_{1}

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Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying
ζ and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function
,
,
by geometric analysis, which has the symmetry: v=0 if
β=0, and basic expression
. We show that |u| is single peak in each root-interval
of
u for fixed
*β* ∈(0,1/2]. Using the slope u
_{t}, we prove that
v has opposite signs at two end-points of I
_{j}. There surely exists an inner point such that , so {|u|,|v|/
*β*} form a local peak-valley structure, and have positive lower bound
in I
_{j}. Because each
t must lie in some I
_{j}, then ||
*ξ*|| > 0 is valid for any
t (
i.e. RH is true). Using the positivity
of Lagarias (1999), we show the strict monotone
for
β >
β
_{0} ≥ 0 , and the peak-valley structure is equiva-lent to RH, which may be the geometric model expected by Bombieri (2000). This research follows Liuhui’s methodology: “Computing can detect the un-known and method”.

Riemann conjecture concerns two functions ζ ( s ) and ξ ( s ) . In 1737, Euler proved the product formula over all prime numbers p

ζ ( s ) = ∑ n = 1 ∞ 1 n σ = ∏ p ∈ p r i m e ( 1 − 1 p σ ) − 1 (1.1)

to be convergent for real σ > 1 , but divergent for σ ≤ 1 . In 1859, Riemann considered the complex variable s = σ + i t , σ > 1 , using Gamma function Γ ( s / 2 ) , and got

ζ ( s ) = ∑ n = 1 ∞ n − s = π s / 2 Γ − 1 ( s 2 ) ∫ 0 ∞ x s / 2 − 1 ψ ( x ) d x , ψ ( x ) = ∑ n = 1 ∞ e − n 2 π x .

Using the functional equality of Jacobi (1828)

2 ψ ( x ) + 1 = x − 1 / 2 ( 2 ψ ( 1 x ) + 1 ) , (1.2)

and transforming the integral by z = 1 / x

∫ 0 1 z s / 2 − 1 ψ ( z ) d z = 1 s ( s − 1 ) + ∫ 1 ∞ x − s / 2 − 1 / 2 ψ ( x ) d x ,

Riemann derived the first expression

ζ ( s ) = π s / 2 Γ − 1 ( s 2 ) { 1 s ( s − 1 ) + ∫ 1 ∞ ( x s / 2 − 1 + x − s / 2 − 1 / 2 ) ψ ( x ) d x } , (1.3)

which is already analytically extended to the whole complex plane except for s = 0 , 1 . Clearly, the pole points s = − 2 , − 4 , − 6 , ⋯ of Γ ( s / 2 ) are the trivial zeros of ζ ( s ) .

Furthermore, Riemann introduced the entire function

ξ ( s ) = 1 2 s ( s − 1 ) π − s / 2 Γ ( s 2 ) ζ ( s ) , ξ ( s ) = ξ ( 1 − s ) . (1.4)

Through replacing by ζ and integrating by parts twice, it follows that

ξ ( s ) = 1 2 + s ( s − 1 ) 2 ∫ 1 ∞ ( x s / 2 − 1 + x − s / 2 − 1 / 2 ) ψ ( x ) d x = r 1 + ∫ 1 ∞ ( x s / 2 − 1 + x − s / 2 − 1 / 2 ) ( 2 x 2 ψ ″ + 3 x ψ ′ ) d x , (1.5)

where r 1 = 1 2 + ψ ( 1 ) + 4 ψ ′ ( 1 ) = 0 derived by (1.2). Riemann had gotten the second expression

ξ ( s ) = ∫ 1 ∞ ( x s / 2 − 1 + x − s / 2 − 1 / 2 ) f ( x ) d x , f ( x ) = 2 x 2 ψ ″ + 3 x ψ ′ , (1.6)

which is symmetric with respect to s = 1 / 2 . He took σ = 1 / 2 , then Im ( ξ ) = 0 (I think this is the most important symmetry for ξ ).

Riemann thought that a number of zeros of ζ ( s ) in the critical region Ω = { s = σ + i t : 0 ≤ σ ≤ 1 , 0 ≤ t < ∞ } have an estimate

N ( T ) = 1 2 π ( T ln T 2 π − T ) + O ( ln T ) , t ≤ T , (1.7)

(proved by Mangoldt in 1905) then (see our basic theorem and

Riemann Hypothesis (RH). All non-trivial zeros of ζ ( s ) lie on the critical line σ = 1 2 .

RH is a very difficult problem, which has stimulated the untiring research in the areas of the analytic number theory and the complex functions, even the scientific computation. Smale [

There have been many theoretical researches for RH, e.g. the reviews [

We list some important progressions as follows, see [

1) Hardy (1914) for the first time proved that ξ ( s ) has the infinite number of zeros on critical line, by Mellin transform and an important property ψ ( z 2 ) → 0 , as z → e i π / 4 + 0 . Later Selberg (1942) introduced the correct function near z 2 = i and proved the number of zeros to be about cT if 0 < t ≤ T , where c ≈ 0.01 . Later Levinson (1974) improved with c = 1 / 3 and Conrey (1989) with c = 2 / 5 . But they are far less than N ( T ) in (1.7).

