1. Introduction
D. Hilbert [1] (1900) reported 23 problems in mathematics and stated RH in the problem 8 as “it still remains to prove the correctness of an exceedingly important statement of Riemann, viz., that the zero points of the function
defined by the series
(1.1)
all have the real part 1/2, except the well-known negative integral real zeros......”
But RH has not been solved in 20th century. Entering new era, S. Smale’s report [2] (2000), reviews [3] (2000), [4] (2003) and books [5] [6], all have cited Hilbert’s statement. J. Conrey [4] pointed out that “It is my belief, RH is a genuinely arithmetic question that likely will not succumb to methods of analysis”. E. Bombieri [3] expected that “For them, we do not have algebraic and geometric models to guide our thinking, and entirely new ideas may be needed to study these intriguing objects”. These advices make us realize that the analysis of the infinite series is hopeless and we should pay more attention to the algebraic and geometric analysis. We have also noted a new trend to give up
and turn to
. P. Sarnak [7] (2004) pointed out that “Riemann showed how to continue zeta analytically in s and he established the Functional Equation:
(1.2)
being the Gamma function. RH is the assertion that all the zeros of
are on the line of symmetry for the functional equation, that is on
.” Where
has the symmetry and alternative oscillation, which for the first time turns to
. Besides, J. Haglund [8] (2011) discussed (another form of (1.6))
(1.3)
which is the first paper to study the equivalence of
and RH. He computed
and proposed a guess: if any part summation has the monotone zeros, then RH holds. He thought that the study of the
-function was the right approach to RH. This is very important.
We have computed the Riemann
-function and other continuations of Euler
-function, and found that only
has the symmetry and alternative oscillation, which intuitively implies RH. Whereas others
have no the properties, and proving RH is hopeless.
We reread the original paper of Riemann (see [5] ) and found his thought to study
-function. We list 4 terms concerning RH and the important progression as follows.
1) Riemann converted Euler series (1.1) into analytical function (no symmetry)
(1.4)
where
is Jacobi’s function.
2) Introduced an entire function (which is a symmetrization)
(1.5)
In critical domain
and
have the same zeros. Taking
, using integration by parts twice and Jacobi equality
, Riemann had gotten a real function
(1.6)
Remark 1. In present point of view, using translating
, it should directly get Riemann’s general formula [5]
(1.7)
On critical line
,
has the symmetry
.
3) Riemann said, “The number of roots of
whose real parts lie between 0 andT is about
” (it is proved by Mongoldt, 1905), and
pointed out that, “One finds in fact about this many real roots within these bounds and it is very likely that all of the roots are real. One would of course like to have a rigorous proof of this” (i.e. RH).
4) He guessed a multiplication formula of
(it is proved by Hadamard, 1893)
(1.8)
From these we see that Riemann had emphasized
, rather than
.
5) Siegel (1932) found a formula unpublished in Riemann’s manuscript(now called R-S formula, which was derived by
, and is large scale computing formula on critical line up to now), and Riemann had already computed the first several roots (due to the inspiration of R-S formula, we shall propose a new computing formula in next paper).
6) Lagarias [9] (1999) found the positivity
for
(a new
property undiscussed by Riemann), which is the most essential progression since 1932, also the first equivalence to RH for
. Its proof requires the properties 3) and 4).
7) C. M. Chen [10] [11] (2020) regarded
as curve family
with parameter
, and found that in each root-interval
of u,
form a peak-valley structure (a new property undiscussed by Riemann) and
(i.e. RH). The framework of geometric analysis used in [11] is correct and should be kept in this paper, but there are two defects in its proof. a) At one end of root-interval
, think
to imply
is not strict, which is strictly proved in Lemma 2 of this paper. b) To prove the single peak of u, assume that u has multiple peaks,
for small
is proved, but this only is local result and not sufficient for using Lagarias’ theorem. This paper directly proves the alternative oscillation of
and derives the single peak of u (Theorem 3). Our main result is
Basic theorem (RC). All the zeros of Riemann
-function lie on symmetric line.
