Generalized Bombieri–Lagarias’ theorem and generalized Li’s criterion with its arithmetic interpretation
We show that Li’s criterion equivalent to the Riemann hypothesis, i.e., the statement that the sums kn=Σρ(1−(1−1ρ)n) over zeros of the Riemann xi-function and the derivatives are nonnegative if and only if the Riemann hypothesis is true, can be generalized and the nonnegativity of certain derivative...
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irk-123456789-1653132020-02-14T01:27:21Z Generalized Bombieri–Lagarias’ theorem and generalized Li’s criterion with its arithmetic interpretation Sekatski, S.K. Статті We show that Li’s criterion equivalent to the Riemann hypothesis, i.e., the statement that the sums kn=Σρ(1−(1−1ρ)n) over zeros of the Riemann xi-function and the derivatives are nonnegative if and only if the Riemann hypothesis is true, can be generalized and the nonnegativity of certain derivatives of the Riemann xi-function estimated at an arbitrary real point a, except a = 1/2, can be used as a criterion equivalent to the Riemann hypothesis. Namely, we demonstrate that the sums kn,a=Σρ(1−(ρ−aρ+a−1)n) for any real a such that a < 1/2 are nonnegative if and only if the Riemann hypothesis is true (correspondingly, the same derivatives with a > 1/2 should be nonpositive). The arithmetic interpretation of the generalized Li’s criterion is given. Similarly to Li’s criterion, the theorem of Bombieri and Lagarias applied to certain multisets of complex numbers is also generalized along the same lines. Показано, що критерій Лі є еквівалентним гіпотезі Рімана, тобто твердження, що суми kn=Σρ(1−(1−1ρ)n) нулях ріманової хі-функції та похідні, є невід'ємними тоді і тільки тоді, коли справедлива гіпотеза Рімана, може бути узагальнене, а невід'ємність деяких похідних ріманової хі-функції, що оцінюються у довільній точці a, крім a=1/2, може бути застосована, як критерій, еквівалентний гіпотезі Рімана. А саме, показано, що суми kn,a=Σρ(1−(ρ−aρ+a−1)n) для будь-яких дійсних a та будь-яких a<1/2 є невід'ємними тоді і тільки тоді, коли справедлива гіпотеза Рімана (відповідно такі ж похідні з a>1/2 повинні бути недодатніми). Наведено арифметичну інтерпретацію узагальненого критерію Лі. Подібно до критерію Лі теорема Бомбієрі та Лагаріаса, у застосуванні до деяких мультимножин комплексних чисел, також може бути узагальнена аналогічним чином. 2014 Article Generalized Bombieri–Lagarias’ theorem and generalized Li’s criterion with its arithmetic interpretation / S.K. Sekatski // Український математичний журнал. — 2014. — Т. 66, № 3. — С. 371–383. — Бібліогр.: 13 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165313 512.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Sekatski, S.K. Generalized Bombieri–Lagarias’ theorem and generalized Li’s criterion with its arithmetic interpretation Український математичний журнал |
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We show that Li’s criterion equivalent to the Riemann hypothesis, i.e., the statement that the sums kn=Σρ(1−(1−1ρ)n) over zeros of the Riemann xi-function and the derivatives are nonnegative if and only if the Riemann hypothesis is true, can be generalized and the nonnegativity of certain derivatives of the Riemann xi-function estimated at an arbitrary real point a, except a = 1/2, can be used as a criterion equivalent to the Riemann hypothesis. Namely, we demonstrate that the sums kn,a=Σρ(1−(ρ−aρ+a−1)n) for any real a such that a < 1/2 are nonnegative if and only if the Riemann hypothesis is true (correspondingly, the same derivatives with a > 1/2 should be nonpositive). The arithmetic interpretation of the generalized Li’s criterion is given. Similarly to Li’s criterion, the theorem of Bombieri and Lagarias applied to certain multisets of complex numbers is also generalized along the same lines. |
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| author_facet |
Sekatski, S.K. |
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Sekatski, S.K. |
| title |
Generalized Bombieri–Lagarias’ theorem and generalized Li’s criterion with its arithmetic interpretation |
| title_short |
Generalized Bombieri–Lagarias’ theorem and generalized Li’s criterion with its arithmetic interpretation |
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Generalized Bombieri–Lagarias’ theorem and generalized Li’s criterion with its arithmetic interpretation |
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Generalized Bombieri–Lagarias’ theorem and generalized Li’s criterion with its arithmetic interpretation |
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Generalized Bombieri–Lagarias’ theorem and generalized Li’s criterion with its arithmetic interpretation |
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generalized bombieri–lagarias’ theorem and generalized li’s criterion with its arithmetic interpretation |
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Інститут математики НАН України |
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2014 |
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Generalized Bombieri–Lagarias’ theorem and generalized Li’s criterion with its arithmetic interpretation / S.K. Sekatski // Український математичний журнал. — 2014. — Т. 66, № 3. — С. 371–383. — Бібліогр.: 13 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT sekatskisk generalizedbombierilagariastheoremandgeneralizedliscriterionwithitsarithmeticinterpretation |
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2025-07-14T18:19:01Z |
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2025-07-14T18:19:01Z |
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1837647414981296128 |
| fulltext |
© S. K. SEKATSKII, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3 371
UDC 512.5
S. K. Sekatskii (LPMV, Ecole Polytechnique Fédérale de Lausanne, Switzerland)
GENERALIZED BOMBIERI – LAGARIAS’ THEOREM AND GENERALIZED
LI’S CRITERION WITH ITS ARITHMETIC INTERPRETATION
УЗАГАЛЬНЕНА ТЕОРЕМА БОМБІЄРІ – ЛАГАРІАСА ТА УЗАГАЛЬНЕНИЙ
КРИТЕРІЙ ЛІ ЗІ СВОЄЮ АРИФМЕТИЧНОЮ ІНТЕРПРЕТАЦІЄЮ
We show that Li’s criterion equivalent to the Riemann hypothesis, viz. the statement that the sums kn =
= !" 1# 1# 1
"
$
%&
'
()
n$
%
&
'
(
) over Riemann xi-function zeroes and derivatives !n "
1
(n #1)!
