A general class of Voronoi’s and Koshliakov-Ramanujan’s
summation formulas involving dk (n)
Semyon Yakubovich∗
July 25, 2010
Abstract
By using the theory of the Mellin and Mellin convolution
type transforms we prove a general summaP
tion formula of Voronoi involving sums of the form
dk (n)f (n), where dk (n), k = 2, 3, . . . , d2 (n) ≡
d(n) is the number of ways of expressing n as a product of k factors. These sums are related to
the famous
PDirichlet divisor problem of determining the asymptotic behaviour as x → ∞ of the sum
Dk (x) = n≤x dk (n). In particular, we generalize Koshliakov’s formula and certain identities from
Ramanujan’s lost notebook to the case of hyper-Bessel functions and Jacobi elliptic theta functions.
New examples of Voronoi’s summation formulas involving Bessel, exponential functions and their
products, which are based on a comprehensive Marichev’s table of Mellin’s transforms are given. The
equivalence of these relations to the functional equation for the Riemann Zeta -function is discussed.
An extension of the Koshliakov formula involving the Kontorovich-Lebedev transform is obtained.
Keywords: Voronoi summation formula, Dirichlet divisor problem, Mellin transform, Mellin convolution
type transforms, Fourier-Watson transforms, Riemann Zeta-function, Kontorovich-Lebedev transform,
Modified Bessel and hyper-Bessel functions, Hypergeometric functions, Ramanujan’s lost notebook
AMS subject classification: 44A15, 33C05, 33C10, 11M06, 11M36, 11N 37
1
Introduction and auxiliary results
The main goal of this paper is to extend the Voronoi summation formula, which was proved in [7]
"N
#
Z N
X
lim
d(n)f (n) −
(log x + 2γ)f (x)dx
N →∞
"
= lim
N →∞
1/N
n=1
N
X
Z
N
d(n)g(n) −
#
(log x + 2γ)g(x)dx ,
(1.1)
1/N
n=1
on the sums involving the divisor function dk (n), k = 2, 3, . . . and d2 (n) ≡ d(n). In the latter case to
prove certain extensions of (1.1) (cf. [16]) the theory of Fourier -Watson transformations in L2 (R+ ) [12]
was applied. We will show below that this theory does not work for the sums with dk (n), k = 3, 4, . . . and
we will call the general theory of the Mellin convolution type transforms (see in [6], [12], [14]). As usual
∗ E-mail: [email protected]. Work supported by Fundação para a Ciência e a Tecnologia (FCT, the programmes
POCTI and POSI) through the Centro de Matemática da Universidade do Porto (CMUP)
1
2
Semyon YAKUBOVICH
[13], dk (n) means the number of ways of expressing n as a product of k factors. We note that γ in (1.1)
is Euler’s constant and functions f, g are defined on R+ representing a Fourier-Watson transformation
in L2 (R+ ) of each other. By using similar methods we will give in the sequel various extensions of the
Koshliakov formula [1], [3], [4], [7], [8], [11] involving the modified Bessel and hyper- Bessel functions (cf.
[2], Vol. II, [6]). Further generalizations will be obtained integrating with respect to a pure imaginary
index of the modified Bessel function and employing the Kontorovich-Lebedev transformation [5], [10],
[14], [15].
As it is known [12], the Mellin transform in L2 (R+ ) is defined by the integral
Z
∗
N
f (x)xs−1 dx,
F (s) ≡ f (s) = l.i.m.N →∞
(1.2)
1/N
where s ∈ σ, σ = s ∈ C : Re s = 21 and the convergence of the integral is in the mean square with
respect to the norm of the space L2 (σ). Moreover, the inversion formula takes place
f (x) = l.i.m.N →∞
1
2πi
Z
1/2+iN
F (s)x−s ds,
(1.3)
1/2−iN
where integral (1.3) is convergent in mean with respect to the norm in L2 (R+ ) and the generalized
Parseval equality holds
Z ∞
Z
1
F (s)G(1 − s)x−s ds.
(1.4)
f (xt)g(t)dt =
2πi
σ
0
However, when f (x)xµ−1 ∈ L1 (R+ ), then integral (1.2) exists as a Lebesgue integral and can be written
in the form
Z ∞
F (s) =
f (x)xs−1 dx, Re s = µ.
(1.5)
0
Moreover, if f has a bounded variation in the neighborhood of a point x, then the inversion formula like
(1.3) holds and can be represented by the pointwise limit (see in [12])
1
1
[f (x + 0) + f (x − 0)] =
lim
2
2πi N →∞
Z
µ+iN
F (s)x−s ds,
µ−iN
where the latter integral converges in the principal value sense. It is easy to verify that under condition f (x)xµ−1 ∈ L1 (R+ ), µ0 < µ < µ1 , one can differentiate with respect to s in (1.5) and we get
straightforward the representation of the derivative F 0 (s)
F 0 (s) =
Z
∞
f (x)xs−1 log x dx,
(1.6)
0
where integral (1.6) is absolutely convergent in the open strip {s ∈ C, µ0 < Re s < µ1 }. Further, the
left-hand side of (1.4) is called the Mellin convolution and for instance, considering f as a kernel we get
a class of the Mellin convolution type transforms. In the L2 -case it corresponds to the Fourier- Watson
transformations
Z ∞
k(xu)
d
gk (x) =
f (u)du,
(1.7)
dx 0
u
Z ∞
h(xu)
d
f (u)du,
(1.8)
gh (x) =
dx 0
u
SUMMATION FORMULAS
3
which are automorphisms in L2 (R+ ; dx) and have reciprocal inversion formulas for almost all x ∈ R+ in
the form
Z ∞
h(xu)
d
f (x) =
gk (u)du,
(1.9)
dx 0
u
Z ∞
d
k(xu)
f (x) =
gh (u)du,
(1.10)
dx 0
u
if and only if a continual analog of the biorthogonality for sequences
Z ∞
dx
(1.11)
k(ξx)h(ηx) 2 = min(ξ, η),
x
0
holds. Moreover, in this case the Parseval type equality takes place
Z ∞
Z ∞
2
[f (x)] dx.
gk (x)gh (x)dx =
(1.12)
0
0
Functions k, h are called the conjugate Watson kernels. These kernels possess the following properties:
functions
k(x)
K(s)
∈ L2 (R+ ; dx) and
∈ L2 (σ) ,
x
1−s
H(s)
h(x)
∈ L2 (R+ ; dx) and
∈ L2 (σ)
x
1−s
are reciprocal Mellin pairs in L2 (1.2), (1.3), where
2
1
sup−∞<t<∞ K
+ it ≤ Ck < ∞,
2
2
1
sup−∞<t<∞ H
+ it ≤ Ch < ∞.
2
Furthermore, conditions (1.11) and
K(s)H(1 − s) = 1,
Re s =
1
2
(1.13)
are equivalent. In the case h(x) = k(x) condition (1.13) becomes K 21 + it = 1 and k(x) is called
the Watson kernel. We note that a class of transformations (1.7), (1.8) involves classical sine and cosine
Fourier transforms, the Hankel transform, the Hilbert transform, etc. (cf. [10], [12], [14]).
Let us introduce the Kontorovich-Lebedev transformation in the form (see in [5], [10], [14], [15])
Z N
gλ (x) = l.i.m.N →∞
Kiτ (λx)f (τ ) dτ, x > 0,
(1.14)
−N
where λ > 0 is a parameter. It involves an integration with respect to the pure imaginary index iτ of the
real-valued modified Bessel function Kiτ (x) given by the following Fourier integral [10]
Z ∞
Kiτ (x) =
e−x cosh u cos τ u du.
