On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis
Using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytical function, we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis can be established and...
Saved in:
| Date: | 2012 |
|---|---|
| Main Authors: | , , , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2012
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2568 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508483937370112 |
|---|---|
| author | Beltraminelli, S. Merlini, D. Sekatskii, S. K. Белтрамінеллі, С. Мерліні, Д. Секацький, С.К. |
| author_facet | Beltraminelli, S. Merlini, D. Sekatskii, S. K. Белтрамінеллі, С. Мерліні, Д. Секацький, С.К. |
| author_sort | Beltraminelli, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:29:46Z |
| description | Using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytical function,
we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the
Riemann hypothesis can be established and present some of them as an example. It is shown that all earlier known equalities of this type, viz., the Wang equality,
Volchkov equality, Balazard-Saias-Yor equality, and an equality established by one of the authors, are certain particular cases of our general approach. |
| first_indexed | 2026-03-24T02:25:56Z |
| format | Article |
| fulltext |
UDC 517
S. K. Sekatskii (LPMV, Ecole Polytechnique Fédérale de Lausanne, Switzerland),
S. Beltraminelli, D. Merlini (CERFIM, Research Center Math. and Phys., Locarno, Switzerland)
ON EQUALITIES INVOLVING INTEGRALS OF THE LOGARITHM OF THE
RIEMANN ζ-FUNCTION AND EQUIVALENT TO THE RIEMANN HYPOTHESIS
ПРО РIВНОСТI, ЩО МIСТЯТЬ IНТЕГРАЛИ ВIД ЛОГАРИФМA
ζ-ФУНКЦIЇ РIМАНА I ЕКВIВАЛЕНТНI ГIПОТЕЗI РIМАНА
Using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytical function, we
show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann ζ-function and
equivalent to the Riemann hypothesis can be established and present some of them as an example. It is shown that all earlier
known equalities of this type, viz., the Wang equality, Volchkov equality, Balazard–Saias–Yor equality, and an equality
established by one of the authors, are certain particular cases of our general approach.
Показано як за допомогою узагальненої теореми Лiттлвуда про контурний iнтеграл, що мiстить логарифм аналiтич-
ної функцiї, можна отримати нескiнченну кiлькiсть iнтегральних рiвностей, що мiстять iнтеграли вiд логарифма
ζ-функцiї Рiмана i є еквiвалентними гiпотезi Рiмана, i наведено кiлька таких рiвностей у якостi прикладу. Показано,
що деякi вiдомi рiвностi такого типу, а саме, рiвностi Ванга, Волчкова, Балазарда – Сайаса – Йора та рiвнiсть, що
встановлена одним iз авторiв, є частинними випадками нaшого загального пiдходу.
1. Introduction. Among numerous statements shown to be equivalent to the Riemann hypothesis
(RH; see e.g. [1] for details and general discussion of the Riemann ζ-function and Riemann hy-
pothesis) there are a few which state that certain integrals involving the logarithm of the Riemann
ζ-function must have some well defined value. Actually, we know four of them established by Wang
[2], Volchkov [3], Balazard, Saias, Yor [4], and one of us [5]. As it stands today, these equalities
look as isolated and not-connected with each other, and the methods used to obtain them were quite
different. In this paper, which is based on our hitherto unpublished notes posted in ArXiv [6], we,
using generalized Littlewood theorem about contour integrals over the logarithm of an analytical
function, introduce a general recipe how infinite number of integral equalities of the aforementioned
type can be obtained. This recipe is illustrated by presenting a number of such equalities. In particu-
lar, all earlier known integral equalities equivalent to the Riemann hypothesis are obtained as certain
particular cases of our general approach.
2. Generalized Littlewood theorem. Littlewood theorem concerning contour integrals of the
logarithm of an analytical function [7] is well known and widely used in the theory of functions in
general and Riemann-function research in particular, see e.g. [1]. The following generalization of the
theorem is quite straightforward and was used actually already by Wang [2].
Throughout the paper we use the notation z = x+ iy and/or z = σ+ it on the equal footing. Let
us consider functions f(z), g(z) and let F (z) = ln(f(z)). Let us introduce and analyze an integral∫
C
F (z)g(z)dz along the contour C which is the rectangle formed by the broken line composed by
four straight line segments connecting the vertices with the coordinates X1+iY1, X1+iY2, X2+iY2,
X2 + iY1; we suppose that X1 < X2, Y1 < Y2 and that f(z) is analytic and non-zero on C and
meromorphic inside it, and that g(z) is analytic on C and meromorphic inside it. Assume first that the
poles and zeroes of the functions F (z), g(z) do not coincide and that the function f(z) has no poles
c© S. K. SEKATSKII, S. BELTRAMINELLI, D. MERLINI, 2012
218 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON EQUALITIES INVOLVING INTEGRALS OF THE LOGARITHM OF THE RIEMANN ζ-FUNCTION . . . 219
iy
x
X
1 X
iY
2
2
iY 1
X+iY
Contour C using during the proof of generalized Littlewood theorem
and only one simple zero located at the point X + iY in the interior of the contour. Then let us do
the cut along the straight line segment X1 + iY, X + iY and consider a new contour C ′ which is the
initial contour C plus the cut indenting the point X+iY (see Figure). Cauchy residue theorem can be
applied to the contour C ′ which means that the value of
∫
C′
F (z)g(z)dz is 2πi
∑
ρ
F (ρ) res(g(ρ)),
where the sum is over all simple poles ρ of the function g(z) lying inside the contour.