dn
dzn
(zn#1 ln ($(z)))
z=1
, where
n = 1, 2, 3,… , are nonnegative if and only if the Riemann hypothesis is true, can be generalized and the nonnegativity of
certain derivatives of the Riemann xi-function estimated at an arbitrary real point a , except a = 1/2 , can be used as a
criterion equivalent to the Riemann hypothesis. Namely, we demonstrate that the sums kn,a = !" 1#
" # a
" + a #1
$
%&
'
()
n$
%
&
'
(
) for
any real a and any a < 1/2 are nonnegative if and only if the Riemann hypothesis is true (correspondingly, the same
derivatives with a > 1/2 should be nonpositive). The arithmetic interpretation of the generalized Li’s criterion is given.
Similarly to Li’s criterion, the theorem of Bombieri and Lagarias applied to certain multisets of complex numbers is also
generalized along the same lines.
Показано, що критерій Лі є еквівалентним гіпотезі Рімана, тобто твердження, що суми kn = !" 1# 1# 1
"
$
%&
'
()
n$
%
&
'
(
) по
нулях ріманової хі-функції та похідні !n "
1
(n #1)!
dn
dzn
(zn#1 ln ($(z)))
z=1
, де n = 1, 2, 3,… , є невід’ємними тоді і
тільки тоді, коли справедлива гіпотеза Рімана, може бути узагальнене, а невід’ємність деяких похідних ріманової хі-
функції, що оцінюються у довільній точці a , крім a = 1/2 , може бути застосована, як критерій, еквівалентний
гіпотезі Рімана. А саме, показано, що суми kn,a = !" 1#
" # a
" + a #1
$
%&
'
()
n$
%
&
'
(
) для будь-яких дійсних a та будь-яких
a < 1/2 є невід’ємними тоді і тільки тоді, коли справедлива гіпотеза Рімана (відповідно такі ж похідні з a > 1/2
повинні бути недодатніми). Наведено арифметичну інтерпретацію узагальненого критерію Лі. Подібно до критерію
Лі теорема Бомбієрі та Лагаріаса, у застосуванні до деяких мультимножин комплексних чисел, також може бути
узагальнена аналогічним чином.
1. Introduction. In 1997, Li has established the following criterion equivalent to the Riemann hy-
pothesis concerning nontrivial zeroes of the Riemann ! -function (see, e.g., [1] for standard defini-
tions and discussion of the general properties of this function) and now bearing his name (Li’s crite-
rion) [2]:
Li’s criterion. Riemann hypothesis is equivalent to the nonnegativity of the following numbers:
!n " 1
(n #1)!
dn
dzn
zn#1 ln ($(z))
z=1( ) (1)
for any nonnegative integer n .
372 S. K. SEKATSKII
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
Here !(z) is the Riemann xi-function related with the Riemann ! -function by the well-
known relation [1]
!(z) = 1
2
z(z "1)#"z /2$(z/2)%(z) . (2)
Two years later, Bombieri and Lagarias generalized Li’s criterion [3]. If ! = 1/2 + iT , T real
and i = !1 , than (! "1)/! = 1 and hence can be written as exp (i!i ) , where !i =
= arctan T
T 2 !1/4
. Let us introduce the sum kn = !" 1# 1# 1
"
$
%&
'
()
n$
%&
'
()
= !" 1# " #1
"
$
%&
'
()
n$
%&
'
()
over
nontrivial Riemann function zeroes ( n is nonnegative integer, zeroes are counting taking into ac-
count their multiplicity, for n = 1 contributions of complex conjugate zeroes should be paired
when summing). For two complex conjugate “correct” Riemann function zeroes ! = 1/2 ± iT we
easily see that their contribution to sum kn is 2(1! cos (n"i )) , and hence nonnegative; corre-
spondingly, the sum kn is also nonnegative. Quite the contrary, if some nontrivial Riemann func-
tion zero with Re ! " 1/2 exists, for large enough n we will have an arbitrary large (by module)
negative contributions from these zeroes, and it is straightforward to show that for infinitely many
n this contribution can not be compensated by all other “correct” 1! cos (n"i ) terms of the sum
[3], whence infinitely many sums kn are to be negative.
