elements of gamma, bessel and zeta function theory 1...
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1
Elements of Gamma, Bessel and Zeta function theory
1. Definitions
The reference to the below are modified Bessel-Hankel functions
)(/)(arctan:)( xJxYx
and
2)1(22
2 )(2)()(2:)2( xHxYxJx ,
which are linked by the relation
x
dxdx )2(
4
12
.
The functions are used to define and investigate the function
)1
(1
)2
()2(1
:)( 2222xxx
xx
x
for 1)2Re(1 and 0)2Re( s with its Mellin transform
)2
()2
()2
(
2cos
22
cos
2)(2
0
22
2/1
sss
s
dxx ss .
2
2. Gamma function theory
An equivalent formulation to Jacobi’s theta function for
1
2
)( xnex (see [12] H.
Hamburger, [13] N. Nielsen, chapter 13, §71) is the series
(2.1)
122
1
2 12121cot
nx
x
xexii nx
It further holds (see [13] N. Nielsen, chapter 1,11,13, §4, 60, 68, 71, 73, [1] B. C. Berndt,
chapter 6, example 2, chapter 8, entry 4/5/17(v))
(2.2) x
dx
x
xx
x
x
x
xx
ss
1
0
1
1)(
)(
)1(
)1(cot for 1)Re(0 s
(2.3) and with
)111
)1(
)1(
1 xnnnx
x
xn
(2.4) )21(1
2sin2)1
1
1cot
122
2
11
nx
x
xkx
xnxnx
.
The symmetric Beta function (see [13] N. Nielsen, chapter 11, §60, chapter X, §53)
(2.5) x
dx
x
x
x
dx
x
xx
sssB
sss
0
1
0
1
11sin)1,(
for 1)Re(0 s
with its relations to [25] E. C. Titchmarsh, 2.11)
(2.6)
x
dx
x
x
s
s s
0
2)1(sin
)1(
for 1)Re(0 s
plays a key role in the Mellin’s inverse theorem, with its relation to Ramanujan series ([1] B.
C. Berndt, chapter 4 ff.) in the form
(2.7) x
dxnxxs
s
ns
1
0 0
)()()(sin
and its link to the Zeta function
(see [8] H.M. Edwards 10.10) representations (integration by
parts ss xsxdx
d )1()( 1 )
0
1
2
2 1
)1(
)1()(
sin
)1(dxx
xx
x
dx
ds
s
s s
,
1)Re(0 s
0
1
)1(
)1(log)(
sin x
dxx
x
xx
dx
ds
s
s
, 1)Re(0 s
3
and its relation to Ramanujan’s infinite series analysis in the form
(2.8)
0
1
0
))(1(log)(sin x
dxxxnxs
s
sn
, 1)Re(0 s ,
leading to the functional equation in the form ([25] E. C. Titchmarsh, 2.9)
)1()1()2()(
2sin
sss
s
s
.
Remark From [13] N. Nielsen, chapter 13, 14 § 68, 78) we recall
1
00
)(logsinlog2
1)
2(log)
2
1(log)
2,
2
1(
2
1log dxxxdx
ssssB
0
32
)1(2
1log)
2,
2
1(
2
1log dt
et
eeeesB
t
txttt
as starting point for Kummer’s series representation of
1
)2sin(log
sinlog2
1log)1()2log)(
2
1()(log nx
n
nxxxx
.
Asymptotically it holds (see [8] H.M. Edwards 10.10, [13] N. Nielsen, chapter 13, §73, [25] E.
C. Titchmarsh, 2.15.8)
0log2
1
log)1(loglog1
)(
)(log
)1(
)1(
x
x
xxxxx
xx
xx
x
x .
1
0
1
1
1
)(
)()(glo dt
t
t
x
xx
x
With reference to (1.20) we further mention (see [13] N. Nielsen, chapter 9, §51)
x
dx
x
xxs
ss
1
0
1
1cot
for 1)Re(0 s
x
dx
x
xx
s
s ss
1
0
1
log1log for 1)Re(0 s
x
dxxxarc
ss
s
1
0
)cot(
2cos
for 1)Re(0 s
4
x
dxx
xss
s)1
1log(
2sin
2
1
0
for 2)Re(0 s
xx
dx
x
xxs
ss
log1)
2tan(log
1
0
1
p. 153
1
1
0
2/31 )2cos(2log
log)1(
2)sin(log
n
ns
t
dt
tt
xxxs
ss
10 s
1
)cos()
2sin(2log
n
nss
see [13] N. Nielsen, Integrallogarithmus, chapter 1, §7 .
