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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e55b3be4b1bf299b5afefde35dccfab21b0d0608)
![{\ displaystyle \ zeta (s, q) = \ sum _ {n = 0} ^ {\ infty} {\ frac {1} {(q + n) ^ {s}}} \ qquad \ quad \ mathrm {Re } (s)> 1 {\ text {and Re}} (q)> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02ff2af6b38417a72a84edc5378d595bbec1be7d)
The series converges absolutely and can be expanded to a meromorphic function for all
The Riemann zeta function is then
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c65f974817241008129f27b8ad2d3d83f080ed3)
![s = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/bac386d8f227fb823cede9b3e33d706cad3ed306)
It then applies
![{\ displaystyle \ lim _ {s \ to 1} \ left [\ zeta (s, q) - {\ frac {1} {s-1}} \ right] = {\ frac {- \ Gamma '(q) } {\ Gamma (q)}} = - \ psi (q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8c060c6ab7fbd1478eaf1383071b2fae825439)
using the gamma function
and the digamma function
.
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87dab234a5628223556903322fe6ff35531c51c9)
for and .
![{\ displaystyle q> -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4896ed73da5935fb901ba6ad6794a77230494094)
![{\ displaystyle s \ in \ mathbb {C} \ setminus \ {1 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1be3d30caaa037b6818ff8660bbf3045c8643e)
Laurent development
The Laurent development around is:
![s = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/bac386d8f227fb823cede9b3e33d706cad3ed306)
![{\ displaystyle \ zeta (s, q) = {\ frac {1} {s-1}} + \ sum _ {n = 0} ^ {\ infty} {\ frac {(-1) ^ {n} \ gamma _ {n} (q)} {n!}} (s-1) ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/241fc27f9fe20ded88e3398bbe0804a0dc5a1b83)
with . are the generalized Stieltjes constants :
![{\ displaystyle 0 <q \ leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/735026d6a7979a96242b1563f394ac2980134834)
![{\ displaystyle \ gamma _ {n} (q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d57b68ed1955479acbe4143fac703535529097b4)
![{\ 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e351c8aa4810f23abbfb9a8a06c3f556455782b7)
For
![{\ 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64cf3452a00e252e942c8f929e93c2c4720f25f8)
with .
![{\ displaystyle \ mathrm {Re} (s) <1 {\ text {and}} 0 <a \ leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9307427add77288ab6bed78ef961053241c6ef00)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40e65cdffc0ad5b7a5db2bbe80b2340f0a2c5ce6)
where and![{\ displaystyle \ mathrm {Re} (s)> 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb34081987b3a377a3c6025a351386ad5bca5126)
Hurwitz formula
Hurwitz's formula is a representation of the function for and you reads:
![0 \ leq x \ leq 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/30810e06ad49f3a837bd2193d4392eda1f74e7ab)
![{\ displaystyle s> 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7076293bc41507753fe8cffc3b66a7123f16088c)
![{\ 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7618fc3a8a3bd1ced1a9774ae1435a8ee621c3fd)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abcc94c6aafdec9e856e367dd06d735fa8e847e6)
It denotes the polylogarithm .
![{\ displaystyle {\ mbox {Li}} _ {s} (z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad5b2faab5c3fb19d68ca3da7a3358f9c34cc08)
Functional equation
For everyone and applies
![s](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632)
![{\ displaystyle 1 \ leq m \ leq n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f604b1cfaa7b94cc1314405664698ec0fdefde20)
![{\ 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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b75ab58bac451bc8aa0e87fea24834c49d709335)
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 .
![q = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/785938d022f0b0b0bf4b3afa5e1cedceab7a3874)
![q = \ tfrac12](https://wikimedia.org/api/rest_v1/media/math/render/svg/c680c415da368d867658c0099cc2902bab6fd118)
![s](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632)
For these , the Hurwitz zeta function has no zeros with a real part greater than or equal to 1.
![q](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)
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 .