This consideration immediately shows that the nonnegativity of the sums kn is equivalent to
the Riemann hypothesis. Li also demonstrated that these sums are equal to derivatives presented in
eq. (1) (certainly, this is the most technically difficult part of his work; another derivation of this re-
lation will be given shortly below).
2. Generalized Li’s and Bombieri – Lagarias’ criteria. Now we note that for ! = 1/2 + iT
and any real a ! " a
! + a "1
= " a +1/2 + iT
a "1/2 + iT
= 1 and introduce the sum
kn,a = 1! " ! a
" + a !1
#
$%
&
'(
n#
$%
&
'("
) = 1! 1! 2a !1
" + a !1
#
$%
&
'(
n#
$%
&
'("
) .
To demonstrate that on RH all these sums are nonnegative, just replace !i = arctan
T
T 2 "1/4
given
above by !i = arctan
T (2a "1)
T 2 " a2 + a "1/4
and repeat all the abovesaid. To demonstrate the inverse
implication, let us briefly reproduce a slightly modified argument of Bomberi and Lagarias [3]; see
their original paper for some more details.
Let a < 1/2 . We observe that for any Riemann zero ! = " + iT , ! " a
! + a "1
2
=
= 1+ (1! 2a)(2" !1)
# + a !1 2
, and thus if ! > 1/2 we may find at least one zero for which
GENERALIZED BOMBIERI – LAGARIAS’ THEOREM AND GENERALIZED LI’S CRITERION … 373
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
! " a
! + a "1
> 1 . Because (1! 2a)(2" !1)
# + a !1 2
tends to zero when !k tends to infinity, maximum of
this expression over ! is achieved and there are only finitely many, say K , zeroes !k for which
! " a
! + a "1
= 1+ t = max , for all others ! " a
! + a "1
# 1+ t " $ for some fixed positive ! . Clearly,
taking n large enough, the term 1! "k ! a
"k + a !1
#
$%
&
'(
n
= 1! (1+ t)n exp (in)k ) !k
"
#$
is an argument
of !k " a
!k + a "1
#
$%
&
'(
&
'(
can be made very large by module and negative. Then, due to the Dirichlet’s
theorem on simultaneous Diophantine approximation, the sum of 1! "k ! a
"k + a !1
#
$%
&
'(
n
over all !k
can be made arbitrary close to K (1! (1+ t)n ) while the sum over all other zeroes is of the order of
O(n2(1+ t ! ")n ) , just due to their known density. The case a > 1/2 is quite similar, so we have
proven the following theorem.
Theorem 1. Riemann hypothesis is equivalent to the nonnegativity of sums
kn,a = 1! " ! a
" + a !1
#
$%
&
'(
n#
$
%
&
'
(
"
) = 1! 1! 2a !1
" + a !1
#
$%
&
'(
n#
$
%
&
'
(
"
)
taken over the Riemann xi-function zeroes for any real a , except a = 1/2 . Here n is nonnega-
tive integer, zeroes are counting taking into account their multiplicity, for n = 1 contributions of
complex conjugate zeroes should be paired when summing.
Indeed, we proved this statement not only for the Riemann ! -function zeroes but for certain
multisets of complex numbers, see [3]. For completeness, here we formulate this result as a follow-
ing theorem.
Theorem 2 (Generalized Bombieri – Lagarias’ theorem). Let a and ! are arbitrary real
numbers, a < ! , and R be a multiset of complex numbers ! such that
(i) 2! " a #R ;
(ii) 1+ Re !( )/(1+ ! + a " 2# 2 )
!$ < +% .
Then the following conditions are equivalent:
(a) Re ! " # for every ! ;
(b) Re 1! " ! a
" ! 2# + a
$
%&
'
()
n$
%
&
'
(
)"* + 0 for n = 1, 2, 3,… ;
(c) for every fixed ! > 0 there is a positive constant c(!) such that
Re 1! " ! a
" ! 2# + a
$
%&
'
()
n$
%
&
'
(
)"* + ! c(,)e,n , n = 1, 2, 3,… .
374 S. K. SEKATSKII
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
If at the same conditions a > ! is taken, the point (a) is to be changed to
(a′) Re ! " # for every ! ,
points (b), (c) remain unchanged.
As you see, the statement of Theorem 2 is formulated for any ! , not only for ! = 1/2 , pro-
vided ! " a . To demonstrate this, just note that for ! = " + iT , ! " a
! + a " 2#
= # " a + iT
a " # + iT
= 1
and for ! = q + iT , ! " a
! + a " 2#
2
= 1+ 4(# " a)(q " #)
! + a " 2# 2 , and then repeat all the abovesaid.
If, additionally to the aforementioned conditions of the generalized Bombieri – Lagarais’ theo-
rem, also the following takes place:
(iii) If ! "R , than ! "R with the same multiplicity as ! , one can omit the operation of
taking the real part in (b), (c), the expressions at question are real. (Here, as usual, ! means a
complex conjugate of ! .)
Following again the paper of Bombieri and Lagarais [3], we conclude this section with the fol-
lowing theorem.