2) Poissin (1899) proved no zero of ζ on line σ = 1 , by an interesting equality and singularity decomposition ζ ( σ ) = 1 1 − σ + g ( σ ) near σ = 1 . But

it is very hard to extend this conclusion to σ < 1 . Up to 1958, Vinogradov and Korobov independently proved no root of ζ in σ ≥ 1 − c ( α ) / ( l n | t | + 1 ) α , where α > 2 / 3 .

3) The moment method of ζ . Levinson (1975) proved that a number of the roots on critical line attains 34.74%. Conrey (1989) improved to 40%, and then he [

4) The Müntz method. A. Durmagambetrov [

We have seen that except for critical line, most works focus to ζ . We think that the appointing ζ in formulation of RH is a historic misguide, because which has gone against the original thinking of Riemann (Actually, he focused to ζ , rather than ζ , see

In recent twenty years, many new methods appeared and the research of RH has taken some important progressions. But RH has not been solved yet.

We can see from (1.7) that the average spacing between two zeros is less than 2 π / l n T . To study the distribution of these zeros, there were lots of large scale numerical experiments, e.g. Lune et al. in [

1) There are a high peak in each segment of the graph and 1 - 9 smaller peaks between two high peaks. They found that the ratio of the high-peak and low-peak can reach 1000 times.

2) There are 1 - 8 roots between two high-peaks. They found a pair of large zeros, these two zeros are very close to each other, and look like a double zero. e.g. in Fig.7, p.678, [

To face so terrible micro-structures near critical line, we has always met a wide gap: how to prove no zeros of the infinite series, analysis method is powerless. Corney [

Although ζ diverges for R e ( s ) ≤ 1 , but can be estimated as follows, see [

| ζ ( σ + i t ) | ≤ C t 1 / 4 − β / 2 l n t , 0 ≤ σ ≤ 1, β = σ − 1 / 2 , | ζ ( 1 2 + i t ) | = O ( t λ ) , λ = 1 / 6 or λ = 19 / 116 , (1.8)

which are possibly expressed in the form

| ζ ( σ + i t ) | ≤ C t 1 / 6 − β / 3 ln t , 0 ≤ β = σ − 1 / 2 ≤ 1 / 2 . (1.9)

Denote β = σ − 1 / 2 . Using an asymptotic expansion

Γ ( s 2 ) = 2 π ( t 2 ) β / 2 − 1 / 4 e − t π / 4 e i ϕ ( 1 + O ( t − 1 ) ) , (1.10)

(1.4) and (1.10), there has an important estimate with exponential decay [

| ξ ( s ) | ≤ C ( t 2 ) 23 / 12 + β / 6 e − t π / 4 l n t , if | β | ≤ 1 / 2 . (1.11)

Due to the decay e − t π / 4 , it is very hard to compute ξ ( s ) for large t. Probably this is the reason why there are few work to discuss ξ . With the same reason, analysis method is also powerless for ξ . Bombieri [

We know that ξ has the most important symmetry on critical line. We point out that the positivity R e ( ξ ′ / ξ ) > 0 of Lagarias [

progression for ξ , which also is a unique result to be cited in our proof for RH, see

R e ( ξ ′ ξ ) = R e ( ξ ′ ξ ¯ | ξ | 2 ) = ψ ( t ) / | ξ | 2 > 0, for β > 0,

where positive quadratic form

ψ ( t ) = ξ ′ ξ ¯ = u u σ + v v σ = u v t − v u t > 0 , for β > 0. (1.12)

plays an important role in our proof.

From these difficulties and advices of Conrey and Bombieri, we should give up ζ -function and pure analysis methods, while turn to geometric analysis. What is geometric analysis? We no longer regard the summation process of series, while prefer the geometric property and structure of ξ -curve itself. That is, “Explain the essence by figure” (Liuhui’s words). This is a big change of our recognition. We compute and study by Liuhui thinking, i.e. “computing can detect the unknown and method” (see

1. Root-interval. For any fixed β ∈ [ 0, 1 / 2 ] , the sub-interval I j = [ t j , t j + 1 ] is called the root-interval, if the real part u ( t j , β ) = 0 , u ( t j + 1 , β ) = 0 and | u ( t , β ) | > 0 inside I j .

2. Single peak. If | u | in each root-interval I j only has one peak, called single peak, else called multiple peak (It is proved that the multiple peak case does not exist, see theorem 2).

The single peak u has the following geometry property.

3. Slope u t . For single peak u and any β ≥ 0 , there are u t > 0 from negative peak to positive one, and u t < 0 from positive peak to negative one.

Using Newton-Leibnitz formula, v ( t , 0 ) = 0 and C-R condition v β = − u t , we have

4. Analytic property. The imaginary part v has a basic expression

v ( t , β ) = − ∫ 0 β u t ( t , r ) d r , β ∈ ( 0, 1 / 2 ] . (1.13)

Because u t has opposite signs at two end-points of I j , then v also has opposite signs.

