilon}$${\ displaystyle q}$${\ displaystyle q}$

### Rational arguments

The Hurwitz zeta function occurs in connection with the Euler polynomials : ${\ displaystyle E_ {n} (x)}$

${\ displaystyle E_ {2n-1} \ left ({\ frac {p} {q}} \ right) = (- 1) ^ {n} {\ frac {4 (2n-1)!} {(2 \ pi q) ^ {2n}}} \ sum _ {k = 1} ^ {q} \ zeta \ left (2n, {\ frac {2k-1} {2q}} \ right) \ cos {\ frac {( 2k-1) \ pi p} {q}}}$

and

${\ displaystyle E_ {2n} \ left ({\ frac {p} {q}} \ right) = (- 1) ^ {n} {\ frac {4 (2n)!} {(2 \ pi q) ^ {2n + 1}}} \ sum _ {k = 1} ^ {q} \ zeta \ left (2n + 1, {\ frac {2k-1} {2q}} \ right) \ sin {\ frac {( 2k-1) \ pi p} {q}}.}$

Furthermore applies

${\ displaystyle \ zeta \ left (s, {\ frac {2p-1} {2q}} \ right) = 2 (2q) ^ {s-1} \ sum _ {k = 1} ^ {q} \ left [C_ {s} \ left ({\ frac {k} {q}} \ right) \ cos \ left ({\ frac {(2p-1) \ pi k} {q}} \ right) + S_ {s } \ left ({\ frac {k} {q}} \ right) \ sin \ left ({\ frac {(2p-1) \ pi k} {q}} \ right) \ right]}$

with . And are defined with the legendary Chi function as follows : ${\ displaystyle 1 \ leq p \ leq q}$${\ displaystyle C _ {\ nu} (x)}$${\ displaystyle S _ {\ nu} (x)}$ ${\ displaystyle \ chi _ {\ nu}}$

${\ displaystyle C _ {\ nu} (x) = \ operatorname {Re} \, \ chi _ {\ nu} (e ^ {\ mathrm {i} x})}$

or.

${\ displaystyle S _ {\ nu} (x) = \ operatorname {Im} \, \ chi _ {\ nu} (e ^ {\ mathrm {i} x}).}$

### Further

The following applies (selection):

${\ displaystyle \ zeta (s, -1) = \ zeta (s) +1 \,}$
${\ displaystyle \ zeta (s, 2) = \ zeta (s) -1 \,}$
${\ displaystyle \ zeta (s, 0) = \ zeta (s, 1) \,}$
${\ displaystyle \ zeta \ left (s, {\ frac {m} {n}} \ right) = {\ frac {1} {n}} \ sum _ {k = 1} ^ {n} n ^ {s } \ cdot \ mathrm {Li} _ {s} \ left (e ^ {\ frac {2 \ pi \ mathrm {i} k} {n}} \ right) e ^ {- {\ frac {2 \ pi \ mathrm {i} km} {n}}} \ qquad \ qquad m, n \ in \ mathbb {N} ^ {+} {\ text {and}} m \ leq n}$
${\ displaystyle \ zeta (0, a) = {\ frac {1} {2}} - a}$
${\ displaystyle \ zeta (2, {\ tfrac {1} {4}}) = \ pi ^ {2} + 8G}$
${\ displaystyle \ zeta (2, {\ tfrac {1} {2}} + {\ tfrac {x} {\ pi}}) + \ zeta (2, {\ tfrac {1} {2}} - {\ tfrac {x} {\ pi}}) = {\ frac {\ pi ^ {2}} {\ cos ^ {2} x}}}$

## Derivatives

It applies

${\ displaystyle {\ frac {\ partial ^ {n} \ zeta (s, a)} {\ partial s ^ {n}}} = {\ frac {(-1) ^ {n}} {2 ^ {n }}} \ sum _ {k = 0} ^ {\ infty} {\ frac {\ log ^ {n} \ left ((a + k) ^ {2} \ right)} {\ left ((a + k ) ^ {2} \ right) ^ {s / 2}}}}$

with as well as and . ${\ displaystyle -a \ notin \ mathbb {N}}$${\ displaystyle \ mathrm {Re} (s)> 1}$${\ displaystyle n \ in \ mathbb {N}}$

The derivatives after result in ${\ displaystyle a}$

${\ displaystyle {\ frac {\ partial ^ {n} \ zeta (s, a)} {\ partial a ^ {n}}} = (1-ns, n) \ sum _ {k = 0} ^ {\ infty} {\ frac {1} {(a + k) ^ {n} \ left ((a + k) ^ {2} \ right) ^ {s / 2}}}}$

for and using the Pochhammer symbol . ${\ displaystyle a \ notin \ mathbb {N}}$${\ displaystyle n \ in \ mathbb {N}}$ ${\ displaystyle (x, n)}$

## Relationships with other functions

### Bernoulli polynomials

The function defined in the Hurwitz formula section generalizes the Bernoulli polynomials : ${\ displaystyle \ beta}$ ${\ displaystyle B_ {n} (x)}$

${\ displaystyle B_ {n} (x) = - \ mathrm {Re} \ left [(- \ mathrm {i}) ^ {n} \ beta (x; n) \ right]}$