An appropriate choice of the branches of the logarithm function assures that the difference be-
tween two branches of the logarithm function appearing after the integration path indents the point
X+ iY is 2πi (see below for an exact definition), which means that the value of an integral along the
initial contour C is 2πi
(∑
ρ F (ρ) res(g(ρ))−
∫ X+iY
X1+iY
g(z)dz
)
. Remind, that the integral is taken
here along the straight segment parallel to the real axis hence for this integral Im(z) = const = Y
and dz = dx. Analogously, a single pole of f(z) function occurring at (other) point X + iY con-
tributes 2πi
∫ X+iY
X1+iY
g(z)dz to the contour integral value. Thus by summing all terms arising from
different zeroes and poles of the function f(z) one obtains the following generalization of the Little-
wood theorem (modification for the case when poles/zeroes of f(z) with an order m > 1 are present
is evident):
Theorem 1. Let C denotes the rectangle bounded by the lines x = X1, x = X2, y = Y1,
y = Y2 where X1 < X2, Y1 < Y2 and let f(z) be analytic and non-zero on C and meromorphic
inside it, let also g(z) is analytic on C and meromorphic inside it. Let F (z) = ln(f(z)), the logarithm
being defined as follows: we start with a particular determination on x = X2, and obtain the value
at other points by continuous variation along y = const from ln(X2 + iy). If, however, this path
would cross a zero or pole of f(z), we take F (z) to be F (z± i0) according as we approach the path
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
220 S. K. SEKATSKII, S. BELTRAMINELLI, D. MERLINI
from above or below. Let also the poles and zeroes of the functions f(z), g(z) do not coincide. Then
∫
C
F (z)g(z)dz = 2πi
∑
ρg
res(g(ρg)F (ρg))−
∑
ρ0f
X0
ρ+iY
0
ρ∫
X1+iY 0
ρ
g(z)dz +
∑
ρpolf
Xpol
ρ +iY polρ∫
X1+iY
pol
ρ
g(z)dz
where the sum is over all ρg which are simple poles of the function g(z) lying inside C, all ρ0f =
= X0
ρ + iY 0
ρ which are zeroes of the function f(z) counted taking into account their multiplicities
(that is the corresponding term is multiplied by m for a zero of the order m) and which lye inside C,
and all ρpolf = Xpol
ρ + iY pol
ρ which are poles of the function f(z) counted taking into account their
multiplicities and which lye inside C. For this is true all relevant integrals in the right-hand side of
the equality should exist.
Of course, exact nature of the contour is actually irrelevant for the theorem as it is for the
Littlewood theorem. As for the last remark concerning the existence of integrals taken along the
segments [X1 + iYρ, Xρ + iYρ] parallel to the real axis (the inexistence might happen e.g. if a pole
of the g(z) function occurs on the segment), the corresponding integration path often can be modified
in such a manner that one will be able to handle these integrals. The case of the coincidence of poles
and zeroes of the functions f(z), g(z) often does not pose real problems and can be easily considered.
We will deal with a few such cases below.
This is also clear that the theorem remains true if there are poles or zeroes of f(z) on the left
border of the contour x = X1: they contribute nothing to the contour integral value but the integral
over the line x = X1 is to be understood as a principal value. Occurrence of poles/zeroes of f(z) on
the right border of the contour x = X2 can be treated similarly.
3. Application of the generalized Littlewood theorem to establish equalities equivalent to
the Riemann hypothesis. All integrals involving the logarithm of the Riemann ζ-function below
are understood in the sense of a generalized Littlewood theorem.