(1.15)
0
The transform (1.14) forms an isometric isomorphism
dx
dτ
↔ L2 R+ ;
,
gλ : L2 R;
τ sinh πτ
x
4
Semyon YAKUBOVICH
where integral (1.14) converges with respect to the norm in L2 R+ ; dx
x . Furthermore, the reciprocal
inversion is given by the formula
Z ∞
dx
1
Kiτ (λx)gλ (x) ,
(1.16)
f (τ ) = l.i.m.N →∞ 2 τ sinh πτ
π
x
1/N
dτ
where integral (1.16) converges with respect to the norm in L2 R; τ sinh
and the Parseval equality
πτ
holds
Z ∞
Z ∞
|f (τ )|2
2 dx
|gλ (x)|
= π2
dτ.
(1.17)
x
0
−∞ τ sinh πτ
However, integral (1.14) exists in the Lebesgue sense for a wide class of functions. Indeed, this fact can be
easily verified using the Schwarz inequality and the uniform inequality for the modified Bessel functions
[14], [15]
x
, x > 0, τ ∈ R.
(1.18)
|Kiτ (x)| ≤ e−π|τ |/3 K0
2
Generally, the modified Bessel function Kµ (z) [2], Vol. II satisfies the differential equation
z2
du
d2 u
+z
− (z 2 + µ2 )u = 0,
2
dz
dz
(1.19)
and has the asymptotic behaviour
π 1/2
e−z [1 + O(1/z)],
2z
z → ∞,
(1.20)
z |Reµ| Kµ (z) = 2µ−1 Γ(µ) + o(1), z → 0, µ 6= 0,
(1.21)
K0 (z) = − log z + O(1), z → 0.
(1.22)
Kµ (z) =
and near the origin
It can be represented by the inverse Mellin transform (1.3) of the product of two Euler’s gamma-functions
(see in [6])
Z µ+i∞
1
s+ν
s−ν
Kν (x) =
2s−3 Γ
Γ
x−s ds, x > 0,
(1.23)
πi µ−i∞
2
2
where µ > |Reν|. This integral can be calculated by the Slater theorem
(cf. [6]) as a sum of residues in
.
So
the
result will be written in terms
the left-hand simple poles (ν 6= 0) of the gamma- function Γ s+ν
2
of the hypergeometric functions. However, when ν = 0 we arrive at the so-called logarithmic case [6]
Z µ+i∞
s
1
K0 (x) =
2s−3 Γ2
x−s ds, x > 0,
(1.24)
πi µ−i∞
2
where the value
of the integral will be a sum of residues in the double poles s = −2k, k ∈ N0 of the
function Γ2 2s . Therefore by straightforward calculations employing a series expansion of the gammafunction in the neighborhood of the pole (see in [6]) we get the result
K0 (x) =
∞ 2k
X
x
ψ(1 + k) − log(x/2)
,
2
[k!]2
(1.25)
k=0
d
where ψ(z) = dz
[log Γ(z)] is Euler’s psi-function [2], Vol. I. In the sequel, we will appeal to the hyperBessel functions, which generalize, for instance, integral (1.24), namely
Z µ+i∞
s
1
Uk (x) =
Γk
x−s ds, x > 0, k ∈ N.
(1.26)
2πi µ−i∞
2
SUMMATION FORMULAS
5
An analog of the series expansion (1.25) in this case is quite complicated because we should calculate the
residue of the function Γk (s/2) in the multiple pole s = −2m, m ∈ N0 of the order k. However, we can
represent Uk (x), k = 2, 3, . . . as a k − 1-fold Mellin convolution of the exponential function [12] since each
2
gamma-function Γ(s/2) is the Mellin transform of 2e−x . Thus denoting by u1 = u1 u2 . . . uk−1 , du =
du1 du2 . . . duk−1 and making elementary changes of variables we obtain
!
Z
k−1
X
2
x2 du
, k ∈ N\{1}, U1 (x) = 2e−x .
Uk (x) = 2
(1.27)
exp −
um − 1
1
u
u
Rk−1
+
m=1
Differentiating with respect to x under the integral sign in (1.26) owing to the absolute and uniform
convergence, it is not difficult to deduce a k-th order differential equation whose particular solution is
Uk (x). So we derive
k
x d
−
ω − x2 ω = 0, ω ≡ ω(x).
2 dx
P
As it is known [13], the Dirichlet function Dk (x) = n≤x dk (n), k = 2, 3, . . . can be represented as
the inverse Mellin transform
Z µ+i∞ k
ζ (s) s
1
x ds, µ > 1,
(1.28)
Dk (x) =
2πi µ−i∞
s
where ζ(s) is the Riemann zeta-function satisfying the familiar functional equation
πs ζ(s) = 2s π s−1 sin
Γ(1 − s)ζ(1 − s).
2
It has the following asymptotic behaviour on the critical line [13]
1
+ it = O t1/6 log t , |t| → ∞
ζ
2
and integral representation in the critical strip
Z ∞
[x] − x
ζ(s) = s
dx, 0 < Re s < 1,
xs+1
0
(1.29)
(1.30)
(1.31)
where [ ] denotes the integer part of a number. Its k-th power is represented by the series
ζ k (s) =
∞
X
dk (n)
, Re s > 1.
ns
n=1
(1.32)
Further, the integrand in (1.28) has a pole of order k at s = 1, and the residue is of the form xPk (log x),
where Pk is a polynomial of degree k − 1. Denoting by
∆k (x) = Dk (x) − xPk (log x),
(1.33)
∆k (x) = O x1−1/k logk−2 x , x → ∞, k = 2, 3, . . . ,
(1.34)
∆k (x) = O x logk−1 x , x → 0, k = 2, 3, . . . .
(1.35)
it behaves as (see in [13])
6
2
Semyon YAKUBOVICH
General expansions
Let f (x) be defined on R+ and its Mellin transform (1.2) be such that sF (s) ∈ L2 (σ), σ = {s ∈ C, Re s =
1
1/2
tends to zero when x → 0 and x → ∞,
2 }. Hence it is easily seen that f is continuous on R+ , f (x)x
f ∈ L2 (R+ ), F (s) ∈ L2 (σ) ∩ L1 (σ) and the integral (1.3) can be written in the form of the absolutely
convergent integral (1.6), namely
Z
1
f (x) =
F (s)x−s ds, x > 0.
(2.1)
2πi σ
In fact, we obtain
Z
|F (s)|2 |ds| < 2
σ
Z
|sF (s)|2 |ds| < ∞,
σ
Z
Z
|sF (s)|2 ds
|F (s)ds| ≤
σ
1/2 Z
σ
σ
|ds|
|s|2
1/2
< ∞.
Therefore formula (2.1) yields the estimate
x1/2 |f (x)| ≤
1
2π
Z
|F (s)ds| = O(1),
x > 0.
σ
Moreover it tends to zero when x approaches to zero and infinity via the Riemann- Lebesgue lemma.