Theorem 3 (Generalized Li’s criterion). Let a is an arbitrary real number, a ! " , and R
be a multiset of complex numbers ! such that
(i) 2! " a #R , a !R ;
(ii) 1+ Re !( )/ 1+ ! + a " 2# 2( )!$ < +% , 1+ Re !( )/ 1+ ! " a 2( )!# < +$ ;
(iii) if ! "R , than 2! " # $R .
Then the following conditions are equivalent:
(a) Re ! = " for every ! ;
(b) Re 1! " ! a
" + a ! 2#
$
%&
'
()
n$
%
&
'
(
)"* + 0 for any a and n = 1, 2, 3,… ;
(c) for every fixed ! > 0 and any a there is a positive constant c(!, a) such that
Re 1! " ! a
" + a ! 2#
$
%&
'
()
n$
%
&
'
(
)"* + ! c(,, a)e,n , for n = 1, 2, 3,… .
For clearly, in the conditions of the theorem we for all ! have Re ! " # and Re (2! " #) ≤
≤ ! , whence Re ! = " . If, additionally to the aforementioned conditions of the generalized Li’s
criterion, also the following takes place:
(iv) If ! "R , than complex conjugate ! "R with the same multiplicity as ! ,
one can omit the operation of taking the real part in (b), (c), the expressions at question are real.
Remark 1. Similarly to the Li’s criterion, generalized Li’s criterion can be applied also to nu-
merous other zeta-functions, as this was shown first by Li himself for Dedekind zeta-function [2],
and afterwards was the subject of a number of sequel papers by other authors. We will not pursue
this line of researches here.
3. Connection between generalized Li’s sums and certain derivatives of Riemann xi-
function. Our next aim is to establish relation “of the Li’s type” similar to eq. (1), viz. the relation
GENERALIZED BOMBIERI – LAGARIAS’ THEOREM AND GENERALIZED LI’S CRITERION … 375
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
between sums kn,a and certain derivatives of the Riemann xi-function. For this we will use the
generalized Littlewood theorem concerning contour integrals of logarithm of an analytical function,
recently used by us to establish numerous equalities equivalent to the Riemann hypothesis [4], which
for completeness we reproduce below. The proof [4] is a straightforward modification of familiar
and well known corresponding Littlewood theorem (or lemma) proof (see, e.g., [5]). Actually, this
theorem has been more or less explicitly used in Riemann researches already by Wang who in 1946
established the first integral equality equivalent to the Riemann hypothesis [6].
Theorem 4 (Generalized Littlewood theorem). 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
nonzero on C and meromorphic inside it, let also g(z) is analytic on C and meromorphic in-
side it. Let F(z) = ln ( f (z)) , the logarithm being defined as follows: we start with a particular de-
termination 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 from above or below. Let also the poles
and zeroes of the functions f (z) , g(z) do not coincide.
Then
F(z)g(z)dz
C
! = 2"i res (g(#g ) $F(#g )
#g
% ) & g(z)dz
X1+iY#0
X#
0+iY#0
!
# f0
% + g(z)dz
X1+iY#pol
X#
pol+iY#pol
!
# f
pol
%
'
(
)
)
*
+
,
,
,
where the sum is over all !g which are poles of the function g(z) lying inside C , all ! f
0 =
= X!
0 + iY!0 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 lie inside
C , and all ! f
pol = X!
pol + iY!pol which are poles of the function f (z) counted taking into account
their multiplicities and which lie inside C . For this is true all relevant integrals in the right-hand
side of the equality should exist.
Remark 2. Actually, 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 have dealt with a few
such cases before [4].
The subtle moment related with this generalized Littlewood theorem is the circumstance that the
function arg (F(z)) (imaginary part of the ln ( f (z)) is not continuous on the left border of the
contour (segment X1 + iY1 , X1 + iY2 ) if there are zeroes or poles of the function f (z) inside the
contour. This is explicitly stated in the theorem condition: 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 from above
or below. In practice, this means that when calculating the corresponding part of the contour inte-
gral, viz. the integral ! arg (F(z))g(z)dz
X1+iY1
X1+iY2" (minus sign comes from the necessity to round the
contour counterclockwise), ± 2!il jumps should be added to an argument function at a point
376 S. K. SEKATSKII
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
X1 + iYz, p whenever a zero or a pole of an order l of the function f (z) occurs somewhere at a
point X + iYz, p inside the contour. Corresponding integral should be properly modified if the use
of a continuous argument branch is desirable. See our paper [7] for details, we also would like to
note that the appropriateness of the necessary modification of an argument has been numerically
tested (and confirmed) by us for a number of integrals, e.g., for the integral
t arg (!(1 / 4 + it))
(1/16 + t 2 )2
dt
0
"
# = $ %! (1/2)
!(1/2)
& 9$ & $ 1
tk2 +1/4
'
()
*
+,-,.k>1/4, tk>0
/
!
"#
similar equality in the form
t arg (!(1 / 2 + it))
(1/16 + t 2 )2
dt
0
"
# = $ %! (3/4)
!(3/4)
& 32$
3
is equivalent to the Riemann hypothesis, our Theorem 5 from [4]!"# . However, for what follows the
asymptotic of the function g(z) for large values of X1 tending to minus infinity makes this modi-
fication irrelevant, the value of the integral ! arg (F(z))g(z)dz
X1+iY1
X1+iY2" tends to zero anyway.