Using the abbreviation
(2.9) )()1(:)( xfxfxf
n
nsss
n
s
........*2*1
))....(2)(1(:
1
, 1:
0
1
s
we recall Mellin’s inverse formula from [13] N. Nielsen, chapter 9, §45-53, chapter 16, §86-
92, with its relation to Ramanujan’s Master Theorem ([1] B. C. Berndt, The first quarterly
report, 1.2 Theorem I (Ramanujan’s Master Theorem) and Theorem (Hardy)) in the form
)()()( 1
0
nndxxxF n
for kxk
kxF )(
!
)()(
0
in the neighborhood of 0x .
Lemma 2.1 (Mellin inverse formula):
For a function
(2.10)
0
1)1(:)(
n
sWsW n for 0)Re( s
fulfilling 0)(
lim kn
nW for n
the following series define analytical functions for 2/1)Re( s resp. 2/1)Re( s
(2.11)
0
)1()1(:)( nnn xWsf
and
0
)1()1(:)( nn xnWsg .
If )(sW can be represented in the form
1
0
1)()( dxxxsW s it further holds
(2.12)
1
0)1(1
)()(
sx
dxxsf
and
1
01
)()(
xs
dxxsg
,
5
whereby both functions (1.28) are analytical (probably with the exception on the x-axis for
1x resp. 1x ). With respect to (1.27) it further holds (with variable substitution
t
tx
x
xt
11)
dxxx
gx
dtxt
tf
x
dx
x
xxsW
s
st
s
s1
0
1
0
1
0
)1
(1
1
)1
1(
)1(
)1()(
sin
for
1)Re(0 s
(2.13) dsxs
sW
ix
xf
xg
x
ia
ia
s
sin
)(
2
1
1
)1
1(
)1
(1
for
10 a and )( with iexx
Corollary 2.2 For
0
))(2
1(:)( nxnxg it holds
(2.14)
0 0
1)()1
(1
)2
1(
sin x
dxxxg
x
dxx
xg
xs
s
ss
and it holds ?)(0
divergentdttg
and dtxtt
x
0
22)(sinh
1
sinh
12)(
)
(2.15) x
dxdttg
xnxgxs
ss s
010
1 )(1
)()2
1(
sin)1(
Remark 2.3 From [26] G.N. Watson 13-2 we recall
(2.16) x
dxebxJx
ba
sbax
s
s
s
s
)(
)(
)2
1()2(
1
02
1
22
1
From (2.16) it follows
i) x
dxexJxs x
s
s
)(2)2
1( 1
0
for 0)Re( s
ii) Riemann’s duality equation can be proven by a self-reciprocal function argument (see [25]
E. C. Titchmarsh, 2.7), i.e.
0
)sin()(2
)( dyxyyfxf
which holds for
6
2
1
1
1)(
2 xexf
x
.
For
0
sinh
sinh)( dt
tx
ex
tx
it holds
dtdyxyJeyt
dyxy
xydt
t
eydyxyyx ty
ty
0 0
2/1
sinh
0 0
sinh
0
)(sinh
1)sin(2
sinh)sin()()(
and therefore
dtxtt
xdyxyy
0
22
0)(sinh
1
sinh
12)sin()(
with dyexyJyxt
x ty sinh
2/1
0
2/1
22
2/1
)()(sinh
)2(
and
x
xxJ
sin2)(2/1
.