Finally, condition sF (s) ∈ L2 (σ) implies that f is equivalent to some absolutely continuous function ρ
such that for almost all x > 0 we have
d
d
−x
ρ(x) = −x
f (x) ∈ L2 (R+ ).
dx
dx
In [16] we have proved main theorems characterizing the classical Voronoi summation formula (1.1)
(k = 2), which can be reformulated as the following
Theorem 1. Let sF (s) ∈ L2 (σ). Then the following statements are equivalent:
i) Riemann’s zeta-function ζ(s) satisfies the functional equation (1.29);
ii) There exist two functions f, g ∈ L2 (R+ ) such that f and g are Fourier-Watson transforms (1.7) of
each other
Z ∞
d
χ(xu)
g(x) =
f (u)du,
(2.2)
dx 0
u
Z ∞
d
χ(xu)
f (x) =
g(u)du,
(2.3)
dx 0
u
with the Watson kernel
1
χ(x) = l.i.m.N →∞
2πi
Z
=2
x
h
Z
"
1
2 +iN
π
1
2 −iN
1−2s
#2
Γ 2s
x1−s
ds
1−s
1−s
Γ 2
√
√ i
2K0 4π t − πY0 4π t dt,
0
and for all x > 0 the following identity holds
Z
Z
2
−s
ζ (s)F (s)x ds =
ζ 2 (s)G(s)xs−1 ds,
σ
σ
(2.4)
SUMMATION FORMULAS
where
"
G(s) = π
1−2s
7
#2
Γ 2s
F (1 − s)
Γ 1−s
2
is the Mellin transform (1.2) of g in L2 , sG(s) ∈ L2 (σ) and both integrals are absolutely convergent;
iii) An analog of the Voronoi formula holds
"N
Z
X
lim
d(n)f (xn) −
N →∞
#
N
(log t + 2γ)f (xt)dt
1/N
n=1
"N
#
n Z N
X
t
1
−
(log t + 2γ)g
lim
dt , x > 0,
d(n)g
=
x N →∞ n=1
x
x
1/N
(2.5)
where f, g are reciprocal Fourier-Watson transforms (2.2), (2.3) and the convergence in (2.5) is pointwise.
Corollary 1. Let in addition to conditions of Theorem 1 series in (2.5) be convergent and f, g ∈
L1 (R+ ; | log x| dx). Then it can be rewritten in a form involving Mellin’s transforms of functions f, g and
their derivatives, which are evaluated at the point s = 1
∞
X
d(n)
n=1
1
1 n
[(2γ − log x) F (1) + F 0 (1)]
=
f (xn) − g
x
x
x
−(2γ + log x) G(1) − G0 (1), x > 0.
(2.6)
Corollary 2. Formula (2.6) holds if F (µ + it), G(µ + it) ∈ L1 (R) are analytic in an open vertical
strip containing the line µ = 1 and tend to zero when |t| → ∞ uniformly in its closure.
Proof. In fact, in this case we get that inverse Mellin transforms (2.1) f, g are continuous on R+ and
the estimates
Z
x−µ µ+i∞
|f (x)| ≤
|F (s)ds| = O(x−µ ), x → ∞, µ > 1,
2π µ−i∞
Z
x−µ µ+i∞
|G(s)ds| = O(x−µ ), x → ∞, µ > 1
|g(x)| ≤
2π µ−i∞
are valid. Therefore conditions of Corollary 1 are easily satisfied and series in (2.6) converge absolutely
(see (1.32)).
In order to extend formulas (2.5), (2.6) for the sums with dk (n), k = 2, 3, 4, . . . , we will appeal to the
theory of Mellin’s transform (1.5) and the Mellin convolution type transforms in the weighted Lebesgue
spaces. We begin with
Lemma 1. Let 0 < µ < k1 , k = 2, 3, . . . , . Then
ζ k (1 − s)
=
1−s
∞
Z
∆k (x)xs−2 dx, s = µ + it,
(2.7)
0
where integral (2.7) converges absolutely.
Proof. In fact, the absolute convergence easily follows from estimates (1.34), (1.35) and a restriction
on µ. Further, using (1.33) we write
Z
∞
s−2
∆k (x)x
0
Z
dx =
1
s−2
∆k (x)x
0
Z
dx +
1
∞
∆k (x)xs−2 dx
8
Semyon YAKUBOVICH
1
Z
Pk (log x)xs−1 dx +
=−
0
Z
∞
∆k (x)xs−2 dx
1
Z
=
∞
∆k (x)xs−2 dx −
1
k−1
X
j=0
aj
dj −1
[s ],
dsj
(2.8)
Pk−1
where we use notation Pk (z) = j=0 aj z j . But the right-hand side of the latter equality is analytic for
µ < 0. Thus it gives the analytic continuation of the integral (2.7) into the left half-plane µ < 0.
In the meantime, appealing to (1.32), splitting the range of integration (1, ∞) into (1, 2), (2, 3), . . .
and making straightforward calculations we get (µ < 0)
Z ∞
Z ∞
Z ∞
∆k (x)xs−2 dx =
Dk (x)xs−2 dx −
Pk (log x)xs−1 dx
1
=
1
∞
X
1
k−1
X
k−1
1
ζ k (1 − s) X dj −1
dj
dk (n)ns−1 +
+
aj j [s−1 ] =
aj j [s ].
1 − s n=1
ds
1−s
ds
j=0
j=0
Substituting the latter expression into (2.8) and returning to the strip 0 < µ < k1 by analytic continuation
we come out with (2.6). Lemma 1 is proved.
From Lemma 1 we obtain that ζ k (1 − s)/(1 − s) is uniformly bounded in the strip 0 < µ < k1 .
Moreover, integration by parts in (2.7) yields the equality
Z N
1
k
(2.9)
ζ (1 − s) = lim
xs−1 d [∆k (x)] , s = µ + iτ, 0 < µ < ,
N →∞ 0
k
where the convergence is uniform by τ ∈ R.
Lemma 2. Let sF (s) ∈ L1 (1 − µ − i∞, 1 − µ + i∞), 0 < µ < k1 , k = 2, 3, . . . , . Then for all x > 0
"N
#
Z 1−µ+i∞
Z N
X
1
k
−s
ζ (s)F (s)x ds = lim
dk (n)f (nx) −
Qk (log t)f (xt)dt ,
(2.10)
N →∞
2πi 1−µ−i∞
0
n=1
where f is defined by the inverse Mellin transform (2.1), Qk (z) = Pk (z)+Pk0 (z), 0 denotes a first derivative
of Pk and the integral in the left hand-side of (2.10) is absolutely convergent.
Proof. Indeed, the absolute convergence of the integral in the left hand-side of (2.10) easily follows
from Lemma 1 and condition of the present lemma. Moreover, it yields that F (s) is integrable and
therefore by (2.1) f (x) is continuous for all x > 0. Hence using the uniform convergence in (2.9),
equalities (1.33), (2.1), the condition of the lemma and the Lebesgue- Stieljes integration we find for all
x>0
Z 1−µ+i∞
Z N
Z µ+i∞
1
1
k
−s
ζ (s)F (s)x ds = lim
d [∆k (t)]
F (1 − s)(xt)s−1 dsdt
N →∞ 2πi 0
2πi 1−µ−i∞
µ−i∞
"N
#
Z N
X
0
= lim
dk (n)f (nx) −
[Pk (log t) + Pk (log t)] f (xt)dt .
N →∞
n=1
0
Thus it leads to (2.10) and completes the proof of Lemma 2.
Now calling the functional equation (1.29) for the Riemann zeta-function and the supplement and
duplication formulas for gamma-functions [2], Vol. I, we write the left-hand side of (2.9) in the form
#k
Z 1−µ+i∞
Z µ+i∞ "
1
Γ (s/2)
1
k
−s
1/2−s
ζ (s)F (s)x ds =
π
ζ(s) 1−s F (1 − s)xs−1 ds.