First, as an exercise, we use this theorem to establish the Li’s relation (1). For this, let us con-
sider the rectangular contour C with vertices at ±X ± iX with real X ! +" , if some Riemann
zero occurs on the contour just shift it a bit to avoid this, and consider a contour integral
g(z) ln (!(z))dz
C" where
g(z) = n
(z !1)2
z
z !1
"
#
$
%
n!1
! n
(z !1)2
. (3)
Known asymptotic of the logarithm of the xi-function for large z , ! O(z ln z) guaranties the
“disappearance” of the contour integral value (it tends to zero when X ! " due to the asymptotic
g(z) ! O(1/z3) ) thus we get, after division by 2!i ,
n 1
(n !1)!
dn
dzn
(zn!1 ln ("(z)))
z=1
! n " '
"
(1) ! 1! 1! 1
1! #
$
%&
'
()
n$
%&
'
()#
* ! n 1
#
= 0
#
* . (4)
Complex
!
"#
conjugate zeroes are to be paired when calculating 1
!!" and
1! 1! 1
1! "
#
$%
&
'(
n#
$%
&
'(") for n = 1.
!
"#
Here the first term is the contribution of the n +1 order pole
GENERALIZED BOMBIERI – LAGARIAS’ THEOREM AND GENERALIZED LI’S CRITERION … 377
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
of g(z) at z = 1 , second term is the contribution of the second order pole arising from the term
! n
(z !1)2
occurring in (3), the third and fourth terms are the integrals ! g(z)dz
! "+iTi
#i$ . Clearly,
!
"#
n
(z !1)2
z
z !1
"
#$
%
&'
n!1
= d
dz
1! 1! 1
1! z
"
#$
%
&'
n"
#$
%
&'
which explains while function g(z) in form (3) is used; the term ! n
(z !1)2
is added just to en-
sure the asymptotic g(z) ! O(1/z3) necessary to bring the contour integral value to zero. Note also
evident
1! 1! 1
1! "
#
$%
&
'(
n#
$
%
&
'
(
"
) = 1! 1! 1
"
#
$%
&
'(
n#
$
%
&
'
( .
"
)
&
'
((
We know that !"
"
(1) = # 1
$$% [1], so that we have
n 1
(n !1)!
dn
dzn
(zn!1 ln("(z)))
z=1
= 1! 1! 1
#
$
%&
'
()
n$
%
&
'
(
)
#
*
which is a relation we have searched for.
Quite similar consideration is applied to our case, where now we introduce the function
!g(z) = ! n(2a !1)(z ! a)
n!1
(z + a !1)n+1
+ n(2a !1)
(z + a !1)2
(5)
and consider contour integral
!g(z) ln (!(z))dz
C" taken round the same contour as above. Applica-
tion of Theorem 4 (generalized Littlewood theorem) gives
! n(2a !1)
(n !1)!
dn
dzn
((z ! a)n!1 ln ("(z)))
z=1!a
+ n(2a !1) #"
"
(z) z=1!a –
– 1! " ! a
" + a !1
#
$%
&
'(
n#
$
%
&
'
(
"
) + n(2a !1) 1
" + a !1
= 0
"
) . (6)
(Again, complex conjugate zeroes are to be paired whenever necessary.) Using well known
!"
"
(z)
z=1#a
= # 1
$ + a #1$
% [1] and reminding our Theorem 1 we have the following theorem.
378 S. K. SEKATSKII
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
Theorem 5. Riemann hypothesis is equivalent to the nonnegativity of all derivatives
1
(n !1)!
dn
dzn
((z ! a)n!1 ln ("(z)))
z=1!a
for all nonnegative integers n and any real a < 1/2 ; cor-
respondingly, it is equivalent also to the nonpositivity of all derivatives
1
(n !1)!
dn
dzn
((z ! a)n!1 ln ("(z)))
z=1!a
for all nonnegative integers n and any real a > 1/2 .
Remark 3. Another possibility to arrive to the same conclusions is to see the formula
! " a
! + a "1
= " a +1/2 + iT
a "1/2 + iT
= 1 as a precursor for the conformal mapping s = z ! a
z + a !1
. For
a < 1/2 and Re z ! 1/2 , module of s is always less or equal to 1; this equality is realized only
on the line z = 1/2 + it . Correspondingly, on RH the function ln ! z " a
z + a "1
#
$%
&
'( is analytic in the
interior of the discus s < 1 . We will not pursue this line of researches here, see again [2, 3] and
our paper [8] where similar idea was used to generalize Balazard – Saias – Yor’s criterion equivalent
to the Riemann hypothesis [9]. Similarly, for more general case s = z ! a
z + a ! 2"
, if a < ! and
Re z ! " , module of s is always less or equal to 1, and if a > ! and Re z ! " , this module
also is always less or equal to 1. This illustrates again our Theorem 2 (the generalized Bombieri –
Lagarias’ theorem).
Remark 4. Along similar lines, analogous formulae connecting generalized Li’s sums and cer-
tain derivatives of the logarithm, can be established for numerous other zeta-functions. We will not
pursue this line of researches here.
Now we want to prove the following minor theorem.