Alternatively, you can say that

${\ displaystyle \ zeta (-n, x) = - {\ frac {B_ {n + 1} (x)} {n + 1}}.}$

For that results ${\ displaystyle n = 0}$

${\ displaystyle \ zeta (0, x) = {\ frac {1} {2}} - x.}$

### Jacobian theta function

With , the Jacobian theta function applies ${\ displaystyle \ vartheta (z, \ tau)}$

${\ displaystyle \ int \ limits _ {0} ^ {\ infty} \ left [\ vartheta (z, \ mathrm {i} t) -1 \ right] t ^ {s / 2} {\ frac {\ mathrm { d} t} {t}} = \ pi ^ {- (1-s) / 2} \ Gamma \ left ({\ frac {1-s} {2}} \ right) \ left [\ zeta (1- s, z) + \ zeta (1-s, 1-z) \ right]}$

where and . ${\ displaystyle \ mathrm {Re} (s)> 0}$${\ displaystyle z \ in \ mathbb {C} \, \ setminus \, \ mathbb {Z}}$

Is whole, this simplifies itself too ${\ displaystyle z = n}$

${\ displaystyle \ int \ limits _ {0} ^ {\ infty} \ left [\ vartheta (n, \ mathrm {i} t) -1 \ right] t ^ {s / 2} {\ frac {\ mathrm { d} t} {t}} = 2 \ \ pi ^ {- (1-s) / 2} \ \ Gamma \ left ({\ frac {1-s} {2}} \ right) \ zeta (1- s) = 2 \ \ pi ^ {- s / 2} \ \ Gamma \ left ({\ frac {s} {2}} \ right) \ zeta (s).}$

( with one argument stands for the Riemann zeta function ) ${\ displaystyle \ zeta}$

### Polygamma function

The Hurwitz zeta function generalizes the polygamma function to non-whole orders : ${\ displaystyle s}$

${\ displaystyle \ psi _ {s} (z) = {\ frac {1} {\ Gamma (-s)}} \ left ({\ frac {\ partial} {\ partial s}} + \ psi (-s ) + \ gamma \ right) \ zeta (s + 1, z)}$

with the Euler-Mascheroni constant . ${\ displaystyle \ gamma}$

## Occur

Hurwitz's zeta functions are used in various places, not just in number theory . It occurs in fractals and dynamic systems as well as in Zipf's law .

In particle physics , it occurs in a formula by Julian Schwinger , which gives an exact result for the pair formation rate of electrons in fields described in the Dirac equation .

## Special cases and generalizations

A generalization of the Hurwitz zeta function offers

${\ displaystyle \ Phi (z, s, q) = \ sum _ {k = 0} ^ {\ infty} {\ frac {z ^ {k}} {(k + q) ^ {s}}}}$,

so that

${\ displaystyle \ zeta (s, q) = \ Phi (1, s, q).}$

This function is called Lerch's zeta function .

The Hurwitz zeta function can be expressed by the generalized hypergeometric function :

${\ displaystyle \ zeta (s, a) = a ^ {- s} \ cdot {} _ {s + 1} F_ {s} (1, a_ {1}, a_ {2}, \ ldots a_ {s} ; a_ {1} + 1, a_ {2} +1, \ ldots a_ {s} +1; 1)}$

With ${\ displaystyle a_ {1} = a_ {2} = \ ldots = a_ {s} = a {\ text {and}} a \ notin \ mathbb {N} {\ text {and}} s \ in \ mathbb { N} ^ {+}.}$

In addition, with Meijer's G function :

${\ displaystyle \ zeta (s, a) = G \, _ {s + 1, \, s + 1} ^ {\, 1, \, s + 1} \ left (-1 \; \ left | \; {\ begin {matrix} 0,1-a, \ ldots, 1-a \\ 0, -a, \ ldots, -a \ end {matrix}} \ right) \ right.}$

with . ${\ displaystyle s \ in \ mathbb {N} ^ {+}}$

## Individual evidence

1. Helmut Hasse: A summation method for the Riemann ζ series In: Mathematische Zeitschrift. Volume 32, 1930, pp. 458-464.
2. http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/06/03/01/01/0001/
3. Eric W. Weisstein : Hurwitz's Formula . In: MathWorld (English).
4. ^ H. Davenport and H. Heilbronn: On the zeros of certain Dirichlet series . In: Journal of the London Mathematical Society. Volume 11, 1936, pp. 181-185
5. ^ JWS Cassels: Footnote to a note of Davenport and Heilbronn . In: Journal of the London Mathematical Society. Volume 36, 1961, pp. 177-184
6. Đurđe Cvijovic and Jacek Klinowski: Values of the Legendre chi and Hurwitz zeta functions at rational arguments . In: Mathematics of Computation . Volume 68, 1999, pp. 1623-1630.
7. http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/03/ShowAll.html
8. http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/20/02/01/01/0001/
9. http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/20/02/02/01/0001/
10. Oliver Espinosa and Victor H. Moll: A Generalized Polygamma Function on arXiv.org e-Print archive 2003.
11. ^ J. Schwinger: On gauge invariance and vacuum polarization . In: Physical Review . Volume 82, 1951, pp. 664-679.
12. http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/26/01/02/01/
13. http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/26/02/01/01/