Corollary. | v ( t , β ) | / β is uniformly bounded with respect to β ∈ ( 0, 1 / 2 ] .

In numerical experiments we found an important geometry structure as follows.

Peak-valley structure (PVS). For fixed β ∈ ( 0, 1 / 2 ] and in each root-interval I j = [ t j , t j + 1 ] , | u | is a peak. While v ( t , β ) has opposite signs at two end-point of I j and v = 0 at some inner point, | v | / β is a valley. Then { | u | , | v | / β } form a local PVS and have a local positive lower bound ‖ ξ ‖ = | u | + | v | / β ≥ μ ( t j , β ) > 0 in I j (i.e. RH is valid in I j ).

Using 4 items above, we have proved the PVS (see theorems 1-3).

Sequence principle. As the zeros of u do not have finite condensation point, each t must lie in some I j , then ‖ ξ ‖ > 0 is valid for any t.

We have gotten

Basic Theorem. All zeros of Riemann ξ -function lie on the critical line.

Besides, by (1.12) we have

Equivalence theorem. The peak-valley structure and RH are equivalent.

Theorem 4. The strict monotone | ξ ( t , β ) | > | ξ ( t , β 0 ) | ≥ 0 for β > β 0 ≥ 0 .

We think that the strict monotone is the deepest description for RH. The PVS may be the geometric model to be expected by Bombieri, which makes the proof of RH get concise and intuitive, and many difficulties are avoided, e.g. need not discuss the summation process of the infinite series and so on.

The PVS and RH in single peak case were shown in our previous paper [

Denote τ = i t + β , β = σ − 1 / 2 . We consider the Riemann kernel integral K ( f ) to define

ξ ( τ ) = K ( f ) = ∫ 1 ∞ ( x τ / 2 + x − τ / 2 ) x − 3 / 4 f ( x ) d x = u + i v , ξ ′ ( τ ) = K ′ ( f ) = 1 2 ∫ 1 ∞ ( x τ / 2 − x − τ / 2 ) x − 3 / 4 ln x f ( x ) d x = u β + i v β , ξ ″ ( τ ) = K ″ ( f ) = 1 4 ∫ 1 ∞ ( x τ / 2 + x − τ / 2 ) x − 3 / 4 ln 2 x f ( x ) d x = u β β + i v β β , (2.1)

here and below the Cauchy-Riemann conditions are used many times. If β = 0 , obviously

x i t / 2 + x − i t / 2 = 2 cos ( t 2 ln x ) , x i t / 2 − x − i t / 2 = 2 i sin ( t 2 ln x ) ,

we have the following analytic property.

The symmetry. If β = 0 , then

v = 0 , u β = v t = 0 , v β β = − v t t = 0 , ⋯ . (2.2)

These properties are essential. Especially we have the basic expression (1.12).

The norm | ξ | = ( | u | 2 + | v | 2 ) 1 / 2 is used in complex analysis. Now define a strong norm

‖ ξ ‖ = { | u | + | v | / β , if β ∈ ( 0 , 1 / 2 ] , t ∈ [ 0 , ∞ ) , | u ( t , 0 ) | + | u t ( t , 0 ) | , if β → + 0 , t ∈ [ 0 , ∞ ) , (2.3)

where three conditions of norm are satisfied. Its advantage is that | u | and | v | / β are of the same order and ‖ ξ ‖ is stable with respect to β > 0 . Note that if β = 0 , u ( t j , 0 ) = 0 , v ( t j , 0 ) = 0 , then | ξ | = 0 , but probably ‖ ξ ‖ > 0 , if | u t ( t j ,0 ) | ≠ 0 , see

Firstly we compute ξ = u + i v . Take a changing scale M = 8 ( t / 2 ) 23 / 12 + 0 e − t π / 4 (where M is independent of β , different from [

To explain the local PVS, we consider a smaller root-interval I 2 = [ t 2 , t 3 ] in

Nextly we compute the derivative i ξ ′ = i ( u β + i v β ) = u t + i v t . We see that u t and v t have also alternative zeros and a local PVS in

We regard { u ( t , β ) , v ( t , β ) } as a continuous changing process from β = 0 to

β = 1 / 2 . For any β ∈ ( 0, 1 / 2 ] , all zeros t j of u ( t , β ) form an irregular infinite sequence dependent on β

0 < t 1 < ⋯ < t j − 1 < t j < t j + 1 < t j + 2 < ⋯ → ∞ ,

which have not the finite condensation points, else u ≡ 0 . We take them as the base in studying PVS.

Theorem 1 (single peak case). If u ( t , β ) is single peak for any β ∈ ( 0, 1 / 2 ] , then ‖ ξ ‖ > 0 for any ( t , β ) ∈ Ω = [ 0 , ∞ ) × ( 0 , 1 / 2 ] .

Proof. Below it is enough to discuss u > 0 inside the root-interval I j = [ t j , t j + 1 ] . For any fixed β > 0 , using the analytic property (1.12), we consider two cases as follows.