3.1. Power functions. The most “natural” starting point to apply the generalized Littlewood
theorem to establish an equality equivalent to the Riemann hypothesis is to take in the conditions of
the aforementioned theorem f(z) = ζ(z), g(z) =
1
z2
, X1 = b, X2 = b + T, Y1 = −T, Y2 = T
with real b ≥ 1/2, b 6= 1, T → ∞ and consider the contour integral
∫
C
ln(f(z))g(z)dz. Known
asymptotic properties of the Riemann function for large argument values assure that the value of
the contour integral taken along the “external” three straight lines containing at least one of the
points b + T + iT, b + T − iT tends to zero when T → ∞ (this has been shown for a similar
O(1/z2) g(z) function already by Wang [2], and his consideration can be one-to-one repeated for
any O(1/zα), Re(α) > 1, function), and thus we obtain that the value of the contour integral
tends to −i
∫ ∞
−∞
ln(ζ(b+ it))
(b+ it)2
dt
(
minus sign comes from the necessity to describe the contour in
the counterclockwise direction; integral is taken along the line z = b + it where dz = idt and
g(z) =
1
(b+ it)2
)
. Suppose that RH holds and there are no zeroes inside the contour. Then by
Littlewood theorem the value of the integral is equal to the contribution of the simple pole of the
Riemann function occurring at z = 1:
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON EQUALITIES INVOLVING INTEGRALS OF THE LOGARITHM OF THE RIEMANN ζ-FUNCTION . . . 221
−i
∞∫
−∞
ln(ζ(b+ it))
(b+ it)2
dt = 2πi
1∫
b
dx
x2
= −2πi(1− 1/b)
and thus
∫ ∞
−∞
ln(ζ(b+ it))
(b+ it)2
dt = 2π(1 − 1/b) if b < 1 and
∫ ∞
−∞
ln(ζ(b+ it))
(b+ it)2
dt = 0 if b > 1. If
there is a zero ρ = σk+ itk with the real part σk > b, it contributes −2πi
∫ σ+it
b+it
dx
x2
; correspondingly
the joint contribution of two complex conjugate zeroes is equal to 4πi
(
1
σ2 + t2
− 1
b2 + t2
)
and we
have the equality
1
2πi
b+i∞∫
b−i∞
ln(ζ(z))
z2
dz = 1− 1/b− 2
∑
ρ:σk>b,tk>0
(
1
σ2k + t2k
− 1
b2 + t2k
)
.
All terms under the sum sign are definitely negative and thus we have established our first equality
equivalent to the Riemann hypothesis:
Theorem 2. An equality
1
2π
∞∫
−∞
ln(ζ(b+ it))
(b+ it)2
dt = 1− 1
b
, (1)
where 1 > b ≥ 1/2, holds true for some b if and only if there are no Riemann function zeroes with
σ > b. For b = 1/2 this equality is equivalent to the Riemann hypothesis.
The case b = 1/2, viz.
∫ ∞
−∞
ln(ζ(1/2 + it))
(1/2 + it)2
dt = −2π, has been tested numerically.
The same theorem takes more elegant form if we apply the generalized Littlewood theorem to
the function f(z) = ζ(z)(z − 1) thus killing the contribution of the simple pole:
Theorem 2a. An equality
1
2π
∫ b+i∞
b−i∞
ln(ζ(z)(z − 1))
z2
dz = 0, where 1 > b ≥ 1/2, holds true for
some b if and only if there are no Riemann function zeroes with σ > b. For b = 1/2 this equality is
equivalent to the Riemann hypothesis.
Remark 1. The use of inverse square function g(z) = 1/z2 is not necessary for conditions of
Theorem 2 and 2a, very broad set of power functions g(z) = 1/zα, where α is real and greater than
1, although not excessively large, can be used instead. From numerical calculations performed by
Wedeniwski as cited by Ramaré and Saouter [8], it follows that there are no “incorrect” Riemann
zeroes with σ > 1/2 if t < tmin = 3.3 · 109. (There is more recent calculations of Gourdon and
Demichel where it is reported that the first 1013 Riemann zeroes are located on the critical line [9],
but we were unable to get an exact value of T from their paper.) This means that the argument of the
“incorrect” zeroes lies in the very narrow intervals (π/2, ∼= π/2− 1/tmin) for positive t and (−π/2,
∼= −π/2 + 1/tmin) for negative t. Correspondingly, the argument of the integrand in the contribution
of such Riemann zero to the contour integral values,
∫ σk+itk
b+itk
1
zα
dz, lies in the intervals (±απ/2,
∼= ±απ/2 ∓ α/tmin) and so the same sign of the real part of the sum of two paired contributions
of complex conjugate zeroes is certain apart from the case when α takes the form α = 2n + 1 + δ
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
222 S. K. SEKATSKII, S. BELTRAMINELLI, D. MERLINI
where n is an integer and δ is positive and very small, or when α is very large. For example, for
α close to 1 it suffices to take α “a bit larger” than 1 +
2
πtmin
∼= 1 + 2 · 10−10 to ensure the
same signs of all “incorrect” Riemann zeroes contributions. Similarly, from above α is limited by
α <
πtmin
2
∼= 5.18 · 10
9
. All the necessary details can be easily precisely elaborated but we will
not do this here and limit ourselves just with a few examples of other equalities equivalent to the
Riemann hypothesis:
1/2+i∞∫
1/2−i∞
ln(ζ(z)(z − 1))
z1+3·10−10 dz = 0,
1/2+i∞∫
1/2−i∞
ln(ζ(z)(z − 1))
z3
dz = 0,
∫ 1/2+i∞
1/2−i∞
ln(ζ(z)(z − 1))
z5·109
dz = 0,
etc.
