## nt.number theory – Does each prime \$p>541\$ has a quadratic residue \$x^4+y^4

For any prime $$p>5$$, one of the numbers
$$1^2+1=2, 2^2+1=5, 3^2+1=10=2times5$$
is a quadratic residue modulo $$p$$. In 2014 I conjectured that each prime $$p$$ has a primitive root $$g of the form $$k^2+1 (kinmathbb Z)$$ (cf. http://oeis.org/A239957), this is still open.

By a result of Fermat, the equation $$x^4+y^4=z^2$$ has no positive integer solution.

In view of the above, here I aske the following question.

Question 1. Whether for each prime $$p>541$$ there is a number of the form $$x^4+y^4$$ (with $$x,yinmathbb Z)$$ which is not only smaller than $$p$$ but also a quadratic residue modulo $$p$$?

Actually, I even conjecture that for any prime $$p>541$$ with $$pnot=941$$ there is a prime $$q of the form $$x^4+y^4 (x,yinmathbb Z)$$ with $$left(frac q pright)=1$$, where $$(-)$$ is the Legendre symbol. Of course, it is not yet proven that there are infinitely many primes of the form $$x^4+y^4$$.

The following question is similar to Question 1.

Question 2. Whether for each odd prime $$pnotin{7,17,47,103}$$ there is a number $$q of the form $$x^4+y^4 (x,yinmathbb Z)$$ with $$left(frac qpright)=-1$$?

In 2001 Heath-Brown (Acta Math. 186 (2001), 1-84) proved that there are infinitely many primes of the form $$x^3+2y^3$$ with $$x,yinmathbb N={0,1,2,ldots})$$. Motivated by this, here I pose the folowing question.

Question 3. Whether for each odd prime $$p$$ there is a prime $$q with $$left(frac qpright)=-1$$ such that $$q=x^3+2y^3$$ for some $$x,yinmathbb N$$ with $$y+1$$ prime?

I have checked Question 3 for all odd primes $$p<2times10^9$$, see http://oeis.org/A344173 for related data. For example, the prime $$q=3^3+2(3-1)^3=43$$ is a quadartic nonresidue modulo the prime $$p=457$$.

## nt.number theory – Can digits of real irrational number be contained in digits of real irrational number?

We have the following cases:

1]Digits of algebraic real irrational number be contained in digits of transcendental real irrational number.

2]Digits of algebraic real irrational number be contained in digits of algebraic real irrational number.

3]Digits of transcendental real irrational number be contained in digits of algebraic real irrational number.

4]Digits of transcendental real irrational number be contained in digits of transcendental real irrational number.

I answered the question for case 1 as following:

We know that digits of algebraic real irrational number can’t be contained in transcendental real number because for example :

If we want $$sqrt2$$ be contained in $$pi$$ then $$pi=frac{a}{10^n}+frac{sqrt2}{10^{n+1}}$$ for $$a,n in mathbb{N}$$ which is algebraic number … contradicting that $$pi$$ is transcendental .

Also the third case it is not right because algebraic number would equal transcendental number by similar approach of first case.

So what about other cases?

## nt.number theory – An invariant subspace under \$G_Q\$ action , and BSD-rank

Let $$E/Q$$ be an elliptic curve. $$E(bar{Q})$$ is a complicated abelian group, which equals to all closed points of $$bar{E}$$, and also, a $$G_Q$$-Galois module. Its torsion part, $$E(bar{Q})_{tor}$$ is a subgroup ( $$G_Q$$-submodule ) of $$E(bar{Q})$$, as an abstract group is isomorphic to $$(Q/Z)^2$$.

Consider $$E(bar{Q})/E(bar{Q})_{tor}$$. It is naturally a $$Q$$-vector space, as it is uniquely divisible, also, a $$G_Q$$-module. Then we can talk about the invariant subspace under $$G_Q$$ action, $$E(bar{Q})/E(bar{Q})_{tor}^{G_Q}$$. Obviously, $$E(Q)otimes Q$$ is a subspace of $$E(bar{Q})/E(bar{Q})_{tor}^{G_Q}$$.

Are they equal?

## nt.number theory – Simultaneous embeddings of ring of integers into product of rings

This question is about something mentioned in Katz’s “$$p$$-adic $$L$$-functions for CM fields” in section 2.0.

Let $$K$$ be a number field with $$(K:mathbb{Q}) = d$$ and ring of integers $$mathcal{O}_K$$. Let $$Sigma$$ be the set of embeddings $$sigma colon K to overline{mathbb{Q}}$$, and $$K^{gal}$$ the compositum of the fields $$sigma(K)$$ over all $$sigma in Sigma$$, the Galois closure of $$K$$. Then let $$mathcal{O}^{gal}$$ be the ring of integers in $$K^{gal}$$, and $$R$$ an algebra over $$mathcal{O}^{gal}$$. For each $$sigma in Sigma$$, we have a ring homomorphism $$sigma colon mathcal{O}_K otimes R to R$$ given on pure tensors by $$n otimes r mapsto sigma(n)r$$. We can take them all together to produce

$$mathcal{O}_K otimes R to prod_{sigma in Sigma} R, qquad n otimes r mapsto (sigma(n)r)_sigma.$$

Katz then states that this is an isomorphism if the discriminant of $$K$$ is invertible in $$R$$.