As u t > 0 near the left node t j , we have

{ v ( t j , β ) / β = − 1 β ∫ 0 β u t ( t j , r ) d r < 0, l i m β → + 0 v ( t j , β ) / β = − u t ( t j ,0 ) < 0. (3.1)

As u t < 0 near the right node t j + 1 , similarly

{ v ( t j + 1 , β ) / β = − 1 β ∫ 0 β u t ( t j + 1 , r ) d r > 0 , lim β → + 0 v ( t j + 1 , β ) / β = − u t ( t j + 1 , 0 ) > 0. (3.2)

They are valid and numerically stable for β ∈ ( 0, 1 / 2 ] .

Because v ( t , β ) has opposite signs at two end-points in I j , there certainly exists an inner point t ′ j = t ′ j ( β ) such that v ( t ′ j , β ) = 0 . Clearly in I j , | u | is a peak and | v ( t , β ) | / β is a valley, thus { | u | , | v | / β } form a local PVS. We regard ‖ ξ ( t , β ) ‖ as a continuous function of ( t , β ) , which certainly has a positive lower bound independent of t ∈ I j ,

m i n t ∈ I j ‖ ξ ( t , β ) ‖ = μ ( t j , β ) > 0, β ∈ ( 0, 1 / 2 ] . (3.3)

This is a fine local geometric analysis.

Thus in each root-interval I j = [ t j , t j + 1 ] , we can determine a positive lower bound μ ( t j , β ) > 0 , which form the positive infinite sequence

μ ( t 1 , β ) , μ ( t 2 , β ) , ⋯ , μ ( t j , β ) , μ ( t j + 1 , β ) , ⋯ (3.4)

Because each t must lie in some I j , thus ‖ ξ ‖ > 0 for any t. In this way, the summation process of the infinite series ξ is completely avoided. □

Theorem 2. The multiple peak case does not exist for β ∈ [ 0, 1 / 2 ] .

Proof. Assume that u ( t , β ) > 0 for β > 0 inside some root-interval I j = [ t j , t j + 1 ] and has odd number of extreme values u ( t j p , β ) = a j p > 0 at the inner points t j p , p = 1 , 2 , ⋯ , 2 k + 1 , see

Below we prove that the multiple peak case does not appear. For this, we consider the minimum extreme value u ( t j i , β ) = a j i > 0 at some point t ′ = t j i , in

there u t = 0 and u t t > 0 , i.e., u is locally convex downwards, see

v t ( t ′ , β ) = − ∫ 0 β u t t ( t ′ , r ) d r < 0. (3.5)

On the other hand, because RH is valid as before, using (1.12), we have

ψ ( t ) = u v t − v u t > 0 , β > 0. (3.6)

But now, u > 0 , u t = 0 , v t < 0 at t = t ′ , which lead to contradiction ψ = u v t < 0 . Thus u ( t , β ) for any β > 0 is single peak and its limit u ( t ,0 ) is also single peak. □

Why need to deny the multiple peak case? We see in

Theorem 3. The peak curve u ( t ,0 ) in a small root-interval (including double root) will remove in parallel towards its convex direction for β > 0 so that ‖ ξ ( t , β ) ‖ > 0 .

Proof. Assume that u ( t ,0 ) has a solitary small root-interval I j 0 = [ t j 0 , t j + 1 0 ] such that u ( t j 0 , 0 ) = 0 , u ( t j + 1 0 , 0 ) = 0 , and u ( t ,0 ) ≥ 0 in I j 0 . Let the maximum value u ( t ′ , 0 ) = ϵ > 0 at some inner point t ′ ∈ I j 0 , then u t ( t ′ , 0 ) = 0 and u t t ( t ′ , 0 ) < 0 . Consider a little enlarged sub-interval I ⊃ I j 0 , in which u t t ( t , 0 ) < 0 and u ( t ,0 ) ≤ ϵ is convex upwards. See

Take a small β > 0 . By basic expression in I we have

v t ( t , β ) = − ∫ 0 β u t t ( t , r ) d r > 0 , as u t t ( t , r ) < 0 , (3.7)

and

u ( t , β ) − u ( t , 0 ) = ∫ 0 β u β ( t , r ) d r = ∫ 0 β v t ( t , r ) d r , as u β = v t , = − ∫ 0 β ( ∫ 0 r u t t ( t , r ′ ) d r ′ ) d r = d > 0. (3.8)

So u ( t , β ) has removed u ( t ,0 ) in parallel upwards by a distance d > 0 (i.e. towards its convex direction).

