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?

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

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

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

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

Algebraic Geometry – Calculation of the Riemann compact surface genus $ {[x_0,x_1,x_2] in Bbb P ^ 2 | x_2 ^ 2x_0 = prod_ {i = 1} ^ 3 (x_1- lambda_ix_0), lambda_i in Bbb C } $

Calculate the kind of compact surface of Riemann $ X = {[x_0,x_1,x_2] in Bbb P ^ 2 | x_2 ^ 2x_0 = prod_ {i = 1} ^ 3 (x_1- lambda_ix_0), lambda_i in Bbb C }, lambda_i $ are Distince numbers.

I know the formula of the kind, but I want to use the Riemann-Hurwitz formula.


An example:

For homogeneous polynomial $ P (z_0, z_1, z_2) = z_0 ^ n + z_1 ^ n + z_2 ^ n $, $ P = $ 0
defines a compact surface of Riemann.

$ M = {[z_0,z_1,z_2] in Bbb CP ^ 2 | P (z_0, z_1, z_2) = 0 } $. To calculate his gender, comment $ f: M to Bbb CP ^ 1, [z_0,z_1,z_2] mapsto [z_0,z_1]$it is a well-defined meromorphic function, $ f ^ {- 1} ([0,1]) = {[0,1,z_2]| z_2 ^ n = -1 } $, $ f $ is $ n $-sheet, a $ n $ branch points, each point with branching index $ n-1 $. The total number of branching is $ n (n-1). $

For the Riemann-Hurwitz formula, kind of $ N $ is $ frac 12 (n-1) (n-2) $


But for $ X = {[x_0,x_1,x_2] in Bbb P ^ 2 | x_2 ^ 2x_0 = prod_ {i = 1} ^ 3 (x_1- lambda_ix_0), lambda_i in Bbb C } $, how can we find such $ f: X to Bbb CP ^ 1 $ whose leaf number and branch points can be easily found?

Thank you for your time and patience.

Integral Riemann for continuous maps as the limit of a sequence of partitions

Is the following lemma correct? I need it in this form, I have not found it explicitly anywhere.

Lemma Given a continuous map $ f:[a,b] to mathbb {R} $and a sequence of partitions $ (P_n) $ with $ operatorname {mesh} (P_n) to 0 $ as $ n to infty $then
$$
lim_ {n to infty} sum_ {k = 0} ^ {n_ {P_n} -1} f ( xi_k) (x_ {k + 1} -x_k) = int_a ^ bf (x) , dx.
$$

So, the limit exists and is equal to the integral.

Otherwise, I have proved the equivalence of different possible definitions of integral R., which I could use to prove the lemma above: 1) the definition using $ varepsilon / delta $; 2) $ P_ varepsilon $, 3) Darboux.

Aggressive geometry – Can a holomorphic vector beam on a compact Riemann surface be defined by a single transition function?

It is known that any holomorphic beam of any rank on a non-compact Riemann surface is trivial. Evidence of this is given in Forster's "Conferences on Riemann Surfaces", section 30.

Let $ E $ to be a holomorphic vector beam on a compact surface of Riemann $ X $ with gauge group $ G $. A consequence of the above theorem is the restriction $ E | _ {X – {p }} $ for any point $ p in X $ is a trivial package. So $ E $ can be recovered by specifying the transition function $ g: D cap (X – {p }) rightarrow $ G or $ D $ is a small disk containing $ p $.

Is it correct? If no, could you give a counterexample? I am mainly interested in learning the space of holomorphic beam modules $ X $ concretely, for example by using transition functions.

complex analysis – The sum of the orders of zeros of a holomorphic function in a surface of genus $ g $ Riemann is equal to $ 2g-2 $

Let $ X $ to be a compact surface of Riemann and $ f: X to mathbb {C} $ to be a holomorphic function. By the local normal form, for each point $ p in X $ there is a map $ varphi: U to V $ in $ X $ focused on $ p $and an integer $ m $ such as
$$ f circ varphi ^ {- 1} (z) = z ^ m $$
in a neighborhood of $ 0. This integer is called the multiplicity of $ f $ at $ p $. We will note it by $ text {Mult} _p (f) $.

We then know that there is only a finite number of points $ p $ or $ text {Mult} _p (f) $ is not zero. In addition,
$$ sum_ {p in X} text {Mult} _p (f) = 0. $$

I've also seen texts saying that the sum of the orders of zeros of a holomorphic function in a genus $ g $ Riemann surface is equal to $ 2g-2 $. This sum is not exactly
$$ sum_ {p in X} text {Mult} _p (f) : : ๐Ÿ˜• $$

I would like to understand exactly what this sentence "the sum of the orders of zeros of a holomorphic function in a kind $ g $ Riemann surface is equal to $ 2g-2 $"means and, if possible, how to prove it. (If necessary, I understand the Riemann-Hurwitz formula.)

Number Theory – Dynamics of the Riemann Zeta Function

Has the dynamics of the Riemann zeta function been studied? By dynamics, I understand the limiting behavior of the sequence of iterates $ s, zeta (s), zeta ( zeta (s)), zeta ( zeta ( zeta (s))) Points for different starting values โ€‹โ€‹of $ s $ in the complex (extended) plane. In particular, for which $ s in mathbb {C} $ do the iterations $ zeta ^ {(n)} (s) $ to converge at 0?

The Number Theory – On Robin's Inequality and the Zeros of the Riemann Zeta Function

Let $ zeta $ denotes the Riemann zeta function. By Robin's argument http://zakuski.utsa.edu/~jagy/Robin_1984.pdf, we know that $ zeta ( rho) = $ 0 for some people $ rho $ with $ Re ( rho) in (1/2, 1/2 + beta)

$, or 0 $ < beta leq 1/2 $, if and only if there are positive constants $ beta & # 39; $ and $ c $ such as
begin {equation}
sum_ {d | N} \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
end {equation}
for an infinity of positive integers $ N $, or $ gamma $ denotes the Euler-Mascheroni constant and $ beta & # 39; $ can be taken to have a satisfactory value $ 1/2 – beta < beta <$ 1/2.

Note that if it can be shown that the above inequality is false for some $ beta & # 39; $, so it must also be wrong for everything $ beta & # 39; < beta & # 39; $, implying that if $ zeta ( rho) neq $ 0 for $ Re ( rho) = 1/2 + beta $then $ zeta ( rho) neq $ 0 for $ 1/2 < Re ( rho) leq 1/2 + b $, or $ 0 <b < beta $.

Is this a known result?

Calculation – Proving that a function is Riemann Integrable

Let $ f: [0,1] rightarrow Bbb R $ Defined by:

$ f (x) = sin ( frac {1} {x}) $ if $ x notin Bbb Q $ , $ 0 if $ x in Bbb Q $

I have to prove that this function is integrable with Riemann in this interval.

It is delimited in $ [-1,1]$ and it is not continuous.

I tried to see if I could do $ U (f, P) – L (f, P) < epsilon $ for a partition $ P $ , but I could not evaluate $ M_k $ and $ m_k $ as for certain intervals, they could be $ 0, $ 1 or -1 $.

I also know that, if it was just $ sin ( frac {1} {x}) $ , if would be integrable in this interval, as it would be continuous in $ (0,1)

$ , and we could ignore the discontinuity in $ {0 } $ like that's just a point.

Thank you in advance.