I’m trying to figure out what the image of this map is when the discriminant is not invertible. If $$R$$ is flat over $$mathbb{Z}$$, it should be injective, just because each $$sigma$$ is injective. I thought I had found an obstruction to surjectivity using the homomorphism of Abelian groups
$$prod_sigma R to R, qquad T((r_sigma)_sigma) = sum_sigma r_sigma$$
where the composition $$mathcal{O}_K otimes R to prod_sigma R to R$$ is $$n otimes r mapsto operatorname{Tr}_{mathcal{O}_K/mathbb{Z}}(n)r$$. The image is then the subgroup of $$R$$ generated by traces of elements of $$mathcal{O}_K$$. However, the subgroup of $$mathbb{Z}$$ generated by traces of elements of $$mathcal{O}_K$$ is not the different ideal as I had assumed (e.g. $$Tr(1) = d$$ is there regardless of whether any primes dividing $$d$$ ramify), and so it doesn’t have as tight a connection to ramification in $$K$$ as I hoped.

Is there a nice description of the image of this map in general or in certain cases? I am most interested in the case when $$R$$ is the ring of integers in a finite extension of $$mathbb{Q}_p$$.

## nt.number theory – Quadratic character of factorials

Let $$p$$ be a prime number and $$S_p={(n!)^2 bmod p, n=1,2,dotsc,p-1}$$ the set of residues mod $$p$$ of squares of factorials. This set is obviously a subset of the group of quadratic residues mod p. For $$p=3,5,7,13,17,23,29$$ it is also a group for multiplication mod $$p$$, i.e. a subgroup of the group of quadratic residues.

Question: Are there infinitely many primes $$p$$ for which $$S_p$$ is a group modulo $$p$$?

## nt.number theory – On complexity of a particular prime problem

Is the following problem in $$PH$$ and is it complete for any class?

Problem: Is the $$i$$th bit of the $$m$$th prime $$1$$?

It appears to require a counting quantifier which has to demonstrate witness is the $$m$$th prime and since there is an unique witness is it in $$UP^{PP}$$?

## nt.number theory – Strange lacunary Lambert series related to the Liouville function

Although I have my own interest for the Liouville function, I will suppress it here as the question seems to be interesting in its own right.

It occurred to me when I saw an answer by GH from MO to Normal numbers, Liouville function, and the Riemann Hypothesis. That answer mentions the paper by Borwein and Coons where it is proved (among other things) that the functions $$f_lambda(z)=sum_{ngeqslant1}lambda(n)z^n$$ and $$f_mu(z)=sum_{ngeqslant1}mu(n)z^n$$ are both transcendental (here $$lambda$$ is the Liouville function and $$mu$$ is the MÃ¶bius function; just in case, let me recall that $$lambda(n)=(-1)^{Omega(n)}$$ where $$Omega(n)$$ is the number of prime factors of $$n$$ counted with multiplicities, while $$mu(n)=(-1)^{omega(n)}$$ when $$n$$ is the product of $$omega(n)$$ distinct primes and zero otherwise).

Initially I became curious whether these series have the unit circle as the analyticity boundary and, if yes, what can be said about their radial limit values at roots of unity. One natural thing to look at in this respect are the Lambert series for these functions. Although this is not directly related to the question, it is still related, so let me just say without proof, that$$f_lambda(z)=sum_{ngeqslant1}frac{tildelambda(n)z^n}{1-z^n},qquad f_mu(z)=sum_{ngeqslant1}frac{tildemu(n)z^n}{1-z^n}$$where $$tildelambda$$ and $$tildemu$$ are multiplicative with, for $$p$$ a prime, $$tildelambda(p^k)$$ is $$(-1)^ktimes2$$ while $$tildemu(p^k)$$ is $$-2$$ for $$k=1$$, $$1$$ for $$k=2$$ and $$0$$ for $$k>2$$.

What happened next was that I thought about representing these functions as logarithmic derivatives of some functions with nice infinite product expansions, and then modified them slightly thinking about obtaining sort of nicer infinite products. Doing that I stumbled upon the following:$$begin{multline*}z(1+f_lambda(z))=z+z^2-z^3-z^4+z^5+…+lambda(n)z^{n+1}+…\=frac z{1-z}-frac{2z^3}{1-z^3}-frac{2z^4}{1-z^4}+frac{2z^{12}}{1-z^{12}}-frac{2z^{13}}{1-z^{13}}+…end{multline*}$$

Surprised by this strange “jump” from 4 to 12 I looked at the exponents in this Lambert series and found that there are several other jumps of this length (from $$n$$ to $$n+8$$), many shorter jumps, as well as at least one still longer jump, from 4450 to 4459. Note that there are no jumps at all for $$tildelambda$$. So my question is,

is there any explanation for these strange jumps? Are their lengths bounded?