iii) for nx
ss enxJxh
1
1 )(:)( a Dirichlet type series can be defined in the form
(2.17) x
dxxhxss s
s )(2)()2
1(
0
for
1)Re( s
Proof of corollary 2.2 It holds
1
0
2/1
1
0
2/1
1)1(1
)2
1(:)(
tt
dt
t
te
x
dxxessW
s
t
t
sx
leads to
0
))(2
1()( nxnxg with
0
)1
(1
)2
1(
sin x
dxx
xg
xs
s
s
Examples 2.4
i)
n
s
ndxxxdxx
ssW
nss
1
1
)1()(
1:)(
1
0 0
1
0
11
1 , 0)Re( s ,
ii)
0
1
0
1
1
0
11
2 )(11sin
)1,(:)( dxxxx
dx
x
xxdx
x
x
sssBsW s
sss
ii)
1
0
1
1
0
1
3 )(1sin
coscot:)( dxxx
x
dx
x
xx
x
xxsW s
ss
for 1)Re(0 s
iii)
1
0
1
0
1
0
1
4 )(1
1
1
)1(
1
1
)(
)(:)( dxxx
n
s
ndx
x
x
s
ssW s
ns
, 0)Re( s ,
7
1
0
1
)1(
log
)(
)(
sindxx
x
x
s
s
s
s
s
iv) ??)()(:)(
1
0
1
5 dxxxssW s
0
1
0
5 )()1
(1
?)(sin x
dxxxg
x
dxx
xg
xsW
s
ss
with
0 1
4 ))(1()1())(1(?)( nn xnxnWxg .
3. Bessel function theory
The Bessel functions (see [26] G.N. Watson 13-74) in the form
(3.1)
)
2()
2(2)(:)( 2
2
2
22
xY
xJxxNxN with
2
1:
can be used to define an appropriate new measure dxxNxdN )(:)( .
From (3.2) below it’s being deduced (see [26] G.N. Watson 13-74) that )(2
xN is an
increasing function when 1 , using the fact that t tanh is an increasing function of ,
when 0 and therefore the last term in (2.3) below is negative or positive according to
1 or 10 . The analysis below is applicable for 10 ; we restrict ourselves to
the critical value 2
1: .
)(2
xN has a representation in the form (see [26] G.N. Watson 6-22, 6-3, 7-31, 13-74, 13-
75)
(3.2) dttttttxKxN
tanhtanhcoshtanh)sinh(32
)(0
0
2
with
(3.3)
t
dt
t
eedtedt
t
xtdtexK
txxtixtx
21
22
1
1
)cos()( sinh
02
0
cosh
0
from which it’s being deduced (see [26] G.N. Watson 13-74) that )(2
xN is an increasing
function when 1 , using the fact that t tanh is an increasing function of , when
0 and therefore the last term in (2.3) below is negative or positive according to 1 or
8
10 . The analysis below is applicable for 10 ; we restrict ourselves to the critical
value 2
1: .
Lemma 3.1 The function )(xN can be represented as infinite integral in the form
0
sinh
sinh)(
t
dte
xxN tx
Proof of lemma 3.1 We recall from [11] I.S. Gradshteyn, I.M. Ryzhik (6.518) the formula
(3.4)
0
2
222
)sinh2()()(cos
dttxKxYxJ
for
2
1)Re(
2
1 .
Putting 4
1: , using
2
1
4sin
4cos
and
x
exK
x
2
)(2/1
leads to
0
sinh
20 2
12sinh22
12)sinh(4
cos
2)( dttx
exdttxKxxN
tx
(3.5) i.e.
0
sinh
sinh
1)(
t
dtexxN tx
Corollary 3.1 The function
(3.6)
0
sinh2
4
14
12
4
14
1
sinh)
2()()
2()(2:)(
t
dtexYx
xJxx
tx
and its Fourier transform
can be represented as infinite integral
(3.7)
0
sinh
sinh)( dt
t
ex
tx
,
0
sinhsinh)( dtetx tx
(3.8)
0
sinh4
sinh
1
4
1)(ˆ
2
1dte
tx t
x
,
0
2
sinh4
sinh224
1)(ˆ
2
1
t
dtex t
x
.
9
Proof of corollary 3.1: It holds
(3.9)
00
sinh2 ))(,(
sinh2
1)(
2
12
dttxfdtt
ex
tx
whereby 2)(
)(2
1:))(,( yte
ttyf
with tt sinh:)( and its Fourier transform
(3.10) )(4
2
4)(
1),(ˆ te
tf
.
It follows
(3.11)
0
sinh42
2
sinh
1
4
1)(ˆ
2
1dte
tx t
From [26] G.N. Watson 6-22, 13-72 and 13-75 we note
dttexK tix
2
cosh2
1)(
4cos sinh
2
resp.
dttexK tx
0
cosh
22
cosh)(
dttxKxK )cosh2(2:)(0
2
2
.