Note that u ( t , 0 ) < 0 outside I j 0 . For β > 0 , u ( t , β ) has already removed upwards by d > 0 , so there surely exists an enlarged sub-interval I j = [ t j , u j + 1 ] ⊃ I j 0 such that,

at the left node t j = t j ( β ) < t j 0 , u ( t j , β ) = 0 and u t ( t j , β ) > 0 ,

at the right node t j + 1 = t j + 1 ( β ) > t j + 1 0 , u ( t j + 1 , β ) = 0 and u t ( t j + 1 , β ) < 0 ,

i.e. I j = [ t j , t j + 1 ] ⊃ I j 0 is new root-interval of u ( t , β ) , and u ( t , β ) > 0 inside I j , see

Besides, by basic expression and the slope u t , we know

v ( t j , β ) < 0 ( as u t > 0 ) , v ( t j + 1 , β ) > 0 ( as u t < 0 ) . (3.9)

So there surely exists an inner point t j * ∈ I j such that v ( t j * , β ) = 0 , i.e., | v | / β still is a valley. Therefore { | u | , | v | / β } in I j form a local PVS and ‖ ξ ‖ > 0 is valid in I j . □

Lune et al. [

Summarizing three theorems above, our basic theorem is proved. □

Remark. In the proof of Theorem 1 we have seen that the Riemann integral ξ = K ( f ) has the symmetry, which is independent of the speciality of f. So we guess that for the very wide class of the fast decay function f, RH is still valid for K ( f ) . We have two examples. For t ≤ 110 , there are larger low bounds ( | u β | / β + | v β | ) / M ≥ 0.20 and ‖ ξ ″ ‖ / M ≥ 0.28 .

Haglund [

In the proof of RH, we have used a unique new result to be the following.

Lagarias theorem (1999). If RH is true, then R e ( ξ ′ ( τ ) ξ ( τ ) ) > 0 for any β > 0 .

This is a unique equivalence to RH for ξ , we think that this is an essential progression along research line of ξ after Hadamard (1893) and Mongoldt (1905), which cannot be directly derived from the integral form (1.6). The multiplication and division operations for the integral form of ξ are impossible.

A simplified proof. If RH is true, Hadamard (1893) proved a product expression

ξ ( s ) = e A + B s ∏ ρ ( 1 − s ρ ) e s / ρ , (4.1)

here ρ runs over all roots of ξ ( ρ ) = 0 , and A and B are some constants.

Now, we transform τ = s − 1 / 2 = β + i t , their roots are conjugate, τ j = i t j , τ ¯ j = − i t j , i.e. τ / τ j + τ / τ ¯ j = 0 . All positive zeros t j form an infinite series

0 < t 1 = 14.134 < t 2 < t 3 < ⋯ < t j < ⋯

Lune et al [

ξ ( τ ) = e A + B τ ∏ j ( 1 − τ τ j ) ( 1 − τ τ ¯ j ) , τ = β + i t .

If β = 0 , ξ is real, then B = 0 . So we get a product expression

ξ ( τ ) = ξ ( 0 ) ∏ j = 1 ∞ { ( 1 − τ τ j ) ( 1 − τ τ ¯ j ) } , 0 < β ≤ 1 / 2 , 0 ≤ t < ∞ . (4.2)

Taking logarithm and derivation, we have

ξ ′ ξ = ∑ j = 1 ∞ { 1 τ − τ j + 1 τ − τ ¯ j } = ∑ j = 1 ∞ { β − i ( t − t j ) β 2 + ( t − t j ) 2 + β − i ( t + t j ) β 2 + ( t + t j ) 2 } ,

then

R e ( ξ ′ ξ ) = β ∑ j = 1 ∞ { 1 β 2 + ( t j − t ) 2 + 1 β 2 + ( t j + t ) 2 } > 0. (4.3)

It remains to explain its convergence. In fact, by the estimate N ( T ) ≈ T 2 π ln ( T 2 e π ) , the average spacing between two adjacent zeros is about 2 π / l n T . For j suitably large, the zero point t j has an approximate estimate

t j ≈ ( j l n j ) / 2 π .

Thus for any fixed t ≥ 0 , the series is convergent. □

Proof of strict monotone.

Denote ξ = u + i v , using the positive quadratic form ψ = u u β + v v β > 0 , we have

| ξ ( t , β ) | 2 − | ξ ( t , β 0 ) | 2 = 2 ∫ β 0 β ( u u β + v v β ) d β > 0 , β > β 0 ≥ 0 ,

then | u ( t , β ) | > | u ( t , β 0 ) | . □

Proof of the equivalence theorem.

Assume that RH is valid and u > 0 inside root-interval I j = [ t j , t j + 1 ] (similarly for u < 0 ). By Lagarias theorem, the quadratic form ψ = u v t − v u t > 0 in I j for β > 0 , and geometric property of u t , we have the following facts.

At the left node t j , u = 0 , u t > 0 and ψ = − v u t > 0 , then v < 0 ;

At the right node t j + 1 , u = 0 , u t < 0 and ψ = − v u t > 0 , then v > 0 .

Thus v has opposite signs at two end-points, there certainly exists an inner point t ′ j ∈ I j such that v = 0 , which implies local PVS. Thus the equivalence of both is proved. □

From the view-point of complex analysis, RH requires | ξ | > 0 , while from the view-point of geometry, the peak-valley structure requires strong norm ‖ ξ ‖ > 0 . Both are equivalent. However, the local geometry property is of extreme importance, which makes the proof be concise and intuitive. I greatly appreciate the mathematical beauty of the symmetry.