Despite their very simple form, above equalities have a drawback that they are “non-symmetrical”
with respect to real and imaginary parts and can not be recast in the form containing integrals of only
an argument or logarithm of the module. This drawback can be eliminated by taken “more symmetric”
functions g(z), of which g(z) =
1
a2 − (z − b)2
is the most natural first example. Here a, b are
arbitrary real positive numbers, 1 > b ≥ 1/2, a+ b 6= 1, and the contour integral
∫
C
ln(f(z))g(z)dz
is reduced to the form I = −i
∫ ∞
−∞
ln(ζ(b+ it))
a2 + t2
dt. Now inside the contour we have a simple pole
at z = b+ a which gives the contribution − iπ
a
(ln (ζ(a+ b))) to the integral value. Contribution of
the Riemann function zero ρ = σk + itk with the real part σk > b is given by an elementary integral∫ σk−b
0
dp
a2 − (p+ itk)2
which real part is equal to
∫ σk−b
0
a2 − p2 + t2k
(a2 − p2 + t2k)
2 + 4p2t2k
dp. Because this
is well known that possible values of tk for the Riemann function zeroes with σk > 1/2, if they
exist, are very large while 0 < p < 1/2, the expression a2 − p2 + t2k is always positive and hence all
contributions of such Riemann zeroes are also positive. Calculation of the contribution of the simple
pole at z = 1 is trivial, and we get the following theorem.
Theorem 3. An equality
a
π
∞∫
−∞
ln |ζ(b+ it)|
a2 + t2
dt = ln
∣∣∣∣ζ(a+ b) (a+ b− 1)
a− b+ 1
∣∣∣∣ , (2)
where a, b are arbitrary real positive numbers and 1 > b ≥ 1/2, a+ b 6= 1, holds true for some b if
and only if there are no Riemann function zeroes with σ > b. For b = 1/2 this equality is equivalent
to the Riemann hypothesis.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON EQUALITIES INVOLVING INTEGRALS OF THE LOGARITHM OF THE RIEMANN ζ-FUNCTION . . . 223
Note that here an integral is taken over the logarithm of the module; corresponding equal-
ity for the integrals involving an argument is trivially fulfilled due to the parity of the functions
involved. Similarly as above, by considering f(z) = ζ(z)(z − 1) it can be recast in the form
a
π
∫ ∞
−∞
ln |ζ(b+ it)(b− 1 + it)|
a2 + t2
dt = ln |ζ(a + b)(a + b − 1)|. This same hint always can be ap-
plied and we will not mention this more. At this stage this is also worthwhile to note that we
have tested numerically not only an equality (2) with a = 1, b = 1/2, but also more general case
a
π
∫ ∞
−∞
ln |ζ(b+ it)|
a2 + t2
dt = ln
∣∣∣∣ζ(a+ b)(a+ b− 1)
a− b+ 1
∣∣∣∣+∑
σk>b
ln
∣∣∣∣a+ σk − b+ itk
a− σk + b− itk
∣∣∣∣ with a = 1, b = 1/4
where first 649 Riemann function zeroes were taken into account [6].
The case a + b = 1 does not pose problems being the transparent limit of above equalities.
Putting a = 1 − b + δ and considering the limit δ → 0 we have
1− b
π
∫ ∞
−∞
ln |ζ(b+ it)|
(1− b)2 + t2
dt =
= ln |ζ(1 + δ) δ| − ln |2− 2b|. Using the known Laurent expansion of the Riemann zeta-function in
the vicinity of z = 1, ζ(1+δ) =
1
δ
+. . . we see that simply
1− b
π
∫ ∞
−∞
ln |ζ(b+ it)|
(1− b)2 + t2
dt = − ln |2−2b|
and thus we have the following theorem.
Theorem 3a. An equality
1− b
π
∞∫
−∞
ln |ζ(b+ it)|
(1− b)2 + t2
dt = − ln(2− 2b), (3)
where b is an arbitrary positive number such that 1 > b ≥ 1/2, holds true for some b if and
only if there are no Riemann function zeroes with σ > b. For b = 1/2 this equality, that is∫ ∞
−∞
ln |ζ(1/2 + it)|
1/4 + t2
dt = 0, is equivalent to the Riemann hypothesis.
The last equality here is of course Balazard, Saias, Yor’s one [4].
Analogously, by considering “impair” functions g(z), of which the most straightforward example
is g(z) = −i z − b
(c2 − (z − b)2)(d2 − (z − b)2)
that takes the form g(t) =
t
(c2 + t2)(d2 + t2)
along the
line z = b + it, we can get integral equalities involving integrals over an argument. In such a way
we obtain the following theorem.
Theorem 4. An equality
∞∫
0
t arg(ζ(b+ it))
(c2 + t2)(d2 + t2)
dt =
π
2(d2 − c2)
ln
∣∣∣∣ζ(b+ d)
ζ(b+ c)
∣∣∣∣+
π
2(d2 − c2)
ln
∣∣∣∣(d2 − (1− b)2)c2
(c2 − (1− b)2)d2
∣∣∣∣ , (4)
where b, c, d are real positive numbers such that c · d ≤ 3.2 · 1019, c 6= d, b+ c 6= 1, b+ d 6= 1 and
1 > b ≥ 1/2, holds true for some b if and only if there are no Riemann function zeroes with σ > b.
