Let $(p_1, p_2)$ be a twin prime pair, where we include $(2, 3)$. If $p_1 \equiv 3$ mod $4$ then we let $t_{(p_1, p_2)} := p_2 ^ 2 / p_1 ^ 2$ otherwise, we let $t_{(p_1, p_2)} := p_1 ^ 2 / p_2 ^ 2$. Also define $t_{(2,3)}:=3 ^ 2/2 ^ 2$.

I conjecture that the product $$ \prod_{(p_1, p_2): \text{twin primes}}t_{(p_1, p_2)} =\tfrac{3 ^ 2}{2 ^ 2} \cdot \tfrac{5 ^ 2}{3 ^ 2}\cdot \tfrac{5 ^ 2}{7 ^ 2}\cdot\tfrac{13 ^ 2}{11 ^ 2} \cdot\tfrac{17 ^ 2}{19 ^ 2} \cdot\tfrac{29 ^ 2}{31 ^ 2} \cdot\tfrac{41 ^ 2}{43 ^ 2} \cdot \tfrac{61 ^ 2 }{59 ^ 2} \cdot \tfrac{73 ^ 2}{ 71 ^ 2}\cdot \tfrac{101 ^ 2 }{ 103 ^ 2}\cdots $$

is equal to $\pi$. (If this is true then twin prime numbers are infinity many.)

Some numerical values of partial products:

3.1887755102040816321 to $10^1$,

3.2055606708805624550 to $10^2$,

3.1290622219773513145 to $10^3$,

3.1364540609918890779 to $10^4$,

3.1384537326021492746 to $10^5$,

3.1417076006640026373 to $10^6$,

3.1417823471756806475 to $10^7$,

3.1415377533170544536 to $10^8$,

3.1415215264211035597 to $10^9$,

3.1415248453830039795 to $10^{10}$,

3.1415126339547108140 to $10^{11}$,

3.1415144504088659201 to $10^{12}$,

3.1415142045284687040 to $10^{13}$,

3.1415144719058962626 to $10^{14}$,

3.1415384423175311229 to $10^{15}$

Can we find a few more decimal places using the extrapolation method?

I am reading this paper about "Numerical approximation of the logarithmic capacity of domains", and there (on the third page) I found simple formulas for logarithmic capacity of simple figures like squares and equilateral triangles.

For a better understanding, I tried to recalculate those by myself several times, but I could not succeed. Is there a way that I can find those calculations or something similar that I can use as a guide?

Is there any irrational number that is known the probability distribution of digits?

Something like 0 appears 10% of time, 1 appears 10% of time, etc.

Probably irrational numbers that are defined by a construction on digits like:

1234567891011121314....

You can prove digits distribution but I am interested in numbers that is not defined this way like pi, e, or square root of prime.

Is there any advance on this topic?

Assume we have a set of equations in $x \in \mathbb{R}^n$

$$|a_i\cdot x|=b_i$$

where $a_i \in \mathbb{R}^n$ and $b_i>0$ are given. Could such a system be solved efficiently?

- In a theoretical machine storing reals with perfect accuracy.
- An approximate solution taking rounding errors into account.

This is equivalent to having $2$ possible values for each linear combination, which is a valid question over any field. Does this equivalent problem over a fixed finite field have an efficient solution? Or is it perhaps known to be NP-complete?

So far, I concluded that squaring both sides of the equation we get linear equations in $y_{ij} = x_i x_j$. However, this helps only if the system is quadratically overdetermined.

**Question:** How to find **the smallest** value $x$ satisfying the equation: $x^2 = a \pmod c$ (known is $a$ and $c$, $c$ is not the prime)?

Using the Tonelli-Shanks algorithm and the Chinese remainder theorem does not always give me the smallest $x$ satisfying condition.

Is there any solution for calculating the smallest $x$?

Does anyone have an idea?