(2.11)
2πi 1−µ−i∞
2πi µ−i∞
Γ 2
SUMMATION FORMULAS
9
Meanwhile the right-hand side of (2.11) can be treated employing representation (1.31), differential properties of the Mellin transform and the value of an elementary integral
Z ∞
1
Γ(s/2)
ts−1 cos(2πt)dt = π −s+1/2
, 0 < Re s < 1.
2
Γ((1 − s)/2)
0
Hence taking into account the uniform convergence by x ≥ x0 > 0 we derive
1
2πi
=
Z
µ+i∞
"
π
1/2−s
µ−i∞
2k−1
πi x
x
d
dx
where
V ∗ (s) = π 1/2−s
1 Γ (s/2)
2 Γ 1−s
2
Γ (s/2)
ζ(s) 1−s Γ 2
k Z
#k
F (1 − s)xs−1 ds
µ+i∞
k
[V ∗ (s)] F (1 − s)xs ds,
(2.12)
µ−i∞
∞
Z
0
Z ∞
1
1
−
xs−1 dx =
V (x)xs−1 dx,
x
x
0
and a function V (x) is defined as a Mellin’s convolution [12] by the convergent integral
Z ∞
[t] − t
V (x) =
cos(2πxt)dt, x > 0.
t
0
On the other hand, calling the Stirling asymptotic formula for gamma-functions [2], Vol. I, we easily find
k
[V ∗ (s)] = O(|s|k(µ−1/2) ), |s| → ∞. Therefore it belongs to L2 (µ − i∞, µ + i∞) ∩ L1 (µ − i∞, µ + i∞)
when 0 < µ < k1 , k = 4, 5, . . . , and for k = 3 when 0 < µ < 16 . The case k = 2 corresponds to the space
L2 (µ−i∞, µ+i∞) when 0 < µ < 14 . Further, the condition sk F (1−s) ∈ L2 (µ−i∞, µ+i∞), k = 2, 3, . . .
will evidently guarantee the integrability of sF (1 − s) on (µ − i∞, µ + i∞). Hence using the generalized
Parseval equality (1.4) we can write the right-hand side of (2.12) in the form
2k−1
πi x
x
d
dx
k Z
µ+i∞
k
[V ∗ (s)] F (1 − s)xs ds =
µ−i∞
where
1
Vk (x) =
2πi
Z
µ+i∞
2k
x
x
k
d
dx
k Z
∞
Vk (t)f
0
[V ∗ (s)] x−s ds, x > 0
t
dt, x > 0,
x
(2.13)
(2.14)
µ−i∞
and the convergence in (2.14) is absolute for 0 < µ < k1 , k = 4, 5, . . . , and for 0 < µ < 16 , (k = 3). In the
case k = 2 integral (2.14) converges in the mean square sense for 0 < µ < 14 .
Thus combining with Lemma 2 we have proved the following result.
Theorem 2. Let sk F (1 − s) ∈ L2 (µ − i∞, µ + i∞), k = 2, 3, . . . , , 0 < µ < k1 . Then for all x > 0
the following Voronoi type formula holds
"N
#
k Z ∞
Z N
X
d
t
−1
lim
dk (n)f (nx) −
Qk (log t)f (xt)dt = x
2x
Vk (t)f
dt,
(2.14)
N →∞
dx
x
0
0
n=1
where f is defined by the inverse Mellin transform (2.1) and Vk (x) is given by (2.14) with the corresponding
convergence of the integral:
i) it converges absolutely for k = 4, 5, . . . , and for k = 3, ifµ ∈ 0, 16 ;
ii) when k = 2, it converges in mean square for all µ ∈ 0, 14 .
10
Semyon YAKUBOVICH
3
Various examples and extensions of the Voronoi and Koshliakov formulas and certain identities from Ramanujan’s lost
notebook
This section will be devoted to examples of general Voronoi’s summation formula and its particular case
such as Koshliakov’s formula [4] associated with the modified Bessel function K0 (x), and as it says in
[1] was proved by Ramanujan about ten years earlier. We will also generalize certain identities from
Ramanujan’s lost notebook recently exhibited in [1]. We note here that this investigation would be
impossible without the use of invaluable comprehensive table of Mellin’s transforms of hypergeometric
functions created by Marichev in 1978 [6] and extended latter in [9], Vol. 3.
Koshliakov proved the following summation formula
∞
∞
X
2πn
1
1
1X
d(n)K0
=
(γ − log 4πz) − (γ − log(4π/z)), Re z > 0.
(3.1)
d(n)K0 (2πzn) −
z
z
4z
4
n=1
n=1
Here we will give another simple proof of (3.1) basing on the equality (2.4) and the inverse Mellin
transform (2.1). In fact, appealing to the relation (2.16.2.2) in [9], Vol. 2
Z ∞
s+ν
s−ν
s−1
s−2 −s
Kν (cx)x dx = 2 c Γ
Γ
, Re c > 0, Re s > |Reν|,
(3.2)
2
2
0
we have reciprocally
Kν (cx) =
Z
1
2πi
α+i∞
2s−2 Γ
α−i∞
s+ν
2
s−ν
Γ
(cx)−s ds, Re s = α,
2
(3.3)
where both integrals (3.2), (3.3) are absolutely convergent via the asymptotic behavior of the modified
Bessel functions (1.20), (1.21) and Euler’s gamma-functions (see [2], Vol. I). Hence, letting ν = 0 and
taking f (x) = K0 (2πzx), we find its Mellin transform (see (3.2), (1.26)) as F (s) = 41 (πz)−s Γ2 (s/2) =
1
−s ∗
U2 (s). Therefore the corresponding function G(s) in (2.4) is equal to G(s) = 41 π −s z s−1 U2∗ (s). So
4 (πz)
reciprocally g(x) = z −1 K0 (2πx/z) and by straightforward calculations of the related absolutely convergent integrals in (2.5) (see (2.6)) we come out with (3.1), which can be written in terms of the function
U2 (x) = 4K0 (2x) (see (1.24), (1.26)) as follows
∞
X
d(n)U2 (πzn) −
n=1
∞
πn 1
1X
d(n)U2
= (γ − log 4πz) − γ + log(4π/z), Re z > 0.
z n=1
z
z
(3.4)
On the other hand, taking (1.27) for k = 2 we write by straightforward calculations for any a > 0
∞
X
√
d(n)U2 (2π an) = 4
n=1
∞
X
√
d(n)K0 (4π an) = 2
n=1
=2
∞ X
∞ Z
X
m=1 k=1
0
∞
e−2π(kv+(am/v))
∞
X
n=1
dv
=2
v
Z
0
∞
Z
d(n)
∞
e−2π(u+(an/u))
0
du
u
dv
.
v(e2πv − 1)(e2πa/v − 1)
So we deduce an identity, which was found in Ramanujan’s lost book and discussed in [1], Theorem 4.1,
namely
Z ∞
∞
X
√
dv
.
(3.5)
2
d(n)K0 (4π an) =
2πv
v(e
− 1)(e2πa/v − 1)
0
n=1
SUMMATION FORMULAS
11
Moreover, the corresponding identity (2.6) for this case can be done dealing with the relation (2.6.5.15)
in [9], Vol. 1. So we have finally the following Koshliakov type formula
∞
∞
X
√
a X d(n) log(a/n) γ
log a
1
log (2π)
2
−
d(n)K0 (4π an) = 2
−
1
+
−
,
(3.6)
2
2
2
π n=1
a −n
2
4
π a
2π 2 a
n=1
which was established for the first time in [11] (see also in [1], [16]).
3/2
2 −s
(2π )
Γ(s/2)
Considering F (s) = Γ(s) = U1∗ (2s) we have correspondingly in (2.4), G(s) = π sin(πs/2)
Γ((1−s)/2) .
