nt.number theory – Equivalent statements of Goldbach's conjecture in terms of Riemann Zeta function properties?

The Riemann hypothesis has many equivalent statements.

Many of them do not concern the first distribution, but rather the properties of the Riemann zeta function, such as the distribution of the zeros of the zeta function or the positivity properties of the zeta functions (ex: criterion Li).

To my knowledge, Goldbach's conjecture does not have many equivalent statements.

First question, why?

Second question: what are some equivalent statements of the Goldbach conjecture in terms of the properties of the zeta function or the Dirichlet-L function?

The only work I know in this direction is the Granville paper:
Refinements of Goldbach's conjecture and Riemann's generalized hypothesis

Can any one point to other references or works in this direction?

Posted on

plot – Riemann surface of incomplete gamma function

Michael Trott, in his book The Mathematica Guide for Symbolics, page 1003, illustrated a good way to visualize the Riemann surface of the incomplete gamma function $$Gamma ( alpha, z)$$. To draw the surface of Riemann, for example, $$Gamma ((3 + i) / 10, z)$$, we have:

With({(Alpha) = 0.3 + 0.1 I, ee = 10^(-12)},
Show(GraphicsArray(
Graphics3D({EdgeForm(Thickness(0.002)),
SurfaceColor(Hue(0.09), Hue(0.18), 2.3),
Table((*split whole graphics into pieces*)
Last /@ Partition(
Cases(ParametricPlot3D((*the sheets*){r Cos((CurlyPhi)),
r Sin((CurlyPhi)), #(
Exp(2 k Pi I (Alpha)) Gamma((Alpha),
r Exp(I (CurlyPhi))) + (1 -
Exp(2 k Pi I (Alpha))) Gamma((Alpha)))}, {r, 0,
2}, {(CurlyPhi), -Pi + ee , Pi - ee},
PlotPoints -> {30, 40},
DisplayFunction -> Identity), _Polygon, Infinity),
2), {k, -2, 2})}, BoxRatios -> {1, 1, 2.5},
PlotRange -> All) & /@(*show real and imaginary part*){Re,
Im})))

where we used the identity: $$Gamma ( alpha, exp (2 k pi i) z) = exp (2 k pi i alpha) Gamma ( alpha, z) + (1- exp (2 k pi i alpha) Gamma ( alpha))$$.

However, Mathematica does not return anything. Any help is appreciated!

Posted on

Dg.differential Geometry – Conventions for the Riemann Curvature Tensor

I know two conventions for the Riemann curvature tensor, namely the expression
$$langle nabla_X nabla_YZ- nabla_Y nabla_XZ- nabla _ {(X, Y)} Z, W rangle$$
is either declared to be $$R (X, Y, Z, W)$$ or $$R (W, Z, X, Y)$$. These differ by a sign. One reason to prefer the first convention is that the order of $$X, Y, Z, W$$ is preserved. One reason for preferring the second convention is that the expression of sectional curvature makes more sense: $$R (a, b, a, b)$$ or $$a$$ and $$b$$ form an orthonormal basis of a plane within the tangent space. Are there any other reasons to prefer one of these conventions to the other? What is the history of these two conventions? Why these and no other orders from $$X, Y, Z, W$$ (The second convention has a weird order, so I wonder if there is a specific reason for this)?

Posted on

Why is the limit of the sum of Riemann an anti-derivative?

I stopped doing mathematics for a while and my brain forgot the answer to my question. Can any one help me understand it ?:

Why is the limit of the sum of Riemann an anti-derivative?

In other words: why is the equation below true? Posted on

complex analysis – Since Riemann maps \$ f: Omega to mathbb D \$, prove that \$ inf_ {z in partial Omega} | za | frac1 {f) (a)} z in partial Omega} | za | \$.

I look at the following problem from an old qualifying exam that I found:

Let $$Omega$$ to be an open subset just connected $$Bbb C$$, let $$a in Omega$$ and suppose that we are given an analytical bijection $$f: Omega to mathbb D: = {z: | z | <1 }$$ satisfactory $$f (a) = 0$$ and $$f (a)> 0$$. Prove $$inf_ {z in partial Omega} | z-a | frac1 {f) (a)} z in partial Omega} | z-a |.$$

To be honest, I do not know where to start. my first thought was that $$1 / f (a) = (f ^ {- 1}) & # 39; (0)$$, and hoped to use this to try to apply the maximum module principle to a function involving $$f$$ or $$(f ^ {- 1}) & # 39;$$ and $$z-a$$but the first problem that I got even before trying to write a function was to realize that $$f$$ and $$(f ^ {- 1}) & # 39;$$ may not even be defined on the respective limits, so it does not appear that I can use the maximum module to determine anything about the limit points in relation to these functions.

I have also assumed, seeing this, that it was simply something proven and used in the demonstration of Riemann's mapping theorem, but that does not really seem to be to be found the.

Posted on

Riemann Integration – Reduction Stage in Theorem 1.9 of Conway's Complex Analysis

Theorem 1.9 Yes $$gamma$$ is smooth in pieces and $$f: [a,b] to Bbb {C}$$ is continuous, then $$int_ {a} ^ {b} fd gamma = int_ {a} ^ {b} f (t) gamma & # 39; (t) dt$$

Here is some of the evidence that confuses me:

Again, we only consider the case where $$gamma$$ is smooth. Also, looking at the real and imaginary parts of $$gamma$$, we reduce the proof in case $$gamma ([a,b]) subseteq Bbb {R}$$

Why is it enough to consider $$gamma$$ actual value? I can not understand that. Presumably, we use the fact that $$gamma = Re ~ gamma + i Im ~ gamma$$ and then use the linearity of the Riemann-Stieltjes integral. But that seems to require that $$Re ~ gamma$$ and $$Im ~ gamma$$ to be fluid functions, which in turn seems to require that the real and imaginary parts be differentiable, because it seems that we would need to use the chain rule somewhere. But I do not believe that the real and imaginary parts are differentiable; they do not seem to satisfy the equations of Cauchy-Riemann.