From [11] I.S. Gradshteyn, I.M. Ryzhik (6.518), (6.544) we note
0
2
2
2
2
2
)sinh2()()(
2cos
dttxKxYxJ
for 1)Re(1
0
sinh2
2
0
2
2
2
2
2 22
sinh
2
1)sinh()(
1)sinh2(
de
x
tdtK
xK
xtxK
tx
From [26] G.N. Watson 15-61 we note for 2
:
)()(4
)( 222
2
2 rYrJreK
i
.
From [26] G.N. Watson 7-15 we further note for 2
1:
mxm
mm
x
xY
xJ 2
222222
0
22
!
))12()...(3)(1()12...(*3*1
4)
2()
2(
.
10
Remark 3.4 An alternative representation of the function )(xN (see [26] G.N. Watson 7-31,
15-5, 15-53) is given by
(3.12)
dx
dxYxJx
1)()(
2
22
resp. )(1
8
)(
8)2(
xxxN
whereby
)(
)(arctan:)(
xJ
xYx
with
0)()(
)/(222
xYxJ
x
dx
d
and
)(
)(arctan:)(
0
0
xP
xQx with
)()(
)/(222
0 0xQxP
x
dx
d
with .......16384
3417
512
33
8
1)(tan)(
53 xxxxx
and )(0 xP and )(0 xQ appropriate polynomials related to )(0 xJ , )(0 xY defined by
(3.13)
)()
4sin()()
4cos(
2)(0 xQxxPx
xxJ
)()
4cos()()
4sin(
2)(0 xQxxPx
xxY
It holds (see [26] G.N. Watson 15-52)
(3.14) )(42
)( xxx
and
0)(
)()(1
)(
)(2
22
0
0
0
0
0
xP
xQxP
xP
xQ
dx
d
with link to e.g. Euler’s investigations of the zeros of
(3.15)
1
00 )1()02()2(n n
xJxJ
and to e.g. the measure
(3.16) xdxQxPxd log)()(:)( 22 .
With reference to (1.31) and (2.2) above we mention (see [26] G.N. Watson 13-21)
(3.17)
xx
t
t
dttK
t
dte )(0 .
With reference to remark 1.1 we mention Hadamard’s formula ([8] H.M. Edwards 2.1)
(3.18)
)1()0()(s
s .
11
We note that for
)(
)(arctan:)(
4/1
4/1
xJ
xYx it holds
(3.19) x
dx
x
dxN
)(
8
1
))(8
1(
x
dx
x
dxN
which leads with proposition 1.3 to the following formula
(3.20) )2
1()
2()
2
1()
1()(
20
1
sss
x
d
xNxNx s
()(
8:)(
xNdx
dxQ
!??!!!!).
Remark A
From [26] G.N. Watson 14-4, 14-42 we recall Hankel’s repeated integral and its inversion, i.e.
Let )(RF be an arbitrary function of the real variable R subject to the condition that
0
)( dRRRF exists and is absolutely convergent; and let the order of of the Bessel
functions be not less than 2/1 . Then
)0()0(2
1)()()()()()(
0 00 0
rFrFRdRuduurJuRJRFRdRurJuRJRFudu
provided that the positive number r lies inside an interval in which )(RF has limited
total fluctuation.
From [26] G.N. Watson 18-24 we recall the sum of the Fourier-Bessel expansion for a given
function, i.e.
Let )(tf be a function defined arbitrarily in the interval )1,0( ; and let
0
)( dtttf exists
and let it be absolutely convergent. Let
01
2)()(
)(
2dttjJttf
jJa m
m
m
where 02
1 . Let x be any interval point of an interval ),( ba such that 10 ba
and such that )(tf has limited total fluctuation in ),( ba . Then the series
1
)( xjJa mm
is convergent and its sum is given by
1
)0()0(2
1)( rfrfxjJa mm
.
12
4. Zeta function theory
The Zeta function )(s can be defined in the critical stripe 1)Re(0 s as a complex-
valued transform of an integral operator with normal distribution measure, i.e. for
(4.1)
1
:)( nxexR
it holds (see [25] E.C. Titchmarsh, 2.11)
(4.2) x
dxdte
xxRxss ts
00
1)()()( for 1)Re(0 s .