Theorem 6. The statement that there are no nontrivial Riemann function zeroes with
Re ! > " > 1/2 is equivalent to the statement that for any a < ! all derivatives
1
(n !1)!
dn
dzn
((z ! a)n!1 ln ("(z)))
z=2#!a
are nonnegative and for any a > 1! " all derivatives
1
(n !1)!
dn
dzn
((z ! a)n!1 ln ("(z)))
z=2!2#!a
are nonpositive.
Proof. From our Theorem 2 (generalized Bombieri – Lagarias’ theorem), we know that the con-
dition that there are no nontrivial Riemann function zeroes with Re ! = " > 1/2 is equivalent to the
statement that for any a < ! all 1! " ! a
" ! 2# + a
$
%&
'
()
n$
%
&
'
(
)"* + 0 for n = 1, 2, 3,… . These sums
are calculated using our Theorem 4 (the generalized Littlewood theorem) exactly as above, with the
only difference that now the function
!!g(z) = n(2! " 2a)(z " a)n"1
(z + a " 2!)n+1
" n(2! " 2a)
(z + a " 2!)2
instead of the
function !g(z) given by (5) is exploited. Such change brings the equality between the sums
GENERALIZED BOMBIERI – LAGARIAS’ THEOREM AND GENERALIZED LI’S CRITERION … 379
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
1! " ! a
" ! 2# + a
$
%&
'
()
n$
%
&
'
(
)"* + 0 and derivatives n(2! " 2a)
(n "1)!
dn
dzn
((z " a)n"1 ln (#(z)))
z=2!"a
which,
consequently, also should be nonnegative.
If there are no zeroes with Re ! > " > 1/2 , there are also no zeroes with Re ! < 1" # and we
are able to apply Theorem 2 with a > 1! " ; all corresponding sums
1! " ! a
" ! 2 + 2# + a
$
%&
'
()
n$
%
&
'
(
)
"
* + 0
and are given by n(2(1! ") ! 2a)
(n !1)!
dn
dzn
((z ! a)n!1 ln ("(z)))
z=2!2#!a
whence the derivatives
1
(n !1)!
dn
dzn
((z ! a)n!1 ln ("(z)))
z=2!2#!a
should be nonpositive.
Theorem 6 is proved.
3. An arithmetic interpretation of the generalized Li’s criterion. In Ref. [3], Bombieri and
Lagarias demonstrated the relation of Li’s criterion with the so called Weil’s explicit formula in the
theory of prime numbers and Weil’s criterion of the truth of the Riemann hypothesis (see [10, 11])
and gave an arithmetic interpretation of the Li’s criterion. Lately, such an interpretation has been
given for some other Zeta-functions (see, e.g., [12]). For completeness, we would like to conclude
this paper establishing an arithmetic interpretation of the generalized Li’s criterion. We closely fol-
low the lines used in [3] here.
For suitable function f , Mellin transform is defined as f̂ (s) = f (x)xs!1 dx
0
"
# while inverse
Mellin transform formula is f (x) = 1
2!i
f̂ (s)x"s ds
Re s=c# with an appropriate value of c . The
following is more or less a repetition of Lemma 2 from [3], which is a particular case corresponding
to a = 1 .
Lemma 1. For n = 1, 2, 3,… , and an arbitrary complex number a the inverse Mellin trans-
form of the function kn,a (s) = 1! 1! 2a !1
s + a !1
"
#$
%
&'
n
is
gn,a (x) = Pn,a (x) if 0 < x < 1 ,
gn,a (x) =
n
2
(2a !1) if x = 1 , (7)
gn,a (x) = 0 if x > 1 ,
where Pn,a (x) = xa!1 Cnj
(2a !1) j ln j!1 x
( j !1)!j=1
n" ; Cnj =
n!
j !(n ! j)!
is a binomial coefficient.
380 S. K. SEKATSKII
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
Proof. We have for Re (s + a) > 1:
Cnj
(2a !1) j
( j !1)!
(ln j!1 x)xs+a!2 dx
0
1
"
j=1
n
# = Cnj
(2a !1) j
( j !1)!
d j!1
ds j!1
xs+a!2 dx
0
1
"
j=1
n
# =
= Cnj
j=1
n
! (2a "1) j ("1) j"1
(s + a "1) j
= 1" 1" 2a "1
s + a "1
#
$%
&
'(
n
.
If a is an arbitrary complex number with Re a > 1 , for the function gn (x) we can apply the so
called Explicit Formula of Weil (see [3, 10, 11]), which is, as given in [3]:
f̂ (!)
!
" = f (x)dx
0
#
$ + !f (x)dx
0
#
$ % &(n)( f (n) + !f (n))
n=1
#
" –
–
(ln ! + " ) f (1) # f (x) + !f (x) # 2
x2
f (1)$
%
&
'
(
)
xdx
x2 #11
*
+ . (8)
Here !(n) is a van Mangoldt function let!
"#
us remind that for Re s > 1 !" (s)
"(s)
= # $(m)
msm=1
%&
[1]!"# , ! = 0.572… is Euler – Mascheroni constant and
!f (x) := 1
x
f 1
x
!
"#
$
%& , thus in our case the
function
!Pn,a (x) = x!a Cnj
(!1) j!1(2a !1) j ln j!1 x
( j !1)!j=1
n" should be used whenever appropriate.