**---- Edit:**

I will describe more accurately my problem:

Let's assume that we are looking for a solution: $x^2 \equiv 1024 \pmod{1302}$.

We need to know the distribution of the factors $1302$, so $1302 = 2 \cdot 3 \cdot 7 \cdot 31$.

Now, using the Tonelli-Shanks algorithm we calculate for all divisors:

**For 2:**

$1024 \equiv k_1 \pmod 2$

$k_1 = 0$

$x_1 ^ 2 \equiv k_1 \pmod{2}$

$x_1^2 \equiv 0 \pmod{2}$

$x_1 = 0$

**For 3:**

$1024 \equiv k_2 \pmod 3$

$k_2 = 1$

$x_2^2 \equiv k_1 \pmod{3}$

$x_2^2 \equiv 1 \pmod{3}$

$x_2 = 1$

**For 7:**

$1024 \equiv k_3 \pmod 7$

$k_3 = 2$

$x_3 ^ 2 \equiv k_3 \pmod{7}$

$x_3^2 \equiv 2 \pmod {7}$

$x_3 = 4$

**For 31:**

$1024 \equiv k_4 \pmod 31$

$k_4 = 1$

$x_4 ^ 2 \equiv k_4 \pmod{31}$

$x_4 ^ 2 \equiv 1 \pmod{31}$

$x_4 = 1$

Then we solve the system of equations from the Chinese remainder theorem. We know the factors and also the values of $x$ from the formula: $x ^ 2 \equiv c \pmod {p}$ where $p$ and $c$ are known.

We solve the system of equations.

$x \equiv 0 \pmod{2}$

$x \equiv 1 \pmod{3}$

$x \equiv 4 \pmod{7}$

$x \equiv 1 \pmod{31}$

The Chinese remainder theorem comes out $x = 32$, and this is the good, smallest solution: $32 ^ 2 \equiv 1024 \pmod {1302}$ - quite trivial case.

The problem, however, is that it does not always agree. And so I write why.

In the above case, for example, for the first factor $31$ I assumed that I found a result equal to $1$. I do not necessarily have to find exactly $1$ as well:

$(31-1)^2 \equiv k_4 \pmod{31}$

$(31-1)^2 \equiv 1 \pmod{31}$

$30^2 \equiv 1 \pmod{31}$

The above is that for $30$ will also be $1$.

For such a system of equations (new value at $31$):

$x \equiv 0 \pmod{2}$

$x \equiv 1 \pmod{3}$

$x \equiv 4 \pmod{7}$

$x \equiv 30 \pmod{31}$

From the Chinese remainder theorem we get $x = 2944$. This also agrees, because $2944 ^ 2 \equiv 1024 \pmod {1302}$ but this is no longer the samllest possible value (smallest possible is $x = 32$).

Knowing the first value ($1$ for factor = $31$), the second one ($30$ for factor = $31$) that fits is easy to calculate as I did here.

However, since all combinations of values will be $2 ^ k$ (where $k$ is the number of prime factors (in different example). I have not found a way to do some search for these combinations in a better way than brutal-force.

Any ideas for that so I'm looking for.

Can someone suggest something?

Let $G=(V,E)$ be a simple, undirected graph such that every vertex has degree at least $2$. Given $n\in\mathbb{N}$, a map $c:E \to \{1,\ldots, n\}$ is said to be a *weak coloring* if for every $v\in V$ the edges adjacent to $v$ do not all have the same color. (More formally, we want the restriction $c|_{E(v)}$ to be non-constant, where $E(v) = \{e\in E: v\in e\}$.)

These two nice posts by Mikail Tikhomirov and Brendan McKay respectively show that for every finite graph there is a weak edge coloring with $3$ colors. I tried to carry through their arguments with transfinite induction to infinite graphs - without success.

**Question.** If $G=(V,E)$ is an infinite simple undirected graph, is there a weak edge coloring $c:E \to \{1,2,3\}$?

I'm not exactly a differential geometer, so I hope this isn't too elementary a question.