![0 <q <1](https://wikimedia.org/api/rest_v1/media/math/render/svg/12a417c5430831d92ef822cbdea64e4a80386e47)
![{\ displaystyle q \ not = {\ tfrac {1} {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ba697d3aa91a55b4b388b596f2679ceb3b65ff)
![{\ displaystyle 1 <\ mathrm {Re} (s) <1+ \ epsilon}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e43f105cc8253b72fa6a29212561d8d2494f42c4)
![\epsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2)
![q](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)
![q](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)
Rational arguments
The Hurwitz zeta function occurs in connection with the Euler polynomials :
![{\ displaystyle E_ {n} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/178dff9f9cc4d23365f95a655effdc665d21608b)
![{\ 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43ef57bcb61d0efb7f01fa1475efb3625bcc2e26)
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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/565ffa7c75d167f277112c14417280cb599afaed)
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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac4d1c422f93bd76554ea0986c83b501485ef28)
with . And are defined with the legendary Chi function as follows :
![{\ displaystyle 1 \ leq p \ leq q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c79a834aa33d2f02c1a839a5eaa9528d0c3a88d)
![{\ displaystyle C _ {\ nu} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68692aa4cf28881c0bd6d86d61c8664d32dfe568)
![{\ displaystyle \ chi _ {\ nu}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5cd287f67bce29bdcd81e8165dad85178ce10f)
![{\ displaystyle C _ {\ nu} (x) = \ operatorname {Re} \, \ chi _ {\ nu} (e ^ {\ mathrm {i} x})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ee1114fa892baf54d6cd744185a6a258483f81)
or.
![{\ displaystyle S _ {\ nu} (x) = \ operatorname {Im} \, \ chi _ {\ nu} (e ^ {\ mathrm {i} x}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c62854b6c1c6b8da7a9d545310327dd3e348cfa0)
Further
The following applies (selection):
![{\ displaystyle \ zeta (s, -1) = \ zeta (s) +1 \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7fe27726451a6ddf26dbadafa00a2f9e5cfa35)
![{\ displaystyle \ zeta (s, 2) = \ zeta (s) -1 \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1d0639a37ebf52ed144e9b2b073ad174cd9214)
![{\ displaystyle \ zeta (s, 0) = \ zeta (s, 1) \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c43bffefcc00dc161e911bcda20e8ac53de87e9)
![{\ 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b45263834177f01962abbba678d6a9e595e5555)
![{\ displaystyle \ zeta (0, a) = {\ frac {1} {2}} - a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5bc40aec769b13672e366440853ff738cd13022)
![{\ displaystyle \ zeta (2, {\ tfrac {1} {4}}) = \ pi ^ {2} + 8G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0454546396c50d1660d89195a22f923b78d56471)
![{\ displaystyle \ zeta (2, {\ tfrac {1} {2}} + {\ tfrac {x} {\ pi}}) + \ zeta (2, {\ tfrac {1} {2}} - {\ tfrac {x} {\ pi}}) = {\ frac {\ pi ^ {2}} {\ cos ^ {2} x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c22d75606b938dc29708511d0b8734a2f7e89c1)
( Riemann zeta function , Catalan's constant )
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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2a3d24a3622f980e53f2656b935819edeeb143)
with as well as and .
![{\ displaystyle -a \ notin \ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d20c6ee601b38e651ad9de4fedd35f2809875b8)
![{\ displaystyle \ mathrm {Re} (s)> 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb34081987b3a377a3c6025a351386ad5bca5126)
![n \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b)
The derivatives after result in
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![{\ 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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e700ee60e25ebce3a0954a8005d3ef2c8ed3e199)
for and using the Pochhammer symbol .
![{\ displaystyle a \ notin \ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8427ff6c993a9d27942aebd58eaeba4d56c05ca)
![(x, n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d039e585a2a613efa6402139d570f56dd0742e3)
Relationships with other functions
Bernoulli polynomials
The function defined in the Hurwitz formula section generalizes the Bernoulli polynomials :
![B_ {n} (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9192cc7ff70d2e7aff04305da16f00c76a42e1bd)
![{\ displaystyle B_ {n} (x) = - \ mathrm {Re} \ left [(- \ mathrm {i}) ^ {n} \ beta (x; n) \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8e1b31a1773a69cf13670620aaf37e3a228aa7)
Alternatively, you can say that
![{\ displaystyle \ zeta (-n, x) = - {\ frac {B_ {n + 1} (x)} {n + 1}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaca0cc1003948ccba90544f63185b400b0668df)
For that results
![n = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae)
![{\ displaystyle \ zeta (0, x) = {\ frac {1} {2}} - x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89a7e10a1d6c95a98ccfb6af1a8f43befdf74698)
Jacobian theta function
With , the Jacobian theta function applies
![{\ displaystyle \ vartheta (z, \ tau)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4af28b40eca681985b0b75572881cddc54bdc6cd)
![{\ 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77071b8d310ded11644273b78400c320e9fdf6f8)
where and .