For b = 1/2 this equality is equivalent to the Riemann hypothesis.
Details of the proof are straightforward and thus omitted here; they can be found in our ArXive
submissions [6]. The appearance of the condition c · d ≤ 3.2 · 1019 is due to the circumstance
that the contribution of the Riemann function zero ρ = σk + itk with σk > b is given by Iρ =
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
224 S. K. SEKATSKII, S. BELTRAMINELLI, D. MERLINI
= −2π
∫ σk−b
0
q + itk
(c2 − q2 + t2k − 2iqtk)(d2 − q2 + t2k − 2iqtk)
dq from which it immediately follows
that the real part of this integral is an integral of the expression
−q
−3t4k − 2t2k(q
2 + c2 + d2)− q2(c2 + d2) + c2d2 + q4
((c2 − q2 + t2k)
2 + 4q2t2k)((d
2 − q2 + t2k)
2 + 4q2t2k)
.
By our choice of b we have 0 ≤ q < 1/2 (all Riemann zeroes lye in the critical strip) while
t < tmin = 3.3 · 109, see Remark 1. Hence, provided the product cd is not very large, the sign of this
expression is certainly positive: it suffices that c · d ≤ 3t2min whence the theorem condition.
As an example one can take b = 1/2, c = 3/2, d = 7/2 and then in view of ζ(2) = π2/6, ζ(4) =
=π4/90 obtain the following rather elegant criterion equivalent to RH:
∫ ∞
0
targ(ζ(1/2 + it))
(9/4 + t2)(49/4 + t2)
dt=
=
π
20
ln(18π2/245). This condition, as some others, has been tested numerically [6].
Again, the case b + c = 1 or b + d = 1 does not create any problem because the corresponding
limit is quite transparent. Proceeding as above, we get the following theorem.
Theorem 4a. An equality
∞∫
0
t arg(ζ(b+ it))
(c2 + t2)((1− b)2 + t2)
dt =
π
2(c2 − (1− b)2)
ln
∣∣∣∣ζ(b+ c)(c2 − (1− b)2)(1− b)
2c2
∣∣∣∣ , (5)
where b, c are real positive numbers such that c(1 − b) ≤ 3.2 · 1019, b + c 6= 1 and 1 > b ≥ 1/2,
holds true for some b if and only if there are no Riemann function zeroes with σ > b. For b = 1/2
this equality is equivalent to the Riemann hypothesis.
As an illustration, we can take, for example, b = 1/2, c = 3/2 and then in view of ζ(2) = π2/6
obtain another rather elegant equality equivalent to RH (this also has been tested numerically):∫ ∞
0
t arg(ζ(1/2 + it))
(9/4 + t2)(1/4 + t2)
dt =
π
4
ln
(
π2
27
)
.
The case of the integrals
∫ ∞
0
t arg(ζ(b+ it))
(a2 + t2)2
dt is considered similarly introducing the function
g(z) = −i z − b
(a2 − (z − b)2)2
and analyzing the contour integral
∫
C
ln(ζ(z))g(z)dz taken round the
same contour C. Now in the interior of the contour we have the pole of the second order at z = a+ b
which, according to the residue theorem, contributes 2πi
d
dz
(
−i z − b
(a+ z − b)2
ln(ζ(z))
) ∣∣∣
z=b+a
=
=
π
2a
ζ ′(a+ b)
ζ(a+ b)
to the contour integral value. Proceeding as above, we have the following theorem.
Theorem 5. An equality
∞∫
0
t arg(ζ(b+ it))
(a2 + t2)2
dt =
π
4a
ζ ′(a+ b)
ζ(a+ b)
+
π
2
(
1
a2 − (1− b)2
− 1
a2
)
, (6)
where a, b are real positive, a+ b 6= 1, a ≤ 5.1 · 109 and 1 > b ≥ 1/2, holds true for some b if and
only if there are no Riemann function zeroes with σ > b. For b = 1/2 this equality is equivalent to
the Riemann hypothesis.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON EQUALITIES INVOLVING INTEGRALS OF THE LOGARITHM OF THE RIEMANN ζ-FUNCTION . . . 225
Again, the case of a+ b = 1 is a transparent limit of above expressions for a+ b = 1 + δ, δ → 0
(see [6] for details):
Theorem 5a. An equality
∞∫
0
t arg(ζ(b+ it))
((1− b)2 + t2)2
dt =
π
4(1− b)
(
γ − 3
2(1− b)
)
, (7)
where b is real 1/2 ≤ b < 1 and γ is the Euler-Mascheroni constant, holds true for some b if and
only if there are no Riemann function zeroes with σ > b. For b = 1/2 this equality is equivalent to
the Riemann hypothesis.