From a naive point of view, it seems as if there are two natural group actions on the space of connections on the tangent bundle of a manifold $X$: the group of diffeomorphisms, which acts by pull-backs, and the group of gauge transformations, which acts by, well, gauge transformations.

Now, there isn't any obvious relationship between the two groups and, as far as I can tell, the gauge group is a much more natural and much more convenient object to study.

However, on the other hand, it is certainly possible that for given diffeomorphism $\phi: X \to X$ and a connection $A$, we have a gauge equivalence between $\phi^{*}(A)$ and $A$. I'm interested in understand when this occurs in general. In particular, for which diffeomorphisms is it true that $\phi^*(A)$ is gauge equivalent to $A$, for *every* connection $A$?

I tried attacking this, at least for diffeomorphisms isotopic to the identity, by viewing the isotopy as coming from a flow on $X$ and writing down a differential equation satisfied by a one-parameter family of gauge transformations inducing the same map on some fixed connection, but it didn't seem to lead anywhere.

Thanks.

Let $G$ be a connected Lie group and $\Sigma$ a closed oriented surface. We know that principal $G$-bundles $P$ can be topologically classified by a characteristic class $c(P)\in H^2(\Sigma,\pi_1G)\cong\pi_1G$.

The following is my question:

Let $G$ be a semisimple connected (or even compact) Lie group, $\beta\colon\pi_1(\Sigma)\to G$ a group homomorphism, and $\Sigma$ a closed oriented surface. Consider the universal cover $\tilde{\Sigma}\to\Sigma$ which is a principal $\pi_1(\Sigma)$-bundle. Form the associated bundle $\tilde{\Sigma}\times_\beta G$ which is necessarily a principal $G$-bundle over $\Sigma$. Hence, it should have a characteristic class $c(\tilde{\Sigma}\times_\beta G)\in H^2(\Sigma,\pi_1G)\cong\pi_1G$. Is there a way to compute this characteristic class in terms of $\beta$?

The difficulty here is that $\pi_1(\Sigma)$ is not connected, and hence I cannot use the functoriality of the characteristic class. Another one is that I am not sure how to compute the induced map $B\beta\colon B\pi_1(\Sigma)\to BG$ concretely.

I've asked this question on MSE and haven't got any answers yet.

Let $\mathcal{R}$ be the Robba ring and $\mathcal{E}^{\dagger}$ the elements of $\mathcal{R}$ that are bounded at 0 (so the coefficients of the powerseries are bounded. Is it true that $x, y\in \mathcal{R}$, $z\in (\mathcal{E}^\dagger)^*$ with $x\cdot y= z$ implies $x,y\in (\mathcal{E}^\dagger)^*$?

If not, is it then true that $x\in \mathcal{R}$, $ y, z\in (\mathcal{E}^\dagger)^*$ with $x\cdot y= z$ implies $x\in (\mathcal{E}^\dagger)^*$?

This kind of seems to be implied in some of the papers of Colmez about $(\varphi, \Gamma)$-modules, for example here https://webusers.imj-prg.fr/~pierre.colmez/triangulines in Proposition 4.2, but I might be mistaken and some other argument is implicitly used.

Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space. Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal. By being microlocal I mean that the wave front sets satisfy $WF(Af)\subset WF(f)$ for all $f$. (For another version, one could consider the singular supports instead of wave fronts, but I assume the answer wouldn't be different.) Does it follow that $A$ is a pseudo-differential operator?

This is a variation on this question about continuous endomorphisms on the Schwartz space. There one only assumed linearity and continuity, and the answer was negative. The other question was about $\mathcal S$ instead of $\mathcal S'$, but I need distributions to allow singularities.

I realize that there are many classes of pseudo-differential operators. The question is whether such a microlocal operator is always a ΨDO of some kind. Any details on what kind would be very interesting, of course.