√
Therefore f (x) = e−x = 21 U1 ( x) and g(x) is calculated with the use of the (1.4) and relation (2.5.9.12)
in [9], Vol. 1. Therefore
h
i
2
2
g(x) = −2 e−4π x Ei(4π 2 x) + e4π x Ei(−4π 2 x) ,
where Ei(z) is the exponential integral. So appealing to the values of the related Mellin transforms and
its derivatives at the point s = 1 in (2.6) we arrive at the following Voronoi summation formula
2 ∞
X
4π n
4π 2 n
1 γ − log x
2 −4π2 n/x
4π 2 n/x
−xn
e
Ei
+e
Ei −
= +
,
(3.7)
d(n) e
+
x
x
x
4
x
n=1
which was proved in [8] by a different approach.
Another Voronoi’s type formula involving exponential function can be obtained employing the function
2
F (s) = Γ(s/2), which forms a Mellin pair with f (x) = U1 (x) = 2e−x . Calculating the corresponding
elementary integrals, the left hand-side of (2.5) in this case becomes
√ ∞
X
2
1
π
1
2
d(n)e−(xn) −
2γ − log x + ψ
, x > 0.
(3.8)
x
2
2
n=1
Meanwhile, G(s) = π 1−2s Γ2 (s/2)/Γ((1 − s)/2) and its inverse Mellin transform can be calculated as a
sum of residues in double poles of the square of a gamma-function. Therefore after a simple change of
variable we get
Z µ+i∞
−s
1
Γ2 (s)
π 4 x2
g(x) =
2π
ds
1
2πi µ−i∞
Γ 2 −s
∞
√ X
(π 2 x)2k
1
2
=2 π
+ k − 2 log π x , x > 0,
2ψ(1 + k) + ψ
(3.9)
[k!]2 (1/2)k
2
k=0
where (a)k is Pochhammer’s symbol [2], Vol. I. Moreover, since G(1) = 0, and G0 (1) = − 21 , ψ(1/2) =
−γ − log 4 we take in mind (2.6), (3.8), (3.9) to arrive at the following summation formula
"
#
√ X
∞
∞
X
π
(π 2 n/x)2k
1
−(xn)2
2
2
d(n) e
+
2ψ(1 + k) + ψ
+ k − 2 log π n/x
x
[k!]2 (1/2)k
2
n=1
k=0
=
√ 1
π 3
+
γ − log(2x) , x > 0.
2
x
2
As it was mentioned in [1], the Koshliakov formula (3.4) can be considered as an analog of the
transformation formula for the Jacobi elliptic theta function [13], namely
1
i
ϑ (0, ix) = √ ϑ 0,
,
x
x
12
Semyon YAKUBOVICH
where
∞
X
ϑ(0; ix) = 1 + 2
2
e−πn x , x > 0.
(3.10)
n=1
It has the following representation [13]
ϑ(0; ix) = 1 +
1
2πi
µ+i∞
Z
π −s/2 Γ
µ−i∞
s
2
ζ(s)x−s/2 ds, µ > 1.
(3.11)
In the sequel we will follow in a similar direction to generalize the Koshliakov formula (3.4), Ramanujan’s
identity (3.5) and Soni’s formula (3.6) to functions Uk (x), k = 2, 3, . . . involving the divisor function
dk (n). We have
Theorem 3. Let a > 0, k = 2, 3, . . . . Then
1
2πi
µ+i∞
Z
h
(2π)−s/2 ζ
µ−i∞
exp
Z
=2
Rk−1
+
µ+i∞
Z
1
2πi
h
2πa
u1 u2 . . . uk−1
π −s/2 ζ(s)Γ
ϑ 0;
Z
Rk−1
+
Z
µ+i∞
µ−i∞
=π
k
h
Rk−1
+
×
−1 k−1
Y
−1 duj
− 1
e2πuj − 1
, µ > 2,
uj
j=1
2
k
−s
a
a−s/2 ds =
j=1
dk (n)Uk (aπ k )1/2 n
k−1
Y
duj
(ϑ (0; iuj ) − 1)
−1
, µ > 1,
uj
j=1
(3.13)
Z ∞
∞
1 X
t
ds = 2
dk (n)
tdt
Uk (nt) Uk
a n=1
a
0
a−1 u1 u2 . . . uk−1 cotanh
k−1
Y
∞
X
(3.12)
n=1
ia
u1 u2 . . . uk−1
π ζ(s)
sin(πs/2)
Z
s ik
µ−i∞
= 21−k
1
2πi
∞
s s ik
X
√ Γ
a−s/2 ds =
dk (n)Uk (2π)k/2 an
2
2
n=1
1
cotanh
uj
π
uj
π
a
i
u1 u2 . . . uk−1 − 1
duj
−1
, 1 < µ < 2.
uj
(3.14)
Proof. Indeed, calling representations (1.27), (1.32), the definition of dk (n) with elementary series
expansions and substitutions, we change the order of integration and summation owing to the absolute
convergence of series and integrals to derive
1
2πi
=2
Z
µ+i∞
−s/2
(2π)
µ−i∞
∞
X
···
m1 =1
=2
h
∞
X
m1 =1
···

k
du
(2π)
a
m
m
.
.
.
m
1
2
k

exp −
uj −
k−1
u
u
.
.
.
u
u
u
.
1 2
k−1
1 2 . . uk−1
R+
j=1

∞ Z
X
mk =1
∞ Z
X
mk =1
∞
s s ik
X
√ −s/2
ζ
Γ
a
ds =
dk (n)Uk (2π)k/2 an
2
2
n=1
k−1
R+
k−1
X

exp −2π
k−1
X
j=1

mj uj −
2πa mk 
du
u1 u2 . . . uk−1 u1 u2 . . . uk−1
SUMMATION FORMULAS
exp
Z
=2
Rk−1
+
2πa
u1 u2 . . . uk−1
13
k−1
Y
−1 duj
−1
.
e2πuj − 1
uj
j=1
This leads to (3.12). In order to prove (3.13) we write similarly, taking into account (3.11)
1
2πi
=2
∞
X
Z
µ+i∞
h
π −s/2 ζ(s)Γ
s ik
2
µ−i∞
···
m1 =1
ϑ 0;
Z
∞
X
dk (n)Uk (aπ k )1/2 n
n=1

k
2
du
aπ
(m
m
.
.
.
m
)
1 2
k 
exp −
uj −
u1 u2 . . . uk−1
u1 u2 . . . uk−1
Rk−1
+
j=1

∞ Z
X
mk =1
21−k
a−s/2 ds =
Rk−1
+
k−1
X
ia
u1 u2 . . . uk−1
k−1
Y
duj
−1
.
(ϑ (0; iuj ) − 1)
uj
j=1
Finally we establish (3.14). Calling series expansion (5.4.5.1) in [9], Vol. 1 and relation (8.4.2.5) in
[9], Vol. 3, we obtain the equalities
Z µ+i∞
∞
π
X
π ζ(s) −s
1
1
1
a ds = 2
=
π
cotanh
−
1
.