For b = 1/2 Theorem 5a gives
1
π
∞∫
0
2t arg(ζ(1/2 + it))
(1/4 + t2)2
dt = γ − 3 (8)
and this is nothing else than Volchkov criterion [3].
Remark 2. Volchkov paper more or less explicitly contains this relation but actually he gave an-
other form of his criterion. After obtaining (8), he considered the function S1(x) =
∫ x
0
arg(ζ(1/2 +
+ it))dt. In view of S1(x) = O(lnx) [1, p. 214], one can integrate by parts to obtain
∞∫
0
arg(ζ(1/2 + ix))g(x)dx = −
∞∫
0
S1(x)
d
dx
g(x)dx.
Then Volchkov introduced the relation
S1(x) =
∞∫
1/2
(ln |ζ(σ + ix)| − ln |ζ(σ)|)dσ
and, noting that
∫ ∞
1/2
ln |ζ(σ)|dσ = const which contributes nothing to the
∫ ∞
0
S(x)g1(x)dx, ob-
tained
32
π
∞∫
0
1− 12t2
(1 + 4t2)3
dt
∞∫
1/2
ln |ζ(σ + it)|dσ = 3− γ;
exactly in this form his criterion was published.
Remark 3. Our above equalities involving integrals containing the function arg(ζ(b + it)) as
a factor under the integral sign, may be interpreted as certain sums over Riemann function zeroes.
For example, for b = 1/2 we have a factor arg(ζ(1/2 + it)) and this same factor is exactly the
“non-trivial” part of an expression describing the number of the Riemann function zeroes lying in
the critical strip and having the imaginary part 0 < t < x:
N(x) = 1− x lnπ
2π
+
1
π
Im
(
ln Γ(1/4) +
ix
2
)
+
1
π
arg(ζ(1/2 + ix)), (9)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
226 S. K. SEKATSKII, S. BELTRAMINELLI, D. MERLINI
see e.g. [1, p. 212] for details. The integral
∫ ∞
0
G(x)dN(x), provided, of course, that it exists,
is equal to the sum
∑
ρ,tk>0
G(tk) taken over all Riemann zeroes. In practice, this calculation is
interesting if the function G(x) is such that the integration by parts can be used:
∞∫
0
G(x)dN(x) = G(x)N(x)|∞0 −
∞∫
0
g(x)N(x)dx;
here, evidently, g(x) = dG(x)/dx. Exactly in this way Volchkov established his criterion.
Of course, an infinite number of other integral equalities equivalent to the Riemann hypothesis can
be obtained in the same way. Note also that by considering the “semi-contour” C with X1 = b, X2 =
= b+ T, Y1 = 0, Y2 = T, a number of interesting equalities relating with each other integrals of the
logarithm of the Riemann function taken along the real axis and the line (b, b+ i∞) can be obtained.
In some cases the consideration of the “square root type” functions like g(z) =
1
(a2 − (z − b)2)3/2
might be interesting as well; for details see our contributions [6].
3.2. Exponential functions. The case of exponential functions g(z), viz. g(z) =
i
cos(a(z − b))
,
g(z) =
−i(z − b)
cos2(a(z − b))
, etc. can be considered along the same lines. Here we will limit ourselves
with the first aforementioned function; for the treatment of the second see [6].
Let us take the same contour as above. For z − b = x+ iy where x, y are real, we have that for
large y | cos(a(z− b))|−1 = O(e−a|y|) provided arg(z− b) 6= 0 and arg(z− b) 6= π. This asymptotic,
together with the known asymptotic of ln(ζ(z)), guaranties the disappearance of the integral taken
along the external lines of the contour (i.e., along lines other than its left border): the problems might
appear only for real positive z − b, but for such a case we have ln(ζ(z)) ∼= 2−z, so it is enough to
avoid such values of X for which g(z) has poles when X →∞.
In the interior of the contour we have simple poles at the points z = b+π/(2a)+πn/a where n is
an integer or zero, and, if b < 1, also a simple pole of the Riemann function at z = 1. If b < 1, in the
interior of the contour we also can have a number of zeroes of the Riemann function, and we definitely
have an infinite number of them if b < 1/2. The contribution of the poles of g(z) to the contour
integral value is, by a residue theorem, equal to 2πi
(
i
∞∑
n=0
(−1)n+1
a
ln(ζ(b+ π/(2a) + πn/a)
)
.