Therefore this paper has amended the defects and given a complete proof of RC.
2. Geometric Properties of ξ
2.1. The Symmetry and Alternative Oscillation of ξ
Denote
. Consider Riemann
-integral and its derivatives
(2.1)
From decomposing expression
we have
Property 1 (symmetry). The
is an even function of
and
is an odd function of
. Taking
, then
, i.e.
(2.2)
Property 2 (alternative oscillation). All the zeros of u and v for
are alternative (proved in theorem 3), see Figure 1. The alternative oscillation intuitively implies RH.
2.2. A Geometric Model of ξ
Definition 1 (root-interval). For any
, a subinterval
called root-interval of u, if the real part
inside
, and
.
Definition 2 (single peak). If
has only one peak in each root-interval
, called single peak, unless called multiple peaks(we shall prove no multiple peaks in theorem 3).
Using Newton-Leibnitz formula, the symmetry
and Cauchy-Riemann conditions
, we have
Lemma 1 (the expression of v). The image part v of
can be expressed by
as
(2.3)
Corollary 1.
is uniformly bounded with respect to
.
Definition 3 (peak-valley structure, PVS). For any
, if
inside the root-interval
, and v has opposite signs at two ends, then there surely exists some inner point
such that
, and
form a peak-valley structure, see Figure 2.
Lemma 2 (the expression of u). The real part u of
can be expressed by
as
(2.4)
Figure 1.
of curve family ABC and
.
Corollary 2. For any
, in the interval of negative peak
, with
(convex downward), then
. In the interval of positive peak
, with
(convex upward), then
. So the peak
will develop toward convex direction (This is the ordering of
in
).
2.3. The Slope
of Function Family
Consider the curve segments ABC and CDE of
for
, where C is a positive peak, A and E are negative peaks, see Figure 1 (the varying scale
is used,
). Let
be the root interval of
, its left end-point
can fall in AB or BC, dependent on relative highs of peaks A and C. We have
Lemma 2 (the slop ut of curve family). For any
, assume that
has a positive peak C inside root-interval
. If
is from negative peak to positive one, then curve family
on the left line
has positive slope
(2.5)
If
is from positive peak to negative one, then curve family
on the right line
has negative slope
for any
. Discussing negative peak is similar.
Proof. For
, the slope
inside ABC,
inside AB,
inside BC and
at B. Decreasing
, the corresponding curve
will continuously vary and the slope
inside
. To prove
for the curve family, we should avoid two peak-points
and
. Denote the upper bound p of
and the lower bound q of
by
Obviously
inside common interval
. We prove the left line
, so (2.5) holds.
For this, rewrite the expressions (2.4) for
and
and their difference
(2.6)
Subdivide ABC and
into three segments with different convexities. We have
1) If the zero
, in which
, by (2.6), we have the ordering
and
, see Figure 1, then
.
2) If the zero
, in which
, by (2.6), we have the ordering
and
, see Figure 1, then
.
3) If the zero
, we can not judge their ordering, but obviously
.
Above three cases prove (2.5). Similarly discuss the curve CEF.
Remark 2. We in [11] intuitively thought that
will imply (2.5), this is not strict. Now (2.5) is strictly proved by three segments of convexity.
3. Geometric Proof of Riemann Conjecture
We shall regard
as a continuously varying process from
to
. The zeros
of
form an irregular infinite sequence
which are single zeros(the double zeros are admitted). For any
, the zeros
of
form an irregular infinite sequence(dependent on
)
The initial
continuously becomes to root-interval
. We prove
Basic theorem. All zeros of Riemann
-function lie on critical line.
Its proof consists of three theorems as follows.
Theorem 1. If
is single peak, then peak-valley structure and RH are valid.
Proof. Consider a root-interval
of
. When increasing
, by corollary 2, the peak
will develop toward convex direction. Assuming
inside
for
, we analyze two cases as follows.
At the left end
,
and
, by Lemma 2 the slop
(author’s remark: in the proof [11] only
is not enough!) we have
(3.1)
At the right end
,
and
, similarly,
(3.2)
They are valid and numerically stable for
.