Müntz’ formula (see [25] E.C. Titchmarsh, 2.11) gives the Zeta function )(s as Mellin
transform of an integral operator in a more general form, i.e. it holds
Lemma 4.1 (Müntz’ formula) For )(),( xx continuous and bounded in any finite interval
with )()( xox and )()( xox for x and 1, it holds
(4.3)
x
dxdtt
xnxx
x
dxxxs ss
0100
)(1
)()(
)(
for 1)Re(0 s .
Proof:
i) because )(x is continuous and bounded in any finite interval with )()( xox it holds
1 0
1 )(1
dxxxn
s
s exists for 1
i.e. the inversion leading to the left hand side of (4.3) is justified.
ii) )1())(()1()()()()(/1
/1
0001
OdtxtOxdtOxdttttxdtxtnxx
x
The first summand is justified, because )(x is continuous and bounded in any finite interval
the second summand is justified, because )()( xox , i.e. it holds
x
cOnx
)1()(1
with dttc
0
)( .
Hence
13
1)()()(
111
1
010
s
c
x
dxnxx
x
dx
x
cnxx
x
dxnxx sss
for 0 except 1s . Also
11
2
s
cdxxc s
for 1
and therefore (4.3) for 1)Re(0 s
Remark 4.2
concerning an application of Polya’s criterion in combination with an application of Müntz
formula it holds from [19] G. Polya, p. 365:
Polya's criterion cannot be applied to Muentz's formula.
Polya's criterion is for an integral over a finite interval and to extend it to an infinite interval it
needs certain conditions, see the notes by R.P. Boas in the second volume of Polya's
collected works.
In order to apply Polya's criterion to Muentz's formula one needs to show that the function
xnxxG
1)(:)(
1
*
is positive and increasing for 0x . It does not suffice to show this only for 1x , because
)(* xG is not the same as )1
(*
xG . However, )(* xG cannot be positive and increasing in the
whole range for x, because otherwise its value at infinity would be positive and not 0, as is
the case. Muentz's formula requires )(x to vanish at infinity to order x with 1 , hence
the corresponding function
dttx
nx
01
)(1
)(
has the value 0 at infinity. Therefore, this expression cannot be both positive and increasing
near infinity and Polya's criterion never applies to a formula of Muentz's type.
Remark 4.3 The standard “measure” in current Zeta function theory is
(4.4)
x
t
t
dtexEix )(:)(
with
14
(4.5) x
dxexd
x
)(
, xx
dxxd
log)(log
,
0
)()( xdxs s .
(3.5) plays a key role in the analysis of Euler, Gauss and Riemann, e.g. Gauss’ Li-function is
defined by
(4.6)
x
xtx
x
xOttddt
exEi
t
dtxLi
log 00
)ln
()(logt
)(loglog
:)(
,
1x .
Euler’s )log(logx divergence (see [8] H.M. Edwards 1.1) can be stated by
(4.7) x
e
x
e
x
tdtt
dt
u
dux
p)(log
log)log(log
1log
11
and Riemann’s estimate of )()( xRx (see remark 3.xx below) is given by
(4.8)
xt
tdxRx
1
)(log)()(
2
(
xt
tdxRx
1
)(log)()(
2
).
Remark 4.4
This remark is about putting the new measure with its underlying Hilbert space in a Hilbert
scale context with reference to Riemann’s duality equation:
Basically the isometric property )(ˆ)( xfxf of the Gauss-Weierstrass density function is
used to prove Riemann’s duality (see [8] H.M. Edwards)
(4.9) )1()2
(2
)1)((:)( 2/ sss
sss s ,
which can be written in the form
(4.10)
0
1
0
)()1()()1)(()( dxxfxssdxxfxsss ss
.
Jacobi’s relation (see also [8] H.M. Edwards 10.3)
(4.11) )1
(1
)(21:21)(ˆ:)( 2
1
222
xG
xxenxfexG xnxn
implies, that the invariant operator is formally self-adjoint (see also [12] H. Hamburger and
[9] D. Gaier for relations to conformal mappings and singular integral operators). But the
operator has no transform at all, because the integral
15
(4.12)
00
1 )()(
nxdFxx
dxxGx ss
does not converge for any s. The integral would converge at if the constant term of (1.27)
above, which is basically )0(f , is absent. Roughly speaking the measure dxxf )( solves
the issue, but the prize to be paid is a scale of higher regularly, jeopardizing adequate self-
adjoint properties.