(3.15)
2πi µ−i∞ sin(πs/2)
1 + (an)2
a
a
n=1
In the meantime, representations (1.26), (1.32) and factorization property of the Mellin’s convolution
immediately imply the first equality in (3.14). On the other hand, invoking with (3.15) and the absolute
convergence of the series and integrals we deduce
Z ∞
∞
∞
∞ Z ∞
X
1 X
1 X
t
t
dk (n)
Uk (nt) Uk
tdt = 2
···
Uk (m1 m2 . . . mk t) Uk
tdt
a2 n=1
a
a
a
0
m =1
m =1 0
1
=
∞
X
···
mk
m1 =1
= 2k
= 2π k−1
Z
k−1
R+
h
···
m1 =1
∞
X
Rk−1
+
mk =1
= πk
1
2πi
=1
∞
X
Z
Rk−1
+
Z
∞
X
"
µ+i∞
µ−i∞
∞
X
k
h s s ik
−s
(m1 m2 . . . mk a) ds
Γ
Γ 1−
2
2
"
1+
mk =1
1+
Z
m1 m2 . . . mk a
u1 u2 . . . uk−1
2 #−1 k−1
Y
j=1
1
duj
1 + u2j uj
2 #−1 k−1
Y 1
π
duj
mk a
cotanh
− 1
u1 u2 . . . uk−1
u
u
uj
j
j
j=1
a−1 u1 u2 . . . uk−1 cotanh
i k−1
Y 1
π
duj
u1 u2 . . . uk−1 − 1
cotanh
−1
.
a
u
u
uj
j
j
j=1
π
This gives (3.14) and completes the proof of Theorem 3.
Remark 1. We note that formulas (3.12), (3.13), (3.14) are k − 1-fold Mellin convolutions of the
−1 (1−k)/k
functions 21/k e2πx − 1
, 2
(ϑ(0; ix) − 1) and π x−1 cotanh πx − 1 , respectively.
Using (3.13) we get straightforward
Corollary 2. The following analog of the Koshliakov- Ramanujan formula (3.4) holds
k/2 ∞
∞
1X
X
π
k/2
dk (n)Uk aπ
n −
n
dk (n)Uk
a n=1
a
n=1
14
Semyon YAKUBOVICH
= 21−k
ϑ 0;
Z
k−1
R+
×
k−1
Y
ia
u1 u2 . . . uk−1
(ϑ (0; iuj ) − 1)
j=1
−1−
1
a
ϑ 0;
i
au1 u2 . . . uk−1
−1
duj
, a > 0, µ > 1, k ∈ N0 \{1}.
uj
Corollary 3. Letting k = 2, moving the contour to the left in the integral with respect to s and taking
into account the residue of the double pole at s = 1, identity (3.12) becomes Soni’s summation formula
(3.6).
Finally in this section we give more new examples of Voronoi’s summation formulas involving Bessel
and exponential functions, which are based on the table of Mellin’s transform in [9], Vol. 3.
2
1−2s
Γ( 2s )
and correspondingly, by (1.4), (2.1) and
Example 1. Let F (s) = s−2 . Then G(s) = π 4
Γ( 3−s
2 )
relations (2.5.24.2) in [9], Vol. 1, (8.4.5.1) in [9], Vol. 3 we find
(
log x1 , if x ∈ (0, 1],
f (x) =
0,
if x ∈ (1, ∞),
g(x) =
1
xπ
1
2 Y0 (4π
√
√ x) + π1 K0 (4π x) . This gives
X
d(n) log(nx) +
n≤ [x−1 ]
r r ∞
1 X d(n) 1
n
n
1
+ K0 4π
Y0 4π
π n=1 n
2
x
π
x
x 3
1
1
(2(1 − γ) + log x) + log
− γ, x > 0.
x
4
16π 8
2
s
s
1
1
Example 2. Let F (s) = Γ 2 Γ ν − 2 , ν > 2 , ν 6= 2 + m, m ∈ N. Then by relation (8.4.2.5) in
Γ(ν− 21 + 2s )
[9], Vol. 3, we obtain f (x) = 2Γ(ν) (1 + x2 )−ν . Meanwhile, G(s) = π 1−2s Γ2 2s
. Hence by
Γ( 1−s
2 )
Slater’s theorem in the logarithmic case we derive
Z
1
s
Γ2 12 − ν
1
1
1
1−2s 2 s Γ ν − 2 +
2
4
−s
4ν−1
2ν−1
2
x ds = 2π
π
Γ
+ ν; + ν, ν; −x π
g(x) =
x
0 F3
2πi σ
2
Γ(ν)
2
2
Γ 1−s
2
=
√
∞
X
2π π
(−4π 4 x2 )k
1
1
2
−
2ψ(1 + k) + ψ
+ k + ψ ν − − k − 2 log π x ,
2
2
cos πν Γ 32 − ν k=0 k! (2k)! 32 − ν k
where 0 F3 (a, b, c; z) is the generalized hypergeometric function [2], Vol. I. Calculating the values in the
right hand-side of (2.6) we arrive at the summation formula for all x > 0
"
2ν 2 1
2 2 !
∞
X
Γ 2 −ν
1 π2 n
1
1
π n
Γ(ν)
4x
d(n)
−
+ ν; + ν, ν; −
0 F3
2 )ν
(1
+
(xn)
πn
x
Γ(ν)
2
2
x
n=1
−1 2k √
∞ X
π π
3
π2 n
1
+
k! (2k)!
−ν
−2
2ψ(1 + k) + ψ
+k
2
x
2
cos πν Γ 32 − ν k=0
k
2 √
1
π n
1
+ψ ν − − k − 2 log
= πΓ(ν − 1/2) 3γ − 2 log(2x) − ψ ν −
+ xΓ(ν).
2
x
2
SUMMATION FORMULAS
15
In particular, the case ν = 1, x = a−1 with the use of relation (2.5.37.2) in [9], Vol. 1 leads to the known
identity
∞
√ X
√ 1
a
π
−
2π
K
4π
d(n) 2
ina
+
K
−4π
ina
= πγ + log a + ,
0
0
2
a
+
n
2
4a
n=1
which is proved in [8] by another method.
2
Example 3. Let F (s) = π5/2
Γ3 2s Γ 1−s
. Then by relation (8.4.20.35) in [9], Vol. 3, we ob2
tain the so-called Nicholson kernel [2], Vol. II f (x) = 2 J02 (x) + Y02 (x) . Correspondingly, G(s) =
3
2π − 2 −2s Γ3 2s Γ 1−s
and g(x) = 2π J02 π 2 x + Y02 π 2 x . It’s easily seen by asymptotic of the Bessel
2
functions that f, g ∈ L2 (R+ ) and not L1 (R+ ). Therefore the Voronoi formula for this case will be written
in the form (2.5). Precisely, we have the identity
"N
#
Z N
X
π
2
2
2
2
lim
d(n) J0 (nx) + Y0 (nx) −
(log t + 2γ) J0 (xt) + Y0 (xt) dt =
N →∞
x
1/N
n=1
"
× lim
N →∞
N
X
d(n) J02
n=1
π2 n
x
+ Y02
which is trivial when x = π.
Example 4. Let F (s) = Γ3
π2 n
x
Z
N
(log t + 2γ) J02
−
1/N
π2 t
x
+ Y02
π2 t
x
#
dt , x > 0,
. Then by (1.26) f (x) = U3 (x) and G(s)
=π 1−2s Γ2 2s Γ 1−s
.
2
√ (πx)2 /2
π 2 x2
Hence relation (8.4.23.5) in [9], Vol. 3 says that g(x) = 2π πe
K0
∈ L2 (R+ ) and not
2
L1 (R+ ) (see (1.20), (1.21), (1.22)). Thus it gives
√ "N
2π π X
1 πn 2
1 πn 2
lim
d(n) exp
K0
N →∞
x
2 x
2 x
n=1
Z
s
2
N
−
(log t + 2γ) exp
1/N
1
2
πt
x
2 !