**Edit:** According to the interesting comments of Michael Albanese and Nick L we revise the question as follows:

By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ which contains $M$ as an open dense subset.

Assume that $M$ is an open connected manifold which admits a manifold compactification. Does $M$ necessarily admit a manifold compactification with zero Euler characteristic?

Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions. Does $\Psi$ admit a retraction $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$, such that $\lVert \Phi \rVert = \lVert \Phi \rVert_{\mathrm{cb}}$?

(This question is a follow-up to a previous question, in which it was established that $\Psi$ may have retractions which do *not* have this property.)

I know some theory of "classical" modular forms, that is functions in the complex upper-half plane satisfying

$f(\frac {az+b} {cz+d})=(cz+d)^kf(z)$

I know one can study modular forms on finite-index subgroups of $SL(2,\mathbb Z)$. But I have not seen much theory of modular forms on arbitrary Fuchsian groups. Which are the most interesting cases of such groups? Can somebody recommend a good reference?

I have also come across Hilbert and Siegel modular forms, but I don't have these in mind as an answer to this question. I wonder whether one can use arbitrary Lie group instead of $SL(2,\mathbb R)$, which is what the Wikipedia page about automorphic forms suggests, but I am not on a level to tackle the theory.

Write $\mathbb{Z}^{a\times b}$ for the $a\times b$ integer matrices. Let $Q\in\mathbb{Z}^{n\times n}$. Let $G=O(Q)$ be the orthogonal group of $Q$. For $X_0\in \mathbb{Z}^{n\times 2}$, set $$ R'(T,Q,X_0) = \{ X\in X_0G(\mathbb{Z}) : |X_{ij}|\leq T\}. $$

**Dubious conjecture:** If $Q$ is indefinite then
$$
\#R'(T,Q,X_0) \ll_{Q,\epsilon}
T^{\max\{0,2n-6\}+\epsilon}
\tag{$\star$}
$$
for all $\epsilon>0$, with an implicit constant depending only on $Q$ and $\epsilon$.

**Question:** Is this true? How about simple (?) examples like $Q=I_3$ or $Q=I_4$?

Let $B\in\mathbb{Z}^{2\times 2}$ be nonsingular and symmetric. I am interested in the number of solutions $$ R(T,Q,B) = \{ X\in \mathbb{Z}^{n\times 2} : XQX^T =B, \, |X_{ij}|\leq T\} $$ with height up to $T$, when $Q$ is indefinite. A naive guess based on the circle method would be that $$ \#R(T,Q,B) \ll_Q 1+T^{2n-6} \tag{$\ast$} $$ with an implicit constant depending at most on $Q$. The set $R(T,Q,B)$ is the union of finitely many $G(\mathbb{Z})$-orbits $R'(T,Q,X_0)$, so the conjecture above is a weak version of this.

Note that ($\ast$) is true for large $n$. For instance when $Q$ is indefinite and $n\geq 12$ this (and much more) follows from the asymptotic result of Brandes. (The condition $n\geq 12$ follows from the comment at the end of section 1 in the subsequent paper.) On the other hand ($\ast$) is certainly false for small $n$, though I don't know of a good example in the indefinite case.

If $B$ is positive definite and $Q$ has signature $(p,q)$ then it is a special case of Theorem 1.4 of Eskin, Mozes and Shah that $$ \# R(T,Q,B) \sim C T^{\min\{2,q\}(n-1-\min\{2,q\})} $$ for some $C= C(Q,B)$, unless a certain volume is infinite. In the latter case, the comments after formula (1.4) of Duke, Rudnick and Sarnak suggest that the right-hand side of ($\ast$) just changes by a factor of $\log T$. Perhaps a proof of ($\star$) in this case is implicit in their work, but I have not been able to verify this.

can anybody here help me, please?I really fed up. I am a master student and I have a numerical analysis course and it's very difficult for me cause I have not had experience in this subject. whatever I read I just mix-up more. Thank you.