Surely,
!Pn,a (x) is inverse Mellin transform of kn,a (1! s) = 1! 1! 2a !1
a ! s
"
#$
%
&'
n
.
Such an application is justified because this is easy to check that for Re a > 1 the functions
gn,a (x) do possess the following necessary properties for eq. (8) can be used for a function f (x)
[3, 10, 11]:
(A) f (x) is continuous and continuously differentiable everywhere except at finitely many
points ai , in which both f (x) and !f (x) have at most a discontinuity of the first kind, and in
which one sets f (ai ) =
1
2
f (ai + 0) + f (ai ! 0)[ ] ;
(B) there is ! > 0 such that f (x) = O(x! ) as x! 0 + and f (x) = O(x!1!" ) as x! +" .
The use of eq. (8) gives
1! " ! a
" + a !1
#
$%
&
'(
n#
$
%
&
'
(
"
) = 1! " + a !1
" ! a
#
$%
&
'(
n#
$
%
&
'
(
"
) =
GENERALIZED BOMBIERI – LAGARIAS’ THEOREM AND GENERALIZED LI’S CRITERION … 381
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
= Cnj
(2a !1) j
( j !1)!j=1
n
" xa!1 ln j!1 x
0
1
# dx + (!1) j!1 x!a ln j!1 x dx
1
$
# ! (!1) j!1 %(m) ln j!1 m
ma
m=1
$
"
&
'
(
)(
*
+
(
,(
–
! n
2
(2a !1)(ln " + # ) ! Cnj
(!1) j!1(2a !1) j!1
( j !1)!j=1
n
$ x!a ln j!1 x ! n
x2
(2a !1)
%
&
'
('
)
*
'
+'
xdx
x2 !11
,
- . (9)
Now, in the second and third integrals in the right-hand side of (9) we make a variable transform x
to 1/x , after what these integrals take the forms I2 = xa!2 ln j!1(x)dx
0
1
" and
I3 = Cnj
ln j!1 x
( j !1)!
(2a !1) j xa!1
j=2
n
" + n(2a !1)(xa!1 ! x)
#
$
%
&%
'
(
%
)%
dx
1! x20
1
* .
(Note, that when writing I3 we move the summation term corresponding to j = 1 from the sum
to the second term under the integral sign.) The first two integrals are handled by virtue of an
Example 4.272.6 of GR book [13]:
lnµ!1(1/x)x"!1 dx
0
1
# = 1
"µ
$(µ) ; Reµ > 0 and Re ! > 0 .
Adopting for our case, we get
ln j!1(x)xa!1 dx
0
1
" = (!1)
j!1
a j
( j !1)! , ln j!1(x)xa!2 dx
0
1
" = (!1)
j!1
(a !1) j
( j !1)!.
The “second part” of the third integral I3 is, by virtue of an Example 3.244.3 of GR book [13],
equal to
I32 = n(2a !1)
xa!1 ! x
1! x2
dx
0
1
" = ! n
2
(2a !1)(# + $(a/2)) ;
! is a digamma function. In the first part of this integral we make the variable change
x = exp (! t) :
I31 = Cnj
ln j!1 x
( j !1)!
(2a !1) j xa!1 dx
1! x2j=2
n
"
0
1
# = Cnj (!1) j!1
(2a !1) j
( j !1)!
t j!1 e!at
1! e!2t
dt
0
$
#
j=2
n
" .
Applying Taylor expansion (1! e!2t )!1 = 1+ e!2t + e! 4t + e!6t +… we get further
I31 = Cnj (!1) j!1
(2a !1) j
( j !1)!
( j !1)!
(2m + a) jm=0
"
#
j=2
n
# = Cnj (!1) j!12! j (2a !1) j$( j, a/2)
j=2
n
# ,
where !(s, a) := 1
(m + a)sm=0
"# is Hurwitz zeta-function.
382 S. K. SEKATSKII
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
Using the relations
Cnj (!1) j!1(2a !1) j a! j
j=1
n
" = 1! Cnj (!1) j
2a !1
a
#
$%
&
'(
j
= 1! !1+ 1
a
#
$%
&
'(j=0
n
"
n
,
Cnj (!1) j!1(2a !1) j (a !1)! j = !1! !1! 1
a !1
"
#$
%
&'j=1
n
(
n
,
and collecting everything together we have proven the following theorem.
Theorem 7. For n = 1, 2, 3,… and an arbitrary complex a with Re a > 1 we have
1! " ! a
" + a !1
#
$%
&
'(
n#
$
%
&
'
(
"
) = 1! " + a !1
" ! a
#
$%
&
'(
n#
$
%
&
'
(
"
) = 2 ! !1+ 1
a
#
$%
&
'(
n
! !1! 1
a !1
#
$%
&
'(
n
+
+ Cnj (2a !1) j
(!1) j
( j !1)!
"(m) ln j!1 m
ma
m=1
#
$
j=1
n
$ +
+ n
2
(2a !1)("(a/2) ! ln #) + Cnj (!1) j2! j (2a !1) j$( j, a/2)
j=2
n
% . (10)
Remark 5. The case n = 1 of the Theorem 1 gives well known equality
1
a ! ""
# = 1
a
+ 1
a !1
! $(m)
ma
m=1
%
# + 1
2
(&(a / 2) ! ln ')
(see, e.g., [1]).