Riemann’s duality can be derived from the representation
0
2/
1
2/)1(2/2/)1(2/ )()1(
1)()
2
1()1()
2()(
x
dxxx
ssx
dxxxx
ss
ss sssss
with x
ex xn
2
1:)(
1
2
whereby
)1
(1
)(
4/1
4/1
xxxx
.
Remark 4.5
Referring to Tauberian Theorems the integrals
0
)(xdx s and
0
)(xdx s show the same
divergence behavior for 0s
as )1( , which can be seen from
(4.13)
)2
2/1()
2
)2/1(
2
11)(
)(
0
sssdx
x
xex
xs
We note the relations ([26] G.N. Watson 13.6)
i
i
s
xs
dsx
ss
s
idt
t
tJ
)2
1(
)2
1(
2
1)(0
and
2
1
)!
11(
!
)1(
t
)t(2:)(
kk
x
kdt
Jex
kk
x
o
t
with
.…0.57721566)1
log(log(log)1()0(
1
00
dtt
tdte x .
The Mellin transforms of )(0 xJ , )(0 xY , )(xKo are given in [26] G.N. Watson 13.21, 13-24,
13-3). We mention the formulas
2loglog)(0
0
tdttJ
,
0
2
2log dtt
ee tt
,
0)(0
0
dttY .
16
Remark 4.6
We mention the following equivalent formulation for the Riemann conjecture (see [8]
Edwards, chapter 5)
(4.14) )()()ln()()( 2
1
xOxLixxOxLix
and Euler’s formula
(4.15) )(log!
)1()(
1
1
x
eOx
k
x
kt
dtex
xkk
x
t
resp. )ln
()(x
xOxLi ,
see [1] B. C. Berndt, Ramanujan’s corollary 2, chapter 4 . Riemann’s estimate is essentially
based on the analysis of the function:
0)Re(
0)Re(
.....
....
)(log
)(log
)1log(1
2
1)(
for
for
tdt
tdt
dsxs
siH
x
o
x
ia
ia
s
The new “measure” (x.y) motivates the alternative function
0)Re(
0)Re(
.....
....
)(log
)(log
:)(*
for
for
tdt
tdt
Hx
o
x
to be put into the context of Gauss’ Li-function (1.22) and Riemann’s formula (see [8] H.M.
Edwards 1.14)
(4.16) s
dsxs
idt
t
txLixLixLixJ s
ia
ian x
n
)(log2
1)0(log
log)()()()(
121
0)Im(
Analyzing analogue convergence behavior to
(4.17) )(11
0 0
log)1(1
xLidrr
edr
r
x xrr
(4.18)
0)Im(
1
0
log
)()(
xLixLidt
t
edr
r
xx tr
,
and to the density
17
(4.19)
0
log
1drx
x
r .
(3.19) is roughly speaking the density of the primes. Note that due to Euler it holds
0
)(logt
dteex txt for 0)Re( x .
Riemann analyzed the expression
(4.20) dsxs
s
ds
d
xis
dsxs
ixJ
ia
ia
s
ia
ia
s
)(log
log
1
2
1)(log
2
1)(
for 1a
to prove the convergence estimate
(4.21)
x
n
xt
tdt
ttt
dtxRx
log
log
log)1()()(
1
2
2
applying the Fourier inverse technique (see [8] H.M. Edwards 1.14 ff.). He used the following
Lemma: For dsxs
sds
d
xiH
ia
ia
s
)1log(
1
log
1
2
1:)(
with C
it holds
0)Re(
0)Re(
.....
....
)(log
)(log
)1log(1
2
1)(
for
for
tdt
tdt
dsxs
siH
x
o
x
ia
ia
s
For )1,( it follows
(4.22) )1log(log)()(0)Im(
1
sxLixLi
(4.22) is only conditionally convergent, it must be summed in order of increasing )Im( .
(4.23) is the critical term concerning an appropriate convergence behavior like (x.y), due to
its oscillating behavior. Using the new measure with the alternative Li-function (x.y) the
(convergence) damping behavior of Bessel functions first and second kind to infinite and to
zero should provide significant contribution to overcome this issue.
18
Remark 4.7
In [5] D. Bump et.al. it’s shown that the zeros of the transforms of the Hermite polynomials lie
all on the critical line. The Hermite polynomials are the orthogonal polynomial system related
to the normal distribution, building the eigenfunctions )(xn of the quantum harmonic
oscillator with its ground state )()( 210 xcfcx .
