1. Introduction
Riemann hypothesis (RH) is one of the most difficult problems in mathematics, which is reviewed in [1] [2] [3] [4] [5]. We shall consider
. Although
and
have the same zeros, but their properties are quite different.
has the symmetry, i.e.
on the critical line, and
are alternative oscillation with single peak outside critical line, which geometrically implies RC true. Whereas the property of
is bad, even if on critical line
are not alternative oscillation, sometimes almost tangent and multiple peak. Studying
is very hard.
To study Riemann conjecture (RC), we have proposed a framework of geometric analysis for
in previous papers. If three theorems are proved, then RC holds, also see section 3. Firstly we have proved theorems 1 and 2 by the symmetry of
. But to prove theorem 3: “on critical line
is single peak”, we have met essential difficulty. The symmetry is not enough and the stronger tool is needed. Thus we have to focus our attention on
and re-investigate Riemann’s thought. We have found two mysteries in it finally proved RC by method of analysis.
Denote
. Riemann
-function has an integral expression [4], p.17,
(1.1)
But Riemann did not use (1.1), he had directly taken
to get the real function [4], p. 301,
(1.2)
Why Riemann preferred (1.2) rather than (1.1)? This is the first mystery.
Riemann also regarded t as a complex variable (very important!). Taking
and using the uniqueness of analytic function, we get
(1.3)
Thus the first mystery is formulated as:
Equivalence.
is uniquely determined by its initial value
, i.e., two dimensional problem is reduced to one dimension.
We also consider the initial value problem of Cauchy-Riemann system
(1.4)
As
is analytic, Cauchy-Kovalevshkaya theorem confirms that it has a unique analytic solution. This solution just is
. Actually, by direct verification,
then (1.3) and the equivalence hold yet. Here
resembles a traveling-wave solution of the wave equation, where
as an initial value. We had used it in previous papers.
We see that Riemann had studied
rather than
, his thought can be formulated as:
Riemann conjecture(RC). All zeros of
lie on critical line
.
To study
, we find the second mystery in Riemann’s paper [4] (pp. 301-302).
Riemann conjecture 2 (RC2). Using all zeros
of
can uniquely determine
(1.5)
For k-ple zeros, should take k-ple products.
The greatest mystery is that RC2 implies RC. Actually, if
, replacing t by
in (1.5), each factor
, then
and RC holds.
We rigorously have proved RC2 by the method of analysis in section 2. Here we briefly formulate our basic idea as follows. Consider functional equation
(1.6)
It is known that on critical line
we get the upper bound of growth
(1.7)
To prove RC2, by contradiction. If
has conjugate complex roots
,
, by symmetry
, then
do yet. Thus by equivalence
must contain four factors
Letting
,
contains a real factor
(1.8)
and then
contains a term (the lower bound)
(1.9)
which contradicts (1.7). So
can not have complex roots and
does not have the factor
. Therefore both RC2 and RC are proved.
Our main contribution is that we for the first time have regarded RC as an initial value problem, found these mysteries and proposed the newest method to prove Riemann conjecture.
Similar work has not been found in other papers and books.
We shall continue to complete the geometric analysis of
in section 3.
2. Analytical Proof of RC and RC2
2.1. Two Holes in Riemann’s Analysis
Riemann denoted
by
in his paper. He pointed out, see [4] (pp. 301-302).
“If one denotes by
the roots of the equation
, then one can express
as
(R)
because, since the density of roots of size t grows only like
as t grows, this expression converges and for infinite t is only infinite like
; Thus it differs from
by a function of t2 which is continuous and finite for finite t and which, when divided by t2, is infinitely small for infinite t. This difference is therefore a constant, the value of which can be determined by setting
.”
Because Riemann did not use
, we must regard α to be real roots, this is extremely important! But then it is misunderstood.
We now explain his analysis. On critical line
is an even entire function and has infinite number of zeros
. We consider formally the remainder of
,
One knows
(We have better
,
,
). Thus
(2.1)
This series (R) converges for finite t. (Riemann said)
has the growth
(This expression is not suitable. We shall use
, which admits
= 0, see (2.6)). The series (R) differs from
by a function of t2, which, when divided by t2, is infinitely small for infinite t. (Riemann said) “This difference is therefore a constant.” This is not rigorous, in general, it should be
rather than a constant. There is a hole of the uniqueness.