Riemann’s paper “On the number of primes less than a given magnitude” is a classic work [

1) In fact

1 n s Π ( s 2 − 1 ) π s / 2 = ∫ 0 ∞ e − n n π x x ( s / 2 ) − 1 d x ;

So when one sets

∑ 1 ∞ e − n n π x = ψ ( x ) ,

it follows that

Π ( s 2 − 1 ) π s / 2 ζ ( s ) = ∫ 0 ∞ ψ ( x ) x ( s / 2 ) − 1 d x

or, because

2 ψ ( x ) + 1 = x − 1 / 2 [ 2 ψ ( 1 x ) + 1 ] ,

that

Π ( s 2 − 1 ) π s / 2 ζ ( s ) = ∫ 1 ∞ ψ ( x ) x ( s / 2 ) − 1 d x + ∫ 0 1 ψ ( 1 x ) x ( s − 3 ) / 2 d x + 1 2 ∫ 0 1 ( x ( s − 3 ) / 2 − x ( s / 2 ) − 1 ) d x = 1 s ( s − 1 ) + ∫ 0 ∞ ψ ( x ) ( x s / 2 − 1 + x − ( s + 1 ) / 2 ) d x .

2). I now set s = 1 2 + i t and

Π ( s 2 ) ( s − 1 ) π s / 2 ζ ( s ) = ξ ( t )

so that

ξ ( t ) = 1 2 − ( t t + 1 2 ) ∫ 1 ∞ ψ ( x ) x − 3 / 4 c o s ( 1 2 t l o g x ) d x

or also

ξ ( t ) = 4 ∫ 1 ∞ d [ x 3 / 2 ψ ′ ( x ) ] d x x − 1 / 4 c o s ( 1 2 t l o g x ) d x .

This function is finite for all finite values of t and can be developed as a power series in tt which converges very rapidly.

3) Now since for values of s with real part greater than 1,

l o g ζ ( s ) = − ∑ l o g ( 1 − p s ) is finite and since the same is true of the other factors of ξ ( t ) , the function ξ ( t ) can vanish only when the imaginary part of t

lies between 1 2 i and − 1 2 i . The number of roots of ξ ( t ) whose real parts lie between 0 and T is about

= T 2 π log T 2 π − T 2 π ,

because the integral ∫ d l o g ξ ( t ) taken in the positive sense around the domain consisting of all values whose imaginary parts lie between 1 2 i and 1 2 i and

whose real parts lie between 0 and T is (up to a fraction of the order of magnitude of 1/T) equal to [ T l o g ( T / 2 π ) − T ] i and is, on the other hand, equal to the number of roots of ξ ( t ) = 0 in the domain multiplied by 2 π i .

4) 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, but I have put aside the research for such a proof after some fleeting vain attempts, because it is not necessary for the immediate objection of my investigation.

5) If one denotes by α the roots of the equation ξ ( α ) = 0 , then one can express log ξ ( t ) as

∑ l o g ( 1 − t t α α ) + l o g ξ ( 0 )

because, since the density of roots of size t grows only like l o g ( t / 2 π ) as t grows, this expression converges and for infinite t is only infinite like t l o g t ; Thus it differs from log ξ ( t ) by a function of tt which is continuous and finite for finite t and which, when divided by tt, is infinitely small for infinite t. This difference is therefore a constant, the value of which can be determined by setting t = 0 .

Remark. 1) It is strange that Riemann had not given a complete integral expression (1.6) of ξ ( s ) . I think this is a neglect, which will bring misunderstanding later. 2) He took s = 1 / 2 + i t and derived a complete expression ξ ( t ) , which is an even real function and I m ( ξ ) = 0 , this is an important symmetry. 3) Riemann had a conjecture on the number N ( T ) of zeros (proved by Mangoldt, 1905). This estimate was applied to the product expression. 4) The RH is the greatest mystery. Later Siegel (1932) had found a computational formula unpublished in Riemann’s manuscript (now called Riemann-Siegel formula, which is still to discuss ξ ) and the first several zeros computed. Siegel had quite surely pointed out that his manuscript had not any steps to go to proof of RH. 5) Riemann had proposed another conjecture on the product expression of ξ (proved by Hadamard, 1893), which is the base to prove the positivity by Lagarias(1999).

Now, when RH is already proved, we would make the following three comments:

1) In these 5 items, Riemann had always discussed ξ , rather than ζ . I don’t know that from what time, RH had become to study ζ . This has gone against the original thinking of Riemann and was a historic misguide. Maybe, there is a possibility that due to the decay of ξ , computing ξ is too hard, while computing ζ is easier, but analysis method for ζ is hopeless. Actually, a unique hope is to study ξ .

2) Except for local PVS to be proposed by us, Riemann had already prepared the main tools needed in proving RH, e.g. the entire function ξ ( s ) , the symmetry, The number of zeros and the product expression and so on. What wise Riemann was!