![{\ displaystyle \ mathrm {Re} (s)> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd987dcdb8d542aebe5402749f279b391b6b5915)
![{\ displaystyle z \ in \ mathbb {C} \, \ setminus \, \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7afff89865d87363088b07a1f003493bd4d974db)
Is whole, this simplifies itself too
![z = n](https://wikimedia.org/api/rest_v1/media/math/render/svg/e57eb18158a74260ef03b71ed63ad5d67d645eea)
![{\ 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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/def92c652b655e11e871a60624dfdbeef0b6c673)
( with one argument stands for the Riemann zeta function )
![\ zeta](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae)
Polygamma function
The Hurwitz zeta function generalizes the polygamma function to non-whole orders :
![s](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632)
![{\ displaystyle \ psi _ {s} (z) = {\ frac {1} {\ Gamma (-s)}} \ left ({\ frac {\ partial} {\ partial s}} + \ psi (-s ) + \ gamma \ right) \ zeta (s + 1, z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c5cf2ce9be84c28b4a6305a871c9432f640559)
with the Euler-Mascheroni constant .
![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
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
-
,
so that
![{\ displaystyle \ zeta (s, q) = \ Phi (1, s, q).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1362cb282713ca260ead3aa6a69bdd6a36002dd9)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a33c27b9f78ff8e22a8cc740969690c2911273f0)
With
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e24b42a99ec3940da37ab1f3f7cd819401b1c3df)
with .
![{\ displaystyle s \ in \ mathbb {N} ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05ef50ea72eba4cf651f489c9f85efa9ce18f0db)
Literature and web links
- Jonathan Sondow, Eric W. Weisstein: Hurwitz Zeta Function on MathWorld and in functions.wolfram.com (English)
-
Milton Abramowitz , Irene A. Stegun : Handbook of Mathematical Functions . Dover Publications, New York 1964, ISBN 0-486-61272-4 . (See paragraph 6.4.10 )
- Victor S. Adamchik: Derivatives of the Hurwitz Zeta Function for Rational Arguments . In: Journal of Computational and Applied Mathematics . Volume 100, 1998, pp. 201-206.
- Necdet Batit: New inequalities for the Hurwitz zeta function (PDF; 115 kB). In: Proc. Indian Acad. Sci. (Math. Sci.) Vol. 118, No. 4, Nov. 2008, pp. 495-503
- Johan Andersson: Mean Value Properties of the Hurwitz Zeta Function . In: Math. Scand. Volume 71, 1992, pp. 295-300
Individual evidence
-
↑ Helmut Hasse: A summation method for the Riemann ζ series In: Mathematische Zeitschrift. Volume 32, 1930, pp. 458-464.
-
↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/06/03/01/01/0001/
-
↑ Eric W. Weisstein : Hurwitz's Formula . In: MathWorld (English).
-
^ 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
-
^ JWS Cassels: Footnote to a note of Davenport and Heilbronn . In: Journal of the London Mathematical Society. Volume 36, 1961, pp. 177-184
-
↑ 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.
-
↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/03/ShowAll.html
-
↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/20/02/01/01/0001/
-
↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/20/02/02/01/0001/
-
↑ Oliver Espinosa and Victor H. Moll: A Generalized Polygamma Function on arXiv.org e-Print archive 2003.
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^ J. Schwinger: On gauge invariance and vacuum polarization . In: Physical Review . Volume 82, 1951, pp. 664-679.
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↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/26/01/02/01/
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↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/26/02/01/01/