The same connection of Li’s criterion with Weil’s criterion of the truth of the Riemann hypothe-
sis that has been discussed in [3], takes place also for the generalized Li’s criterion. This can be
seen as follows.
The multiplicative convolution of functions f (x) , g(x) satisfying conditions (A), (B) given
above, is defined as ( f * g)(x) = f (x /y)g(y) dy
y0
!
" , and Mellin transform of such a convolution is
f̂ (s) ! ĝ(s) . For the multiplicative convolution f *
!f signs!
"# of complex conjugation and definition
!f (x) := 1
x
f 1
x
!
"
#
$ are used!"# , Mellin transform is given by f̂ (s) !
!
f (1" s) — an expression which
clearly is real and positive for Re s = 1/2 . Correspondingly, for any function expressible as f *
!f ,
on RH the sum over the nontrivial Riemann zeroes should be positive. Weil showed that this is also
a sufficient condition for RH to be true.
GENERALIZED BOMBIERI – LAGARIAS’ THEOREM AND GENERALIZED LI’S CRITERION … 383
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
Now let us remind that if h(s) = s ! a
s + a !1
then h(1! s) = 1/h(s) and thus
kn,a (s) ! kn,a (1" s) = kn,a (s) + kn,a (1" s) , (11)
where kn,a (s) = 1!
s ! a
s + a !1
"
#
$
%
n
. By construction, kn,a (s) = ĝn,a (s) and, due to general proper-
ties of the Mellin transform, !̂gn,a (s) = ĝn,a (1! s) thus (11) can be rewritten as ĝn,a (s) ! !̂gn,a (s) =
= ĝn,a (s) + !̂gn,a (s) . Whence, applying inverse Mellin transform, gn,a (x) + !gn,a (x) =
= (gn,a * !gn,a )(x) , and this establishes the aforementioned connection: right-hand side of eq. (8) is
invariant with respect to the change of f (x) into
!f (x) .
4. Conclusion. Thus we see that to judge the truth of the Riemann hypothesis, evaluation of
certain derivatives of the Riemann xi-function can be effected at any point of the real axis apart from
the point z = 1/2 . In particular, such a point can lie arbitrary far to the right from the critical strip:
however large the number b > !1/2 is, all derivatives 1
(n !1)!
dn
dzn
((z + b)n!1 ln ("(z)))
z=b+1
should be nonnegative for RH is true, and vice versa. The present author sincerely hopes that this,
and other related interesting possibilities might be useful for Riemann researches. Finally, we are
also sure that there is a room to use the approach presented in the paper to study other than Riemann
function analytic functions.
1. Titchmarsh E. C., Heath-Brown E. R. The theory of the Riemann Zeta-function. – Oxford: Clarendon Press, 1988.
2. Li X.-E. The positivity of a sequence of numbers and the Riemann hypothesis // J. Number Theory. – 1997. – 65. –
P. 325 – 333.
3. Bombieri E., Lagarias J. C. Complements to Li’s criterion for the Riemann hypothesis // J. Number Theory. – 1999. –
77. – P. 274 – 287.
4. Sekatskii S. K., Beltraminelli S., Merlini D. On equalities involving integrals of the logarithm of the Riemann ! -
function and equivalent to the Riemann hypothesis // Ukr. Math. J. – 2012. – 64, № 2. – P. 218 – 228.
5. Titchmarsh E. C. The theory of functions. – Oxford: Oxford Univ. Press, 1939.
6. Wang F. T. A note on the Riemann Zeta-function // Bull. Amer. Math. Soc. – 1946. – 52. – P. 319 – 321.
7. Sekatskii S. K., Beltraminelli S., Merlini D. A few equalities involving integrals of the logarithm of the Riemann ! -
function and equivalent to the Riemann hypothesis III. Exponential weight functions // arXiv: 1006.0323v2.
8. Sekatskii S. K., Beltraminelli S., Merlini D. A few equalities involving integrals of the logarithm of the Riemann ! -
function and equivalent to the Riemann hypothesis I // arXiv: 0806.1596v1.
9. Balazard M., Saias E., Yor M. Notes sur la fonction de Riemann // Adv. Math. – 1999. – 143. – P. 284 – 287.
10. Weil A. Sur les “formules explicites” de la théorie des nombres premiers // Meddelanden Fran Lunds Univ. Math.
Sem. (dedié a M. Riesz). – 1952. – P. 252 – 265 (see also Weil A. OEvres scientifique. – Collected papers. – New
York: Springer, 1980. – Vol. 2. – P. 48 – 61).
11. Bombieri E. Remarks on Weil’s quadratic functional in the theory of prime numbers I // Rend. Mat. Accad. Lincei. –
2000. – 9. – P. 183-233.
12. Smajlovic L. On Li’s criterion for the Riemann hypothesis for the Selberg class // J. Number Theory. – 2010. – 130. –
P. 828 – 851.
13. Gradshtein I. S., Ryzhik I. M. Tables of integrals, series and products. – New York: Academic, 1990.
Received 22.07.13
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