This is a refined version of Do binary symmetric channels maximize mutual information?, which was answered negatively.

Let the random variables $(X, Y)$ be a doubly symmetric binary source with parameter $0 \le p \le 1/2$, i.e., $X,Y \sim \text{Bernoulli}(1/2)$ and $P(X \neq Y) = p$.

Define the two regions $\mathcal{A}, \mathcal{B} \subseteq \mathbb{R}^3$:

$\mathcal{A}$ consists of all points $(R_0, R_1, R_2)$ such that there exist

**binary**random variables $U,V$, satisfying the Markov chain $U - X - Y - V$ and \begin{align} R_1 &\ge \mathrm{I}(U;X) , \\ R_2 &\ge \mathrm{I}(Y;V) , \\ R_0 &\le \mathrm{I}(U;V) , \end{align} where $\mathrm{I}(\cdot;\cdot)$ is mutual information.$\mathcal{B}$ consists of all points $(R_0, R_1, R_2)$ such that there exist probabilities $a,b \in [0,1]$ with \begin{align} R_1 &\ge 1 - \mathrm{H}(a * p) , \\ R_2 &\ge 1 - \mathrm{H}(b * p) , \\ R_0 &\le 1 - \mathrm{H}(a * p *b) , \end{align} where $a*b := a(1-b)+(1-a)b$ is binary convolution and $\mathrm{H}()$ is the binary entropy function.

My Question: Is $\mathrm{conv}(\mathcal A) = \mathrm{conv}(\mathcal B)$, where $\mathrm{conv}()$ denotes the convex hull?

Some comments:

The definitions of $\mathcal{A}$ and $\mathcal{B}$ are almost the same, except that the channels $X \to U$ and $Y \to V$ are required to be symmetric for $\mathcal{B}$ and hence $\mathcal B \subseteq \mathcal A$.

The question Do binary symmetric channels maximize mutual information? asked whether $\mathcal A = \mathcal B$ and a counterexample was provided for $p=0$, which does not seem to apply here.

Is it true that there exists some constant $A$ such that

$$\int_{2}^x \frac{1}{t}{\Big[\frac{d}{dt}Li(t)\Big]} \mathrm{d}t=\log\log x + A + O(x^{-1/2+\epsilon})$$ for any $\epsilon>0$, where $Li(z)=\int_{2}^z \frac{dz}{\log z} $ ?

For a Banach space $X$, the map $p:X\rightarrow\mathbb{C}$ is called a 2-homogeneous polynomial if there exists a bilinear form $A$ on $X$ such that $p(x)=A(x,x)$, the norm of this polynomial is defined by $sup_{\Vert x\Vert=1}|p(x)|$.

Take $X=l_\infty^n$, a finite dimensional space with maximum norm. I wanna show any 2-homogenous polynomial $p:l_\infty^n\rightarrow\mathbb{C}$ has a norm-preserving extension to $l_\infty$, a complex sequence space with supremum norm, i.e. there is an extension $\hat{p}:l_\infty\rightarrow\mathbb{C}$ so that $\Vert p\Vert=\Vert\hat{p}\Vert$ and $\hat{p}$ is a polynomial on $l_\infty$.

Any related articles are appreciated!

A theorem of Hungerford says that : Every PIR (principal ideal ring , obviously commutative ) is a homomorphic image of a finite direct product of PID s . My question is , is there a similar criteria for Noetherian rings, i.e. : Is every Noetherian ring a homomorphic image of a finite direct product of Noetherian domains ?

This Is every Noetherian Commutative Ring a quotient of a Noetherian Domain? shows that we cannot expect every Noetherian ring to be a homomorphic image of a Noetherian domain.

Recall that the spectrum of an invertible harmonic oscillator is continuous and
covers the real axis with **multiplicity two**.

Thanks a lot