3) How to find RH? I think that Riemann had proposed RH likely based on the finite numerical results and theoretical consideration for the symmetry. In my opinion, Riemann’s thinking, really, looks a little like to the eastern mathematical thinking. Fortunately, I have detected the essence of ξ ( s ) and proved RH by use of Liuhui methodology, see

Finally we recall several key progressions along Riemann thinking as follows.

Riemann (1859) constructed ξ and proposed several conjectures, of key RH;

Hadamrd (1893) proved the product formula of ξ ;

Mangoldt (1905) proved a number N ( T ) of zeros of ξ ;

Siegel (1932) found R-S formula on critical line for ξ ;

Lagarias (1999) found and proved the positive R e ( ξ ′ / ξ ) > 0 outside critical line;

Author (2020) found PVS and proved RH by local geometric analysis for ξ .

It is known that ζ has 2 m-order Euler-Maclaurin evaluation [

ζ ( s ) = ∑ n = 1 N − 1 n − s + N 1 − s s − 1 + 1 2 N − s + B 2 2 s N − s − 1 + ⋯ + B 2 m ( 2 m ) ! s ( s + 1 ) ⋯ ( s + 2 m − 2 ) N − s − 2 m + 1 + R 2 m , (6.1)

with the remainder

R 2 m = − s ( s + 1 ) ⋯ ( s + 2 m − 1 ) ( 2 m ) ! ∫ N ∞ B 2 m ( { x } ) x − s − 2 m d x , (6.2)

where B i ( x ) is i^{th} Bernoulli polynomial, B i is i^{th} Bernoulli number and {x} is fractional part of x. It is an analytic continuation and the most efficient computing formula up to now. Clearly, it is impossible to prove RH by so complicated double series.

Taking 2 m = 10 and N ≥ 10 t and neglecting its remainder, the desired accuracy can be attained. Denote ζ ( s ) = U + i V .

1). U and V are of same order, even if on critical line, V does not disappear, no symmetry.

2). ζ behaves badly near zeros, sometimes U and V are almost tangent. What geometry structure it is? No PVS.

To draw curves, we use the norm | ζ ( s ) | = | U | + | V | . Edwards [

t 2 = 17143.821844 for β = 0 , and t 2 − t 1 = 0.035308 is very small, see the left of

β = 0.005 , 0.01 . This is a mystery to be hidden behind ζ , which consists with theorem 3.

Greek preferred the deduction. To face so difficult RH, where did the idea of proof come from? The Greek mathematics does not give any inspiration. But the eastern mathematics may do it! Because Chinese emphasized the combination of computing and analyzing, in particular, Liuhui thought that “can detect the unknown and method by computing”. We shall show how to reveal the essence of RH by Liuhui methodology.

Chinese “Nine Chapters Mathematics” (at least B.C.4-2 century) and Greek “Geometry Original” (B.C. 3-2 century) are two mathematical classics over the world, and also are different systems of mathematical idea and method. Liuhui (A.D.225-295) made about one hundred remarks in “Nine Chapters” (A.D.263) [

1) Sum and take limit to prove the existence of π.

In B.C.11 century, Shanggao theorem (Pythagoras theorem, in B.C.5 century) was proposed and extensively applied. Liuhui in “cut circle” (1600 words. English version [

“The more finely is cut, the less loss there is. Cut it again and again until one is unable to cut further, that is, when the shape coincides with that of the circle and there is no loss”.

He introduced a small square T (rather than circumscribed polygons) and proved the existence of π by pressing of both sides, see

π 2 n = π n + ∑ T / 2 < π < π n + ∑ T = 2 π 2 n − π n , i.e. linear interpolation

Although Archimedes computed the circumference π 96 = 3.14 , Dauben [

2) Discover extrapolation (Richardson proposed extrapolation in 1927).

Liuhui considered the ratio of two small squares (found by Wang [

r 12 = π 24 − π 12 π 48 − π 24 = 3.95 , theoretical value 4,

and got the extrapolation value π 192 * = π 192 + ( π 192 − π 96 ) / ( r 12 − 1 ) = 3.1416 . But he felt anxious, finally computed π 3072 = 3.1416 and said “the ratio obtained is

the same as the previous one; in this way, the ratio is again verified”. This is the prediction-correction idea.

The extrapolation is an important idea of modern scientific computation. We [

3) New elimination method for linear system of equation.

He used the arrangement (like the present augmented matrix) and elimination. This was a classical algorithm in ancient China (Gauss elimination appeared in 18 century).

4) Compute the area and volume, the seed of defined integral.

Liuhui proposed the “irreducible method” and limit to prove the tetrahedron volume V = 1 3 S H . After 200 years, Zuchongzi (A.D. 429-500) proved the sphere volume V = 4 3 π R 3 by this method. The both play an important role in

completing Chinese mathematical system (although Archimedes has already obtained). While Liu-Zu principle,“the area and hight are same, then their volume is also same”, called Cavalieli principle (1635) in the west.

5) Use “rate” and “multiple difference” to study the ratio of difference, the seed of the “slope”. Ancient Greek did not have these concepts.