Again, these are just speculations on my part, because I do not quite understand the stage of reduction.

Posted on

Difference between the integration of Riemann in the real space and the banach space

Do you know the difference about these ..

Posted on

Simple representations of the Riemann function \$ Xi \$

The Riemann $$Xi$$ function, defined as
$$Xi (z) equiv – frac {1} {2} left (z ^ 2 + frac {1} {4} right) pi ^ { frac {1} {4} + i frac {z} {2}} Gamma left ( frac {1} {4} + i frac {z} {2} right) zeta left ( frac {1} {2} + iz right )$$
has a number of beautiful properties. It's an entire function, unlike the $$Gamma$$ and $$zeta$$ the functions. His reflection formula $$Xi (-z) = Xi (z)$$ is particularly easy to remember. The Riemann hypothesis for $$Xi (z)$$ is also much simpler: all the zeros of $$Xi (z)$$ are real.

On the other hand, this formula, in its definition, is very ugly and it is obvious that it is about all that the zeta function is shifted, rotated and scaled. Is there a better representation, possibly in the form of an integral function or other?

Posted on

cv.complex variables – On the analytical properties of the Riemann zeta function

Note by $$zeta$$ the zeta function of Riemann. For $$Re (s) = sigma> 0$$it is well known that

$$sum_ {n leq x} n ^ {- s} = zeta (s) + frac {x ^ {1-s}} {1-s} + O (x ^ {- sigma}) .$$

But is there infinitely $$x$$ such as

$$Bigg | sum_ {n leq x} n ^ {- s} – zeta (s) – frac {x ^ {1-s}} {1-s} Bigg | gg x ^ {- sigma}?$$

Posted on

Differential geometry – Riemann curvature and Levi-Civita connection on symmetric positive definite matrix collector

Let $$mathcal {S}$$ to be the symmetric matrices and $$mathcal {P}$$ to be positive defined matrices.
$$mathcal {S}$$ naturally carries the structure of a vector space. Indoor product on $$mathcal {S}$$ is given by $$langle A, B rangle = trace (A B)$$ .
$$mathcal {P}$$ is an open set in ( mathcal {S} \$.

the map $$log: mathcal {P} rightarrow mathcal {S}$$
is a diffeomorphism between two varieties. We can identify the tangent space in x $$T _ { text {x}} mathcal {P}$$ with $$text {x} times mathcal {S}$$.
The metric induced on the Tangent space is given by $$langle A, B rangle_ {x} = langle dlog (A) | _ {x}, dlog (B) | _ {x} rangle$$ ,or
$$dlog: T_x mathcal {P} rightarrow T_x mathcal {S}$$
$$An mapsto an X ^ {- 1}$$
Explicitly write the inner product on $$mathcal {P}$$ is given by
$$langle A, B rangle_ {X} = text {trace} (A X ^ {- 1} B X ^ {- 1})$$

Length $$L ( gamma)$$ and energy $$E ( gamma)$$ of a curve $$gamma: [0,1] rightarrow mathcal {P}$$ is given by

$$L ( gamma) = int_a ^ b | dfrac {d gamma} {dt} | _ { gamma (t)} dt$$
$$E ( gamma) = frac {1} {2} int_a ^ b | dfrac {d gamma} {dt} | _ { gamma (t)} ^ 2 dt$$

Geodesics are curves that minimize energy on a collector. They allow us to introduce a distance between two points of a manifold. To calculate them we can simply use euler's lagler equations

$$frac {d} {d}} frac {d} {d dot { gamma}} f (t, gamma (t), dot { gamma} (t)) = frac {d} {d gamma} f (t, gamma (t), point { gamma} (t))$$

write explicitly we have

$$frac {d} {dt} frac {d} {d dot { gamma}} text {trace} ( dot { gamma} gamma ^ {- 1} dot { gamma} gamma ^ {- 1}) = frac {d} {d gamma} text {trace} ( dot { gamma} gamma ^ {- 1} dot { gamma} gamma ^ {- 1})$$

The left side of the equation reduces to

$$frac {d} {d}} gamma ^ {- 1} dot { gamma} gamma ^ {- 1}$$

while the right side of the equation reduces to

$$– gamma ^ {- 1} dot { gamma} gamma ^ {- 1} dot { gamma} gamma ^ {- 1}$$

By solving the differential equation, we obtain two different expressions for geodesics.
$$gamma (t) = P exp (t P ^ {- 1} S)$$
Here, P is located on the variety and S is located on the tangent space. intuitively $$gamma$$ is the curve that starts in P with the direction S.
$$gamma_ {AB} (t) = A (A ^ {- 1} B) ^ t$$
Here, gamma is the geodesic between the points of variety A and B.

Now that the geodesics are defined, the distance on the spd manifold can be defined via the length of the geodesic curve connecting two points. After some calculations we come to
$$d (X, Y) = | log (X ^ {- 1} Y) |$$

So, these are my calculations up to now. I am pretty new to differential geometry. My first question is: Do arguments usually make sense? I'm not really convinced that $$dlog = dX X ^ {- 1}$$ or that the internal product on $$mathcal {S}$$ is $$trace (AB)$$. My ultimate goal is to read the christoffel symbols from the geodesic eq. and calculate the riem. curvature. But I'm not sure how to proceed or if calculations have up to now a meaning

Posted on