To find the corresponding contribution of the pole of the Riemann function we apply the generalized
Littlewood theorem: for b < 1 it is equal to
2πi
1∫
b
i
cos(a(z − b))
dz = −2π
1−b∫
0
1
cos(ax)
dx =
π
a
ln
1− sin(a(1− b))
1 + sin(a(1− b))
;
we take a(1 − b) < π/2 to avoid the problems with this integral. Analogously, the order lk zero of
the Riemann function ρ = σk + itk, for b < σk, contributes
−2πi
σk+itk∫
b+itk
ilk
cos(a(z − b))
dz = 2π
σk−b∫
0
lk
cos(ax) cosh(atk)− i sin(ax) sinh(atk)
dx
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON EQUALITIES INVOLVING INTEGRALS OF THE LOGARITHM OF THE RIEMANN ζ-FUNCTION . . . 227
to the contour integral value. Pairing the complex conjugate zeroes we see that due to the symmetry
of the distribution of the Riemann function zeroes the imaginary part of these contributions vanishes
and thus, collecting everything together, we have the following equality: for −2 ≤ b < 1
∞∫
0
ln |ζ(b+ it)|
cosh(at)
dt = π
∑
ρ,tk>0
2lk
σk−b∫
0
cos(ax) cosh(atk)
cos2(ax) cosh2(atk) + sin2(ax) sinh2(atk)
dx+
+
π
2a
ln
1− sin(a(1− b))
1 + sin(a(1− b))
+ π
∞∑
n=0
(−1)n
a
ln(ζ(b+ π/(2a) + πn/a)). (10)
As usual, the sum here is taken over all zeroes ρ = σk + itk with b < σk taken into account
their multiplicities; when obtaining (10) we paired the contributions of the complex conjugate zeroes
ρ = σk ± itk hence tk > 0.
This is easy to see that if we take a(1− b) < π/2 than the integrand in
σk−b∫
0
cos(ax) cosh(atk)
cos2(ax) cosh2(atk) + sin2(ax) sinh2(atk)
dx
is always positive (the value σk − b can not exceed 1 − b) and thus we have proven the following
theorem.
Theorem 6. An equality
a
π
∞∫
0
ln |ζ(b+ it)|
cosh(at)
dt =
1
2
ln
1− sin(a(1− b))
1 + sin(a(1− b))
+
∞∑
n=0
(−1)n ln(ζ(b+ π/(2a) + πn/a)), (11)
where b, a are real positive numbers such that 1 > b ≥ 1/2, a(1− b) < π/2 holds true for some b if
and only if there are no Riemann function zeroes with σ > b. For b = 1/2 this equality is equivalent
to the Riemann hypothesis.
It is interesting to take b = 1/2 and consider the limit a → π. There are no Riemann function
zeroes with Re s = 1 hence the positivity of the contributions of Riemann function zeroes nonlying
on the critical line is still certain and it remains only to consider the limit
lim
a→π
(
1
2
ln
1− sin(a(1− b))
1 + sin(a(1− b))
+ ln
(
ζ
(
1/2 +
π
2a
)))
.
This can be done without problems and we obtain the following equality equivalent to the Riemann
hypothesis
∞∫
0
ln |ζ(1/2 + it)|
cosh(πt)
dt = ln
π
2
+
∞∑
n=2
(−1)n+1 ln(ζ(n)). (12)
Here those logarithms of the Riemann function which correspond to even n can be expressed
via Bernoulli numbers B2m, m = 1, 2, 3, . . . : ζ(2m) = (−1)m+1(2π)2m
B2m
2(2m)!
, and the fol-
lowing remarkable property of the Riemann ζ-function
∏∞
n=2
ζ(n) = C = 2.29485 . . . , that is
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
228 S. K. SEKATSKII, S. BELTRAMINELLI, D. MERLINI∑∞
n=2
ln(ζ(n)) = lnC = 0.8306 . . .can be also used for calculations, see [10, p. 16]. Here C is
the residue of the pole at s = 1 of the Dirichlet series whose coefficients g(n) are the numbers of
non-isomorphic Abel’s group of order n. Eq. (12) also has been tested numerically.
Conclusions. In this paper we have established a number of criteria involving the integrals
of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis. Our results
include all earlier known criteria of this kind [3, 6, 9] which are certain particular cases of the general
approach proposed. An infinite number of other criteria of the type “
∫ b+i∞
b−i∞
g(z) ln(ζ(z))dz = f(b)
is equivalent to RH” can be constructed following the lines of the present paper, viz. selecting an
appropriate function g(z) and calculating the value of a contour integral exploiting the generalized
Littlewood theorem.
1. Titchmarsh E. C., Heath-Brown E. R. The theory of the Riemann Zeta-function. – Oxford: Clarendon Press, 1988.
2. Wang F. T. A note on the Riemann Zeta-function //Bull. Amer. Math. Soc. – 1946. – 52, – P. 319 – 321.
3. Volchkov V. V. On an equality equivalent to the Riemann hypothesis // Ukr. Math. J. – 1995. – 47, № 3. – P. 422 – 423.
4. Balazard M., Saias E., Yor M. Notes sur la fonction de Riemann // Adv. Math. – 1999. – 143. – P. 284 – 287.
5. Merlini D. The Riemann magneton of the primes // Chaos and Complex. Lett. – 2006. – 2. – P. 93 – 98.
6. Sekatskii S. K., Beltraminelli S., Merlini D. A few equalities involving integrals of the logarithm of the Riemann
ζ-function and equivalent to the Riemann hypothesis I, II, III // arXiv: 0806.1596v1, 0904.1277v1, 1006.0323v1.