“Nine Chapters” discussed practical problems, the “relation” of two quantities just is function, and the piecewise expression of function was used. While the “rate” (called “lv” in ancient China) is difference ratio of function (averaging

slope or velocity) K ( x ) = f ( x ) − f ( a ) x − a . e.g. f ( x ) = A x + B has the rate A.

The linear interpolation was extensively applied in “Nine Chapters”. Fibonacci in “Calculus classic” (1202) had one chapter “Chida algorithm”, i.e. linear interpolation, Chida just is China. Liuhui supplemented “multiple difference” as Chapter 10 in “Nine Chapters”.

Multiple difference is a difference ratio of several values.

e.g. y = A x 2 + B has the rate K ( x ) = A ( x + a ) . But using bi-section method and three data(called “thrice watchings” in ancient China) y 1 = A h 2 + B , y 2 = A ( h / 2 ) 2 + B , y 3 = A ( h / 4 ) 2 + B . One gets their differences y 1 − y 2 = A ( h 2 − h 2 / 4 ) , y 2 − y 3 = A ( h 2 / 4 − h 2 / 16 ) , and multiple difference(now called mathematical invariant)

y 1 − y 2 y 2 − y 3 = 1 − 1 / 4 1 / 4 − 1 / 16 = 4.

which just is Liuhui’s extrapolation coefficient r = 4 . We [

We have seen that Chinese ancient mathematical thinking was very different from Greek. In that period, Liuhui had studied these two “infinitesimal processes”. But nobody could understand his thinking 50 years ago.

Liuhui’s preface (800 words) is a wonderful paper through ages, and expounded his deep mathematical thinking.

1)“Scientific discovery is a recognition process of the prediction and correction”. Liuhui’s extrapolation is a typical prediction method.

2) Mathematical methodology: “Computing can distinguish tiny and detect the unknown and method”. “And analyze the reason by logic, explain the essence by figures”.

This is an open, progressive and creative methodology (do not stipulate conclusion in advance), very different from Greek idea. Chinese academician Z. C. Shi pointed out [

When study a difficult problem, because for lack of understanding, we should detect its properties by computing in various directions, then analyze these results and look for key properties, finally find the idea of proof. This just is Liuhui thinking.

I studied ancient Chinese mathematical thinking for 10 years. I believe that Liuhui methodology has powerful vitality. How to detect the mystery of RH? During four years, my recognition process is as follows.

1) Studying ζ is hopeless, see

2) We found that the real and image parts of ξ have the positive phase-difference, which implies RH. But we don’t know how to describe it. A chance of local geometry analysis is missed. At that time we always wanted to use the asymptotic analysis.

3) To attain the exponential decay O ( e − t π / 4 ) , we used Riemann technique of integration by parts and Jacobi equality to get high-order expression, m = 2 k + 2 ,

{ ξ ( τ ) = ( − 1 ) k P k − 1 ( τ ) ( W m + ( τ ) + W m − ( τ ) ) , τ = β + i t , W m ± ( τ ) = K ± ( f m ) = ∫ 1 ∞ x ± τ / 2 x − 3 / 4 f m ( x ) d x , P k ( τ ) = ∏ j = 0 k ( τ / 2 + 1 / 4 + j ) ( − τ / 2 + 1 / 4 + j ) , f m ( x ) = ∑ n = 1 ∞ ∑ j = k + 1 m d m , j ( − n 2 π x ) j e − n 2 π x ,

where the coefficients d m j are defined by recurrence formula. For any t we can look for suitable m such that P k − 1 ( τ ) ≈ O ( e − t π / 4 ) . Thus need not cancel each other in the integral, while directly use asymptotic analysis. But f m ( x ) for large m is too complicated. We adapted the incomplete gamma function, Laplace integral and saddle point method etc., all fail.

4) We considered the integrals W ± ( τ ) along complex lines z = 1 + ( a ± i b ) r , and found that f m ( z ) condenses into a solitary high peak, I’m excited for this. We have constructed a simple function g m ( z ) = C z A e − B z to simulate the high peak with relative residue 1/80, where A, B and C are some constants independent of τ , and K ± ( g m ) can be approximately expressed by Gamma function, as a dominated function. When t ≤ 100 , using a comparison criterion, it seems RH is proved. But for large t, the asymptotic analysis is also hopeless.

5) In these researches, we have always met a wide gap: How to prove no zero for the infinite series? It is impossible. Finally, in 2019 Oct.2, we have suddenly waked up that

Give up the summation process, study the geometry structure of ξ itself.

Actually we come back to the positive phase-difference once again. But now, we have found the local peak-valley structure and the importance of symmetry, and then proved RH by geometry method, where t is arbitrary.

The author expresses sincere thanks to the reviewer for his valuable comments, suggestion and kind encouragement.

The author declares no conflicts of interest regarding the publication of this paper.

Chen, C.M. (2020) Local Geometric Proof of Riemann Conjecture. Advances in Pure Mathematics, 10, 589-610. https://doi.org/10.4236/apm.2020.1010036