K0
1
2
πt
x
2 !
#
dt =
∞
X
d(n)U3 (nx)
n=1
√
i
π π hγ
− log(8x) , x > 0.
x
2
2 s
Example 5. Let F (s) = −Γ 2 Γ 1 − 2s . Then by relation (8.4.11.3) in [9], Vol. 3, f (x) =
2
, F (1) = −π 3/2 , F 0 (1) = 12 π 3/2 (γ +
2ex Ei(−x2 ), and correspondingly, G(s) = −π 1−2s Γ2 2s Γ 1+s
2
log 4),
√ G(1) = −1, G0 (1) = √1π 3γ
2 + log(2 2π) . Calculating g(x) by Slater’s theorem we find
−
Z
1
4 +i∞
g(x) = i
1
4 −i∞
Γ2 (s) Γ
∞
√ X
1
(−1)k (xπ 2 )2k
+ s (xπ 2 )−2s ds = 2π π
2
[k!]2 (1/2)k
k=0
1
3 3
2
4
2 4
× 2 log xπ − 2ψ(1 + k) − ψ
−k
− 8π x 0 F2
, ; −x π .
2
2 2
So,
2
∞
X
n=1
"
(xn)2
d(n) e
4π 4 n
Ei(−(xn) ) + 2 0 F2
x
2
3 3
, ; −
2 2
nπ 2
x
2 !#
π 3/2
+
x
1
2γ − log x + ψ
2
1
2
16
Semyon YAKUBOVICH
1
= 2γ + log x − √
π
∞
∞
X
√
3γ
2π 3/2 X
(−1)k
+ log(2 2π) +
d(n)
2
x n=1
[k!]2 (1/2)k
k=0
2k n π2
1
n π2
2 log
− 2ψ(1 + k) − ψ
− k , x > 0.
×
x
x
2
√ 2 s 1+s −1
Example
. Then by relation (8.4.23.3) in [9], Vol. 3, f (x) =
2 6.
Let F (s) = πΓ 3 2 Γ 2 −1
x
s
−x2 /2
−2s
2
2e
K0 2 . Hence G(s) = −π 2
Γ 2 Γ 1 − 2s
, F (1) = π 3/2 , F 0 (1) = −π 3/2 γ2 + log 4 ,
G(1) = 1, G0 (1) = − 23 γ − log(8π 2 ). Further,
√
g(x) = −i π
1
4 +i∞
Z
1
4 −i∞
∞
√ X
Γ2 (s)
(xπ 2 )2k (xπ 2 )−2s ds = 2π π
3ψ(1 + k) − 2 log xπ 2 .
3
Γ (1 − s)
[k!]
k=0
Therefore,
2
∞
X
"
d(n) e−(nx)
2
/2
K0
n=1
(nx)2
2
π 3/2
=
x
2 2k 2 #
∞
nπ
π 3/2 X 1
nπ
−
3ψ(1 + k) − 2 log
x
[k!]3
x
x
k=0
2
γ
8π
3
γ − log(4x) − + log
, x > 0.
2
2
x
√
−1
Example 7. Let F (s) = 2π Γ3 2s Γ 1+s
. Then by relation (8.4.23.27) in [9], Vol. 3, f (x) =
2
−1
1−s
1 32 −2s 2 s
2
K0 (x). Hence G(s) = 2 π
Γ 2 Γ 2
Γ 1 − 2s
, and relation (8.4.23.23) in [9], Vol. 3 gives
2
2
immediately g(x) = 2π 2 I0 (xπ 2 )K0 (xπ 2 ). Moreover, F (1) = π2 , F 0 (1) = − π2 (γ + 3 log 2). Consequently,
"
lim
N →∞
N
X
d(n)I0
n=1
=
4
nπ 2
x
K0
nπ 2
x
Z
N
−
(log t + 2γ)I0
1/N
tπ 2
x
K0
tπ 2
x
#
dt
∞
1
x X
d(n)K02 (nx) − (γ − log(8x)) , x > 0.
π 2 n=1
4
An application of the Kontorovich-Lebedev transform
In this section we will use the Kontorovich-Lebedev transformation (1.14) (see also in [16]), which involves
an integration with respect to a pure imaginary index of the modified Bessel function
order
ins−iτ
to obtain
a new class of Voronoi’s summation formulas. In fact, taking Fτ (s) = 2s−2 Γ s+iτ
Γ
, τ ∈ R we
2
2
appeal to (1.23) and we get fτ (x) = Kiτ (x). Then as usual
π
Gτ (s) = (2π 2 )−s
2
!2 Γ 2s
1 − s + iτ
1 − s − iτ
Γ
Γ
2
2
Γ 1−s
2
and after straightforward calculation of additional terms with the use of (2.4) it becomes
∞
X
π
x
d(n)Kiτ (nx) −
2 cosh(πτ /2)
n=1
1 + iτ
2γ − log
+ Re ψ
2
2
x
SUMMATION FORMULAS
=
1
2πi
Z
ζ 2 (s)Gτ (s)xs ds, x > 0, τ ∈ R.
17
(4.1)
σ
Hence, assuming τ 6= 0 and substituting Gτ (s) into the right hand-side of (4.1), we make an elementary change of variable and we move the line of integration to the right, taking the contour
σ̂ = {s ∈ C, Re s = 1}. Doing this we encounter simple poles s = 21 ± iτ2 of the respective gammafunctions. Therefore invoking with (1.32), we write (2.5) for this case as follows
∞
X
x
π
1 + iτ
+ Re ψ
2γ − log
2 cosh(πτ /2)
2
2
n=1
!
2 2 −2s
Z 1+i∞
Γ (s)
1
1
iτ
1
iτ
2π
2
πζ (2s)
Γ
ds
=
−s+
Γ
−s−
2πi 1−i∞
2
2
2
2
x
Γ 21 − s
"
2 −iτ #
x
Γ
((1
+
iτ
)/2)
4π
+ √
Re ζ 2 (1 + iτ )
Γ (−iτ /2)
x
2 π cosh(πτ /2)
"
−iτ #
∞
n
X
x
Γ ((1 + iτ )/2) 4π 2
2
=
d(n) Kτ
+ √
Re ζ (1 + iτ )
,
x
Γ (−iτ /2)
x
2 π cosh(πτ /2)
n=1
x
d(n)Kiτ (nx) −
where
1
Kτ (x) =
2πi
Z
1+i∞
π
1−i∞
Γ (s)
Γ 12 − s
!2
Γ
1
iτ
−s+
2
2
−2s
iτ
1
−s−
2π 2 x
ds
Γ
2
2
2k 2 2π 2 x
1
1
iτ ψ(1 + k) + ψ
=2
Γ
+k+
+k
[k! (1/2)k ]2 2
2 2
k=0
iτ
1
1
+k+
− log 2π 2 x + √
−Re ψ
2
2
2 π cosh(πτ /2)
"
−iτ #
Γ ((1 + iτ )/2) 4π 2
, x > 0, τ ∈ R\{0}.