The relation of the Hermite polynomials to the density )(xdF in relation to the concept of
convolution operators is given in [6] D.A. Cardon.
Both analysis’ could be applied replacing )(xdF by )(xdN .
The Bessel function )(0 xK plays a key role in the analysis of the next section. The analysis
technique of [10] G. Gasper might be applicable using the Mellin transform of )(0 xK (see
[25] G. N. Watson 13-21), which is
(4.24)
)2
(4
1)2( 2
0
0
1 sdxxKx s
for )Re(0 s .
or the relation (2.2) ff. below.
Remark 4.8
A famous usage of Dirichlet’s series is in the context of Planck’s black-body radiation
function
(4.25)
1
/
5
1
/5
1 2
2 1
1),( Tnc
Tce
c
e
c
d
TdR
with 2
1 2 hcc and khcc /2 . The relation to the Zeta function
(4.26)
01
)()(x
dx
e
xss
x
s
is given by
(4.27)
0 1
4
0 1
44
)()()4()4(90 x
dxex
x
dxex x
n
nx
.
(4.42) describes the total radiation and its spectral density at the same time, i.e.
(4.28) dxx
gx
dx
e
x
x
dx
e
xdxxg
xx)
1(
11)(
4
/1
4
.
The new measure dxxN )( resp. the “measure” )(xd allows a modified definition of this
radiation function.
19
Referring to the probability model of the location of an electron we recall Parseval’s equation
(see [8] Edwards 10.7), which G.H. Hardy used (see [8] H.M. Edwards 11.1), to prove that
there are infinitely many roots of 0)( on the line 2/1)Re( s . With lemma 1.2 and
lemma 2.6 it holds
dsss
ssi
dxxNx
i
i
2/1
2/1
2
0
2)
2
2/1()
2
)2/3()1()
2
1(
2
1)(2
(4.29)
dsss
sssi
dxxNx
i
i
s
2/1
2/1
223
0
2
)2
2/1()
2
)2/3()
2
1()
2
3)(
2
1(
2
2)(ˆ2
.
Remark 4.9
A modified norm to the standard inner product
0
2)()(),( dxxfxffff
can be defined in the form
(4.30) dxdtt
tt
td
edxxNxfft
x
0 0 0
sinh4
0
22
2tanh
2
1tanh
2coshtanh
16)()(:
22
where
02
tanh2
1tanh
tt .
The above might be seen as a step forwards “The Road to Reality”, as it’s about complex
number systems in combination with “real” duality (see [15] R. Penrose 34.8) in the context
of location and frequency probability.
The Helmholtz equation with space dimension n
is given by
inuui
0
2
where i represents the Dirac delta function at the source point i corresponding to the
fundamental solution. The domain can be unbounded or bounded with or without
boundary conditions; x denotes the n-dimensional coordinate variable and kk xxr :
The kernel wavelet basis functions are
)()2(4
:)( )2(
12/
2/12/1
kn
n
k
n
kn rHri
rh
, 2n
20
where nh comply with the divergence (conservation) theorem
1)1(lim 1
k
n
n
n
kr
gSr
,
0kr
and nh satisfy the Sommerfeld radiation condition at infinity
0lim
n
k
n gir
gr , kr
The link to the density function (2.1) below is given by
)()()()()( 2
2
2
2
2
2
)2(
12/
)1(
12/
2)1(
12/ rYrJrrHrrHrHr nnnnn
With reference to fractional mathematics we note that 4
1: in (2.1) would correspond to a
fractional dimension of space of 5.2n .
Using the Hankel functions
)()(:)()1( xiYxJxH resp. )()(:)()2( xiYxJxH
and
2)1()2()1(222 )()()()()(:)( xHxHxHxYxJxR
it follows
iexRxH )()()1( resp.
iexRxH )()()2( .
Putting )(:)(4
1 xx ...... can be re-formulated to
x
dxdxR
)(
2
2
resp. x
dxdixKd
x
xN
2
4
1 )(2
8
)2(
With reference to (x.y) and remark x.y below we mention
)(22)1( )()()()()(ri
erYrJriYrJrH
resp. )(1
2
)(
2
4)2()(
)()()()1(
r
e
r
eerNrH
ririri
with 4
1:
which might motivate an alternative or additional “polar” coordinate transformation in the
context of Riemann manifolds.