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, see Figure 1. We consider a continuous function of
which certainly has a positive lower bound independent of
(3.3)
So RH holds in
. This is a refine local geometric analysis.
As each root-interval
will repeat the peak-valley structure, we get a positive irregular infinite sequence
(3.4)
Because all the zeros
of analytical function
do not have the finite condensation point (unless
), then any finite t surely falls in some
. RH holds for any t.
Theorem 2. If two roots of
are very close to each other (including double root), then the peak-valley structure for
and RH still hold.
Proof. Let
have root-interval
,
inside
and
outside
, see Figure 3. Assume for
,
(convex upward) in larger interval
. By (2.4),
i.e.
translates
by a positive distance d upward. So
has a larger root-interval
. The
has a positive peak inside the new interval
. Besides
has opposite signs at two ends of
and surely
at some inner point, then
is valley. Therefore
form a peak-valley structure and
in
.
Remark 3. In large scale computation [12] [13] one found that all the zeros of
are single, no double. Perhaps in the future, double roots can be found. Theorem 2 shows that RH still holds for double roots. Besides one also found
Figure 3. Artificial
.
that
is single peak, no multiple one.
Theorem 3.
is single peak.
Proof. We know that analytic function
is alternative oscillation with single peak, if
monotonously trends to the infinity when t increases. Riemann introduced
where
, its amplitude
(3.5)
is super-linearly increasing with t, so
is alternative high-frequency oscillation. We recall [5], by amplitude principle, the number of the zeros of
in
critical rectangle
is about
, its main part is
, but
has only
. If take an increment
for larger t, we have
and
, then the amplitude
monotonously tends to the infinity. As
is not oscillation function, it is proved that
and then
are alternative oscillation with single peak for
. Below prove that its limit value
still is single peak.
By contradiction, assume that
inside root-interval
has three positive peaks A, B and C, see Figure 4, and
atD with
and
at E with
. So
at four points
has the signs
. By (2.3),
for small
has the signs
at these four points, Therefore
has three different zeros inside
, which
Figure 4. Artificial curve (do not exist).
goes against the alternative oscillation and then
is single peak (Author’s remark: in multiple peak case, [11] proved
for small
, but this is only a local result, which is not sufficient for using Lagarias’ theorem. This paper directly proves theorem 3).
Finally summarizing three theorems above, our basic theorem (RH) is proved.
It is interesting that the proof of RH looks like to solve the Cauchy problem of Cauchy-Riemann system
with analytic initial values
on line
. Follow Riemann, by
, its solution is
.
4. Three Corollaries of Lagarias Theorem
Assume that RH holds, Lagarias (1999) proved the following wonderful result:
Lagarias theorem. RH is equivalent to the positivity
for any
.
Denoting
and
, the positivity can be expressed in the form
(4.1)
This result is sharp (can not be improved), which has three important corollaries.
Corollary 3. The peak-valley structure and RH are equivalent.
Proof. Above RH is proved by peak-valley structure. Now assume that RH holds, then the positivity also is valid, we shall prove peak-valley structure. Let
be root-interval of
,
inside
, and
.
At the left end
,
, then
leads to
.
At the right end
,
, then
leads to
.
Then
is valley in
and
form peak-valley structure.
Therefore the peak-valley structure is the exact geometric description to RH.
Corollary 4. If RH holds, then
is single peak.
Proof. If
has three peaks, see Figure 4, and
and
at extremes point
. For small
, the
at some point
near
, and in its neighbor,
and
At this point
,
is contradiction to the positivity.
So we see that the correctness of our geometric proof of RH can be derived by Lagarias’ theorem.
Corollary 5 (Monotone). If RH holds, then
for
.
Proof. Using (4.1), the positive integral
means the monotone. So
for
is a concise statement of RH.
Remark 4. We recall that Aristotle, an ancient Greek philosopher and mathematician, thought: “order and symmetry are important elements of beauty”. Therefore we can say, the symmetry and monotone of
are mathematical beauty of Riemann conjecture.
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 the discussion.