Some considerations around it. Certainly there are lots of jumps for $$tildemu$$, since it is zero on any number divisible by a cube; but they are much shorter: no longer than $$4$$ up to $$n=5000$$. The analogous “shift” for $$mu$$, that is, the Lambert series for $$z(1+f_mu(z))$$ has slightly longer jumps but still, it seems, essentially smaller than the shift for $$lambda$$ — for example, up to $$n=5000$$ it does not have jumps longer than 6. Maybe all this changes for larger $$n$$, I don’t know.

Another thing: the Wikipedia page on Lambert series that I link to above contains some recent additions about some Factorization theorems that seem to exhibit new exciting links between Lambert series and partition functions. In principle these theorems provide explicit expressions between the Maclaurin and Lambert series in very general situations. However I don’t readily see how to use them to explain these strange jumps.

I found two related questions on MO: Ordinary Generating Function for Mobius where the answers indicate that most likely there are no radial limits at all for $$f_mu$$ (so maybe I will ask a separate question about the other functions that appear here), and Lambert series identity with an answer that might be useful here, maybe also related to those factorization theorems.

## nt.number theory – Any more improvement for \$|pi-22dfrac{4times7^{2n+1}-4times7^{2n-3}+7^{2n-4}}{4times7^{2n+2}}|

This is a part of my research about irrationality measure of $$pi$$ and sum of power divisor function $$sigma$$, I have used the following ratio of odd iteration and even iteration of sum power divisor function $$dfrac{sigma_{2n+1}(p)}{sigma_{2n+2}(p)}$$ for $$p=7$$ which the ratio close to $$1/7$$. Multiply $$dfrac{sigma_{2n+1}(p)}{sigma_{2n+2}(p)}$$ by $$22$$ the value of that sequences would be close to $$22/7$$. It is known that $$sigma_n(p)=n^p+1$$ with $$p$$ is a prime number. Just a small improvement I come up to the following integer sequences $$a_n$$, and $$b_n$$ such that :
$$a_n=22*(4times7^{2n+1}-4times^{2n-3}+7^{2n-4}) quadtext{and}quad b_n=4times7^{2n+2}, quad n>0$$
such that $$a_n/b_n to pi$$ with approximation of $$10 ^{-6}$$. Now my research is the determination of the values of both $$C$$ and $$delta$$ to get a bound for the difference in the below absolute value or LHS of the following inequality:
$$begin{equation} leftlvertpi-22dfrac{4times7^{2n+1}-4times7^{2n-3}+7^{2n-4}}{4times7^{2n+2}}rightrvert

With the choice of $$C=e^{e^{e^{e^{e}}}}$$ the inequality holds up to $$10^{10}$$ using mathematica code with $$delta$$ in the range $$(0,1)$$. Now I have two question regarding the prediction of irrationality measure of $$pi$$ rathar than that the upper bound of LHS of inequality eqref{1}:

Question

1. Must $$delta$$ always lie in $$(0,1)$$ with the choice of large real number $$C$$ and with the choice of integer sequences $$a_n$$ and $$b_n$$?
2. Is it possible to find integer polynomial $$P_n$$ to make the ratio $$dfrac{a_n+p_n}{b_n+p_n}$$ as close as possible to $$pi$$ which lead to have a stronger bound which it self lead to predict the exact irrationality measure of $$pi$$?

## nt.number theory – Riemann-Siegel formula for Dirichlet characters

After unearthing and giving a proof of what is now known as the Riemann–Siegel formula for the Riemann zeta function enabling the computation of $$zeta(1/2+iT)$$ in time $$O(T^{1/2})$$,
in 1943 Siegel published a generalization for L-functions of Dirichlet characters (sorry, I don’t know how to put a link to that paper). If $$m$$ denotes the conductor of the character, it
seems to me that the time is now $$O((mT)^{1/2})+O(m)$$. The $$(mT)^{1/2}$$ term is clearly
necessary, but I do not see how to remove the $$O(m)$$ term, which is quite annoying in practice
when $$m$$ becomes large.

One of the reasons I ask is that in a relatively recent arXiv paper (arXiv:1703.01414v7)
K.~Fisher describes another algorithm which runs in $$O((mT)^{1/2})$$, without the $$O(m)$$.
Implementations of both algorithms seem to show that Siegel’s is considerably faster
(i.e., the implicit big-Oh constant is much smaller), but only when $$m$$ is small, and even
for $$m$$ as small as $$10^5$$ the $$O(m)$$ term becomes a huge problem and Fisher’s algorithm is
much faster. Does anyone have any idea
a) Is there a congruence subgroup $$Gamma leq mathrm{SL}_2(Bbb Z)$$, an integer $$k in Bbb Z$$ and a non-constant modular form $$f in M_k(Gamma)$$ such that $$f$$ has only finitely many non-zero Fourier coefficients $$a_n(f)$$ ?
b) What about $${n geq 0 mid a_n(f) neq 0}$$ having zero density?
One can easily have a non-zero modular form $$f$$ such that $$a_n(f) = 0$$ for every odd integer $$n$$. For part a), I think the answer should be no : $$f$$ is just a trigonometric polynomial and I guess one can come up with some elementary argument, but I don’t know exactly how. Part b) is maybe a more subtle question, I would be glad to have any information about it!