21
The simplest version of the harmonic oscillator is the Hamiltonian system with Hamiltonian
)(2
1),( 222 qpqpH
and pq , qp , qq 2
Identifying CR 2 by putting qipz a solution to 2
2
1),( zqpH
is given in the form
tiCetz )( .
Remark 4.10 (just to kick off a next level of brainstorming --> linkage hyperbolic functions
and strings)
With respect to Lemma 1.2 we mention the somehow “birthday” of the Superstring theory. In
1968 Gabriel Veneziano and Mahiko Suzuki came across using the Euler beta function to
describe interactions of elementary particles:
Consider an elastic scattering process with 2 incoming spinless particles of transverse
momenta 21, pp , outgoing particles of momenta 43 , pp . With a metric with signature
,...,,, the mass squared of a particle is 22 pm . The conventional Mandelstam
variables are defined as
2
21 )( pps , 2
32 )( ppt , 2
31 )( ppu .
which obey the one identity
imuts .
The largest )(sJ value at given s with 22 )2( pms the square of the energy in the center
of mass frame and the angular momentum prr
pJ 2
2 formed the so-called “leading
trajectory”. Experimentally, it was discovered that the leading trajectories were almost linear
in s.
In the field theory of the weak interactions the simplest model amplitude ),( tsA is constructed
as a sum of s-channel & t-channel input diagrams in the form
)(
1)())(),((
))()((
))(())((),(),(
0 sjj
jttsB
ts
tsstAtsA
j
,
that shows poles, where the resonance of the leading (Regge) trajectories )(s is necessarily
linear in s, i.e. )0()( xx with the “daughter trajectories” nss )0()( , (postulated
by Veneziano), to achieve, that the formula is physically acceptable. )0( depends on the
quantum numbers such as strangeness and baryon number, but appeared to be
universal, approximately, i.e.
22
tconsGeV
tan1
1
2
Regge slope
)0()( xx linear Regge trajectory.
A resonance occurs at those s values where )(s is a nonnegative integer (mesons) or a
nonnegative integer plus ½ (baryons).
Ns )( mesons
Ns 2/1)( baryons
which gives some relation to our
0
* )(2)2
1(),1()( dxxxsssBs s .
Remark 4.11
A wavelet transform is similar as a Fourier transform, which delivers the frequency spectrum
of a timely signal f(t) without any loss of information, although the Fourier transform itself
gives the frequencies without any information about the points in time, when the frequencies
occur. The wavelet transform delivers this sort of information in a better distinguishing form:
one gets both the frequency analysis and the points in time, when those frequencies happen,
similar like the written notes, which results into the music of an orchestra, which are
described in form of a wavelet transform on a 2-dimensional paper ([23] M. du Sautoy: “the
primes have music in them”)
A wavelet is a function )()( 2 RLx with a Fourier transform which fulfills
dc
2)(ˆ
2:0 .
The wavelet transform of a function )()( 2 RLxf with the wavelet )()( 2 RLx is the function
dt
a
bt
atf
cdtttf
cbafW ab )(
1)(
1)()(
1:),( ,
, RbRa ,0
For a wavelet )()( 1 RLx its Fourier transform is continuous and fulfills
dt)(2
1)0(ˆ0
The wavelet transform to the wavelet )()( 2 RLx
23
),()(:2
2
22a
dadbRLRLW ,
is isometric and for the adjoint operator
)(),(: 22
2
2
* RLa
dadbRLW
2
* ),()(1
)(1
:),(a
dadbbag
a
bt
atg
cbagW
it holds IdWW
* and
)(
*
Wrange
PWW .
The continuous wavelet transform is known in pure mathematics as Calderón’s reproducing
formula, i.e. for )()( 1
nRLx real and radial with vanishing mean, i.e.
1)(ˆ
0
2
daa
a
It holds for )(1
:)(a
x
ax
na Calderón’s formula, i.e.
a
daff aa
0
** .
Classical Hilbert spaces in complex analysis are examples of wavelets, like Hardy space of
2L functions on the unit circle with analytical continuation inside the unit disk.
We note that )( 2x has a similar structure than the Mexican hut, which is a continuous
wavelet function (see remark 1.16 below) )()1()( 2
2/22/2
22
RLexedx
dx xx
fulfilling
dc
2)(ˆ
2:0 .