Hadamard in 1893 proved product formula for general entire function, see [4] (p. 20), [5] (p. 16). Denoting
,
, the zeros of
are conjugate, one gets
which is even if
, then
. Taking
, then
.
But this Hadamard’s way from
to
implies a serious contradiction. If all zeros are real, then which itself assumes RC. If conjugate complex zeros are admitted, which just denies RC. Hadamard’s theorem was referred by Von Mangoldt “The first real progression in the field in 34 years”, see [4], p. 39. We think this is a misunderstanding. Actually, Hadamard’s way is independent of proving RC and far from Riemann’s thought. Our idea is to consider
at
as an initial value. We shall directly prove RC2 by method of analysis, so this contradiction is cast off.
2.2. Analytical Proof of RC and RC2
We consider the functional equation
(2.2)
Firstly,
has asymptotic expansion
Take logarithm and decompose the real part,
,
Thus for
(2.3)
Besides,
and
, we get
(2.4)
Secondly, there are growths of
for large t, [4], p. 185, p. 200,
(2.5)
We only need the estimate of
, also see Remark 1.
Thus, we get an upper bound of growth (note: not
)
(2.6)
So
is admitted.
Finally, to prove RC2 (or RC), by contradiction. If
has conjugate complex roots
,
,
, by symmetry
, then
do yet. Denoting
and using the equivalence,
must contain four factors
Letting
, then
must contain a real polynomial of fourth degree
(2.7)
and
contains a term (as a lower bound)
(2.8)
Its growth contradicts (2.6). Thus
can not have complex roots and
does not have the factor
. Therefore both RC and RC2 are rigorously proved.
Originally we want to prove only RC2, fortunately, both RC and RC2 are proved.
Remark 1. In period of Riemann, no estimates (2.5), but it is possible to prove RC. As
is already continued analytically to
(actually this is Euler’s method). Thus
and gets an coarse estimate (e.g.
in [5], p. 27)
(2.9)
By (2.8) one can still prove that
does not have complex roots. But nobody noted it. Therefore I feel, Riemann had already approached to prove RC. Our proof is completed to follow Riemann’s thought.
Numerical experiments 1. Using the data of the first 105 zeros in Odlyzko [6], we have computed
and
in (1.2) for
. We draw these curves by variable scale
, here
,
. We see in Figure 1 that
approximates
very well, of course, larger is t, then larger is its deviation.
We have for the first time computed
and Figure 1, which make us believe the correctness of RC2, and then consider its analytic proof as before.
3. Continuation of Geometric Analysis
In previous papers [7] [8] [9], we have proposed geometric analysis and proved three results:
Theorem 1. If
is single peak and single zero, then the peak-valley structure for
and RC hold.
Theorem 2 (old). If two roots of
are very close to each other (including double zeros), then the peak-valley structure for
and RC still hold.
Theorem 3.
is single peak.
RC can be derived by these three theorems. We at present re-examine these theorems. Theorem 1 is correct, see section 3.1. Theorem 2 is also correct but uncomplete, which is generalized in section 3.4. The original proof of theorem 3 [9] has defect (see section 3.2), which is derived by RC2 in section 3.3.
3.1. A Concise Proof of Theorem 1
Consider a root-interval
of
, obviously
and
depend on
. Assume
inside
for
. At the left end
,
and
, then the slop
,
(which was proved in [9] ) we have
At the right end
,
and
, similarly,
Figure 1.
approximates
by
zeros
.
Because
has opposite signs at two ends of
, there surely exists some inner point
such that
. Then
is valley and
form a peak-valley structure. There is a positive lower bound independent of
.
So RC holds in
. Because the zeros
of analytical function
do not have finite condensation point, then any finite t surely falls in some
. RC holds for any t.
3.2. Defect in Original Proof of Theorem 3
To prove
to be single peak, we discussed monotone growth of the argument
of
in [9] and used Riemann’s estimate
. As
is of super-linear growth, when t increases to
, the increment
slowly increases,
whereas
, then claimed that
monotone increases. But here
is not correct, which may be
. We consider Riemann’s symmetrization
(here the decay factor
is reduced) and have
Thus the local argument
in root-interval
.
We see in Figure 2 (also see [6] [10] ) that
jumps by 1 at zero
of
, then linearly decreases in
with the slope
. Whereas
has slope
. It seems
, and difficult to prove
. Besides, as
, this research is not suitable.