×Re
Γ (−iτ /2)
x
∞
X
(4.2)
It can be extended continuously for all τ ∈ R via the absolute and uniform convergence by τ ∈ R of
the integral in (4.2) and its value K0 (x) = 2πK0 (4π 2 x) due to (1.25). So letting x = 2πz we derive the
following extension of the Koshliakov formula (3.1)
∞
X
π
1 + iτ
d(n)Kiτ (2πzn) −
2γ − log (πz) + Re ψ
4πz cosh(πτ /2)
2
n=1
"
−iτ #
∞
n 1 X
1
Γ ((1 + iτ )/2) 4π 2
2
=
d(n) Kτ
+ √
Re ζ (1 + iτ )
, z > 0. (4.3)
2πz n=1
2πz
Γ (−iτ /2)
x
2 π cosh(πτ /2)
Letting τ → 0 in (4.3) and taking into account the value
"
−iτ #
√ Γ ((1 + iτ )/2) 4π 2
π
4π
2
lim Re ζ (1 + iτ )
=−
γ − log
,
τ →0
Γ (−iτ /2)
x
2
z
we easily come out again with the Koshliakov formula (3.1).
18
Semyon YAKUBOVICH
On the other hand, taking an arbitrary function f (τ ) ∈ L2 (R) we multiply by f both sides of equality
(4.1) and integrate with respect to τ . Changing the order of integration and summation via the absolute
convergence (see (1.18)) we call (1.14), where the integral exists in the Lebesgue sense to obtain
∞
X
π
x
d(n)gx (n) −
2
n=1
Z
f (τ )
cosh(πτ /2)
R
=
where
π
(Hf ) (s) = (2π 2 )−s
2
∗
1
2πi
Z
1 + iτ
+ Re ψ
2γ − log
dτ
2
2
x
ζ 2 (s)(Hf )∗ (s)xs ds,
(4.4)
σ
!2 Z
Γ 2s
1 − s + iτ
1 − s − iτ
Γ
Γ
f (τ )dτ.
2
2
Γ 1−s
R
2
(4.5)
Meanwhile, integrals with respect to τ in (4.4), (4.5) can be treated employing the Parseval equality for
the Fourier transform [12]. In fact, appealing to relations (2.5.46.5) in [9], Vol. 1 and (1.104) in [15] with
the duplication formula for gamma-functions we derive
r Z
Z
2
F f )(y)
f (τ )
dτ =
dy,
cosh(πτ
/2)
π
R cosh y
R
s Z
(F f )(y)
π
2 −s 2 s Γ 1 − 2
∗
(4.6)
(Hf ) (s) = √ (2π ) Γ
1−s dy, s ∈ σ,
1−s
2 Γ 2
y
2
R cosh
where
1
(F f )(y) = √
2π
Z
f (τ )eiτ y dτ
(4.7)
R
is the Fourier transform with the convergence in the mean square sense. Further, by definition of the
psi-function and making a change of the integration order, it is not difficult to calculate the following
Fourier transform
2
Z
1 + iτ 1 + iτ
2π
iτ y e Γ
Re ψ
[γ + y + log(2 cosh y)] .
dτ = −
2
2
cosh y
R
Therefore via the Parseval equality for Fourier transforms
r Z
Z
1 + iτ
2
(F f )(y)
f (τ )
Re ψ
dτ = −
[γ + y + log(2 cosh y)] dy
2
π R cosh y
R cosh(πτ /2)
and taking into account our calculations above
Z
R
f (τ )
cosh(πτ /2)
r Z
x
1 + iτ
2
(F f )(y) h
2γ − log
+ Re ψ
dτ =
γ − log
2
2
π R cosh y
2
x
−y − log(2 cosh y)] dy.
Hence the left hand- side of (4.4) becomes
x
∞
X
n=1
r
d(n)gx (n) −
π
2
Z
R
x
i
(F f )(y) h
γ − log
− y − log(2 cosh y) dy.
cosh y
2
SUMMATION FORMULAS
19
In the meantime calculating the inverse Mellin transform of (Hf )∗ (s) we use values above to get summation formula (3.7). So calling the result as a transformation (Hf )(x) we find
r Z
i
2
1 2
(F f )(y) h −4π2 x cosh y
e
Ei(4π 2 x cosh y) + e4π x cosh y Ei(−4π 2 x cosh y) dy. (4.8)
(Hf )(x) = −
π π R cosh y
Moreover, if f is absolutely continuous then similar to [16] involving the inversion theorem for the Fourier
transform [12] we prove from (4.6) that (Hf )∗ (1) = 0, ((Hf )∗ )0 (1) = − π4 f (0). Therefore we find from
(2.6), (4.4)
Z
∞
n πx
X
1
+
ζ 2 (s)(Hf )∗ (s)xs ds =
f (0)
d(n)(Hf )
2πi σ
x
4
n=1
and combining with our calculations above we derive the following Voronoi type summation formula
r Z
∞
x
n πx
X
π
(F f )(y) h
=
γ − log
d(n) xgx (n) − (Hf )
f (0) +
x
4
2 R cosh y
2
n=1
−y − log(2 cosh y)] dy, x > 0.
(4.9)
Thus we prove
Theorem 4. Let f be absolutely continuous on R and belong to L2 (R). Then the summation formula
(4.9) holds for all x > 0, where gx is the Kontorovich-Lebedev transform (1.14), Hf is a transformation
(4.8) and F f is the Fourier transform (4.7).
References
1. B.C. Berndt, Y. Lee and J. Sohn, The formulas of Koshliakov and Guinand in Ramanujan’s lost
notebook, in Surveys in Number Theory, Developments in Mathematics, K.Alladi, ed., Springer,
New York, 17 (2008), 21-42.
2. A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vols.
I and II, McGraw-Hill, New York, London and Toronto (1953).
3. A.L. Dixon and W.L. Ferrar, On the summation formulae of Voronoi and Poisson, Quart. J. Math.
(Oxford) 8 (1937), 66-74.
4. N.S. Koshliakov, On Voronoi’s sum-formula, Mes. Math. 58(1929), 30-32.
5. N.N. Lebedev, Sur une formule d’inversion, C.R.Acad. Sci. URSS 52(1946), 655-658.
6. O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions: Theory and
Algorithmic Tables, Ellis Horwood, Chichester (1983).
7. C. Nasim, On the summation formula of Voronoi, Transactions of the American Mathematical
Society, 163 (1972), 35-45.
8. F. Oberhettinger and K.L. Soni, On some relations which are equivalent to functional equations
involving the Riemann zeta-function, Math. Z. , 127 (1972), 17-34.
9. A.P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series: Vol. 1: Elementary
Functions, Gordon and Breach, New York (1986); Vol. 2: Special Functions, Gordon and Breach,
New York (1986); Vol. 3: More Special Functions, Gordon and Breach, New York (1989).
20
Semyon YAKUBOVICH
10. I.N. Sneddon, The Use of Integral Transforms, McGray Hill, New York (1972).
11. K. Soni, Some relations associated with an extension of Koshliakov’s formula, Proc. Amer. Math.
Soc. 17(1966), 543- 551.
12. E.C. Titchmarsh, An Introduction to the Theory of Fourier Integrals . Clarendon Press, Oxford
(1937).
13. E.C. Titchmarsh, The Theory of The Riemann Zeta- Function, The Clarendon Press, Oxford,
Second edition (1986).
14. S. B. Yakubovich and Yu. F. Luchko, The Hypergeometric Approach to Integral Transforms and
Convolutions. Mathematics and its Applications, 287. Kluwer Academic Publishers Group, Dordrecht (1994).
15. S.B. Yakubovich, Index Transforms, World Scientific Publishing Company, Singapore, New Jersey,
London and Hong Kong (1996).
16. S.B. Yakubovich, Voronoi- Nasim summation formulas and index transforms, Preprint, Center of
Math. Univ. Porto (2010), 19 pp.
S.B.Yakubovich
Department of Mathematics,
Faculty of Sciences,
University of Porto,
Campo Alegre st., 687
4169-007 Porto
Portugal
E-Mail: [email protected]