7. Titchmarsh E. C. The theory of functions. – Oxford : Oxford Univ. Press, 1939.
8. Ramaré C., Saouter Y. Short effective intervals containing primes // J. Number Theory. – 2003. – 98. – P. 10 – 33.
9. Gourdon X. The 1013 first zeroes of the Riemann Zeta Function, and zeroes computation at very large height //
http://numbers.computation.free.fr/Constants/miscellaneous/zetazeroes1e13_1e24.pdf
10. Zagier D. B. Zetafunktionen und quadratische Körper (Eine Einführung in die höhere Zahlentheorie). – Berlin etc.:
Springer, 1981.
Received 29.03.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
|
| id | umjimathkievua-article-2568 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:56Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a6/1e26ec1ecb15bd27b059c50688226fa6.pdf |
| spelling | umjimathkievua-article-25682020-03-18T19:29:46Z On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis Про рiвностi, що мiстять iнтеграли вiд логарифмa ζ-функцiї рiмана i еквiвалентнi гiпотезi Рiмана Beltraminelli, S. Merlini, D. Sekatskii, S. K. Белтрамінеллі, С. Мерліні, Д. Секацький, С.К. Using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytical function, we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis can be established and present some of them as an example. It is shown that all earlier known equalities of this type, viz., the Wang equality, Volchkov equality, Balazard-Saias-Yor equality, and an equality established by one of the authors, are certain particular cases of our general approach. Показано як за допомогою узагальненої теореми Лiттлвуда про контурний iнтеграл, що мiстить логарифм аналiтичної функцiї, можна отримати нескiнченну кiлькiсть iнтегральних рiвностей, що мiстять iнтеграли вiд логарифма ζ-функцiї Рiмана i є еквiвалентними гiпотезi Рiмана, i наведено кiлька таких рiвностей у якостi прикладу. Показано, що деякi вiдомi рiвностi такого типу, а саме, рiвностi Ванга, Волчкова, Балазарда – Сайаса – Йора та рiвнiсть, що встановлена одним iз авторiв, є частинними випадками нaшого загального пiдходу Institute of Mathematics, NAS of Ukraine 2012-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2568 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 2 (2012); 218-228 Український математичний журнал; Том 64 № 2 (2012); 218-228 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2568/1895 https://umj.imath.kiev.ua/index.php/umj/article/view/2568/1896 Copyright (c) 2012 Beltraminelli S.; Merlini D.; Sekatskii S. K. |
| spellingShingle | Beltraminelli, S. Merlini, D. Sekatskii, S. K. Белтрамінеллі, С. Мерліні, Д. Секацький, С.К. On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis |
| title | On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis |
| title_alt | Про рiвностi, що мiстять iнтеграли вiд логарифмa ζ-функцiї рiмана i еквiвалентнi гiпотезi Рiмана |
| title_full | On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis |
| title_fullStr | On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis |
| title_full_unstemmed | On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis |
| title_short | On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis |
| title_sort | on equalities involving integrals of the logarithm of the riemann ζ-function and equivalent to the riemann hypothesis |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2568 |
| work_keys_str_mv | AT beltraminellis onequalitiesinvolvingintegralsofthelogarithmoftheriemannzfunctionandequivalenttotheriemannhypothesis AT merlinid onequalitiesinvolvingintegralsofthelogarithmoftheriemannzfunctionandequivalenttotheriemannhypothesis AT sekatskiisk onequalitiesinvolvingintegralsofthelogarithmoftheriemannzfunctionandequivalenttotheriemannhypothesis AT beltramínellís onequalitiesinvolvingintegralsofthelogarithmoftheriemannzfunctionandequivalenttotheriemannhypothesis AT merlíníd onequalitiesinvolvingintegralsofthelogarithmoftheriemannzfunctionandequivalenttotheriemannhypothesis AT sekacʹkijsk onequalitiesinvolvingintegralsofthelogarithmoftheriemannzfunctionandequivalenttotheriemannhypothesis AT beltraminellis prorivnostiŝomistâtʹintegralividlogarifmazfunkciírimanaiekvivalentnigipotezirimana AT merlinid prorivnostiŝomistâtʹintegralividlogarifmazfunkciírimanaiekvivalentnigipotezirimana AT sekatskiisk prorivnostiŝomistâtʹintegralividlogarifmazfunkciírimanaiekvivalentnigipotezirimana AT beltramínellís prorivnostiŝomistâtʹintegralividlogarifmazfunkciírimanaiekvivalentnigipotezirimana AT merlíníd prorivnostiŝomistâtʹintegralividlogarifmazfunkciírimanaiekvivalentnigipotezirimana AT sekacʹkijsk prorivnostiŝomistâtʹintegralividlogarifmazfunkciírimanaiekvivalentnigipotezirimana |