This defect makes us turn to
on critical line and re-investigate RC2.
3.3. Revision Proof of Theorem 3
By RC2, take logarithm
and derivation to get
(3.1)
Consider root-interval
and decompose the summation into three parts.
Figure 2. Curves
. S jumps by 1 twice in
.
(3.2)
Note that
inside
has same sign. We investigate each sum
(3.3)
For
,
are finite. If t is close to
, then
tends to
. If t is close to
, then
tends to
. There surely exists some inner point
such that
. We show this point
is unique. For this we consider their derivatives
(3.4)
Thus all
are monotone decreasing for
, their sum
does yet. Therefore this zero
of
is unique and then
is single peak. This proves theorem 3.
Using Lagarias’s positivity [11] (RC is assumed), we get monotone growth [8], p. 344,
This is a clear description for RC. Therefore Riemann
-function has mathematical beauty: The symmetry, single peak and monotone growth (i.e. the ordering).
3.4. Theorem 2 Holds for m-ple Zeros
In large scale computations [6] [12] [13] [14] all zeros of
are single, no multiple zeros are found. People believe there are only single zeros, but so far do not prove. We have to skirt round the difficulty to prove theorem 2. We shall extend theorem 2 (old) as:
Theorem 2. If
has m-ple zeros on critical line, then
for small
will bifurcate into m alternative oscillations with single peak, and RC still holds.
Proof. Assume that there are three consecutive zeros
of
on critical line
, which form two root-intervals, and
is a m-ple zero,
. Denote
and
. Denoting
and the origin
, and fixing a small
, we discuss a circle
with
(
to be defined). Assume that the real function
has m-ple zero at O
(3.5)
These coefficients
and
are of same order. Thus
(3.6)
Below we discuss
, and temporarily omit high-order remainder.
Using
and De Moivre formula we have
and discuss
and
respectively.
1) The zeros of
satisfy
, i.e.
which are symmetric with respect to
. If
even, then all
. If
odd, the middle argument
satisfies
, i.e. original point
.
Taking the roots
, i.e.
, we have
and re-arrange the ordering of these m zeros as
(3.7)
2) The zeros of
satisfy
, i.e.,
here no
, as
corresponds a trivial zero
(i.e.
). Thus there are only
nontrivial zeros, whose arguments are symmetric with respect to
. When
even,
for
, i.e.
, which corresponds the original
.
Taking the roots
, i.e.
, we have
and re-arrange the ordering of these
roots as
(3.8)
Comparing (3.7) and (3,8) we see that for fixing
, the maximum of these roots is
, thus the radius of circle
satisfies
, i.e. we should confine
.
For
, we have the following conclusions:
1) The real part
has m zeros
, and the imaginary part
has
zeros
. Due to
obviously
. Hence all zeros of (3.7) and (3.8) are alternatively arranged.
2) At zeros
of
, the peak values
alternatively change their signs. At zeros
of
, the peak values
also alternatively change their signs.
3) Because
at zero of
and
at zero of
, they all are single peak. Thus in the m root-intervals of
, all
form local peak-valley structures, and RH locally holds.
Finally, in
with
suitably small we discuss a general case
(3.9)
The actual zeros of
are small perturbations of these zeros mentioned above, which do not change these m peak-valley structures in
. When
increases, they will continue to develop toward locally convex direction by
Figure 3.
at zero point
bifurcate into 2 or 3 zeros.
theorem 1, so RC holds. Hence theorem 2 is proved.
Numerical experiments 2. With the suggestion of Dr.XM Jiao, we have computed
at the second zero
of
. We see in Figure 3 for
,
indeed bifurcate into m curves of single peak, and
are alternative oscillation. The peak of
develops toward its convex direction.
Remark 2. The author of this paper should sincerely thank Dr. Xiangmin Jiao (Stone Brook University, US). He, on 22 April in 2021, sent e-mail to me to discuss the highest super-convergence (see [15] ), we know for the first time. I told him I’m studying RC, and brings special interests and discuss together. He has carefully verified my papers and proposed valuable comments. He suggested the example
and sent papers [10] [16] to me. If no support from him, I very hard, at least in a shorter time, propose the newest proof.
Acknowledgements
The author expresses sincere gratitude to the reviewer for his careful remark, valuable and constructive comments. Besides, I should thank Prof. Zhengtin Hou and Prof. Xinwen Jiang for their precious opinion in discussion.