**9**

votes

**3**answers

352 views

### Representing a number close to 1 with a sum of reciprocals of natural numbers

For positive integers $n_1, \ldots, n_k$, let $H(n_1, \ldots, n_k)$ denote $1/n_1 + \ldots + 1/n_k$. Let $V(N)$ be the largest possible value of $H(n_1, \ldots, n_k)$ that is less than 1, subject to ...

**3**

votes

**1**answer

47 views

### Holomorphic Hoffstein-Lockhart

In the article Hoffstein, Jeffrey; Lockhart, Paul "Coefficients of Maass forms and the Siegel zero." Ann. of Math. (2) 140 (1994), no. 1, 161–181, it is stablished a good bound for the Petersson norm ...

**11**

votes

**0**answers

146 views

### Functions $f$ on $\mathbb{Z}/N\mathbb{Z}$ with $|f|$ and $|\widehat{f}|$ constant

Let $N$ be a positive integer; for simplicity I'm happy to assume it's an odd prime but I'm interested in the general case too.
Let $f \colon \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}$ and let ...

**-1**

votes

**0**answers

43 views

### An isogeny from a split algebraic torus

Suppose that there is an isogeny (in the category of commutative algebraic groups) from a split algebraic torus to a semi-abelian variety. Does it follows that this semi-abelian variety is also an ...

**28**

votes

**2**answers

599 views

### Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for ...

**9**

votes

**1**answer

1k views

### Go I Know Not Whither and Fetch I Know Not What

Next day: apparently my original question is harder, by far, than the other bits. So: it is a finite check, I was able to confirm by computer that, if the polynomial below satisfies $$ f(a,b,c,d) ...

**0**

votes

**0**answers

32 views

### On the global and local hermitian space

Let $E/F$ be a quadratic extension of number fields and $v$ a finite place of $F$.
Then I am wondering if there is a global hermition vector space over $E$ such that for all finite places $v$, local ...

**0**

votes

**0**answers

72 views

### Squarefree Parts of Mersenne Numbers with prime exponent [on hold]

The $n$-th Mersenne number is $M_n=2^n−1$. Write $M_n=a_n b^2_n$ where $a_n$ is positive and squarefree. In the discussion
Squarefree Parts of Mersenne Numbers , the lower bound of $a_n$ has been ...

**13**

votes

**7**answers

2k views

### How $a+b$ can grow when $a!b! \mid n!$

Let $a,b,n$ be natural numbers such that $a!b! \mid n!$. I am looking for a (somehow best) upper bound of $a+b$ in terms of $n$ (for large values on $n$). For example it is clear to see that we must ...

**-7**

votes

**0**answers

129 views

### Does $\pi$ encode the prime numbers? [on hold]

I have a question regarding whether or not $\pi$ encodes the sequence of primer numbers. It is common knowledge that
$$ \zeta (2) = \sum_{i = 1}^{\infty} \frac{1}{n^2} = \prod_{p \in \mathbb{P}} ...

**13**

votes

**3**answers

533 views

### Probing the generalization of the abc conjecture to more than 3 variables

Browkin and Brzezinski, in "Some remarks on the $abc$-conjecture", Math. Comp. 62 (1994), no. 206, 931–939, state the following generalization of the $abc$ conjecture to more than three variables:
...

**67**

votes

**10**answers

7k views

### Why is the Gamma function shifted from the factorial by 1?

I've asked this question in every math class where the teacher has introduced the Gamma function, and never gotten a satisfactory answer. Not only does it seem more natural to extend the factorial ...

**5**

votes

**1**answer

261 views

### Diophantine equation

I would like to ask the broad community what is known about the solutions of diophantine equation $$\frac{u}{v} +\frac{v}{w} +\frac{w}{u} =t$$ where $t,v,u,w\in \mathbb{N}.$
I read a book of W. ...

**6**

votes

**4**answers

591 views

### A question on non-archimedian Fourier transform

Let $M(n)$ be the vector space of $n\times n$ matrices over a local non-archimedian field $K$.
Let $\mathcal S$ denote the space of locally constant compactly supported functions
on $M(n)$. Similarly, ...

**1**

vote

**0**answers

40 views

### square tiled surfaces: Counting Saddle Connections vs Counting Square-Tiled Surfaces

I am reading Prime arithmetic Teichmuller discs in H(2) where they discuss the Siegel-Veech constant in a stratum $\mathcal{H}(2)$ of surfaces. However I see two definitions of Siegel-Veech constant:
...

**4**

votes

**0**answers

188 views

### $x^2+1$ attaining almost prime values

Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...

**9**

votes

**1**answer

383 views

### Is it easy to prove that $\sum_n |X(\mathbb{F}_{q^n})| t^n$ is rational?

Background: Let $X$ be an algebraic variety over a finite field $\mathbb{F}_q$. One of the successes of Etale cohomology - previously achieved by Dwork- was proving the rationality of the Zeta ...

**2**

votes

**1**answer

359 views

### Fractional Part Problem

Suppose we have coprime integers $a$ and $b$ with $p \mid a$ but $p^2 \nmid a$ for some prime $p\geq 5$. We can write $a=px$ and $b=pr+\hat{b}$. Suppose also that $a$ and $\hat{b}$ are coprime; that ...

**8**

votes

**2**answers

450 views

### Prove that the Dirichlet eta function is monotonic

Let us consider $\eta(p):= \sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n^p}$ for $p>0$. Has anyone come along with an elementary proof that $\eta(x)$ is monotonically increasing on this set? By ...

**2**

votes

**1**answer

126 views

### Growth of $r_k(n)$

What is the best known growth bound of $r_k(n)$, where $$r_k(n)=\#\{(a_1,\dots,a_k\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2=n\}?$$ Please provide some reference if known. Thanks.

**13**

votes

**1**answer

689 views

### Wrong asymptotics of OEIS A000607?

Sequence A000607 in the Online Encyclopedia of Integer Sequences is the number of partitions of $n$ into prime parts. For example, there are $5$ partitions of $10$ into prime parts: $10 = 2 + 2 + 2 + ...

**3**

votes

**0**answers

120 views

### Subgroup cliques in the Paley graph

It is a famous open problem to estimate non-trivially, for a prime $p\equiv 1\pmod 4$, the largest size of a subset $A\subset{\mathbb F}_p$ such that the difference of any two elements of $A$ is a ...

**11**

votes

**0**answers

609 views

### Are the twin primes the only positive double zeros of this real function?

Agno's answer
was extremely helpful.
For $x \in \mathbb{R}, x \ge 1$ define
$$ f(x) = \sin\frac{\pi(\Gamma(x)+1)}{\lfloor x \rfloor}$$
By Wilson's theorem the positive integer zeros of $f(x)$ are ...

**8**

votes

**4**answers

593 views

### The powers of non-empty subset of a group that generate a subgroup

If G is a group and A and B to non-empty subsets of G, then by AB we mean the set consist of all product ab where a is in A and b is in B.(Standard definition) Similarly we can define X^m where X is a ...

**6**

votes

**1**answer

306 views

### Is Gauss sum a p-adic measure?

Let $\Gamma$ be Galois group of cyclotomic $\mathbb{Z}_p$ extension over $\mathbb{Q}$. Consider the function $G$ which sends each finite order character $\chi$ of $\Gamma$ to the Gauss sum $G(\chi)$, ...

**2**

votes

**0**answers

95 views

### quasi-split algebraic group [migrated]

While reading papers, there usually an assumption "quasi-split" for reductive algebraic groups. To use their results I need to know which groups are quasi-split. For the case I am interested in ...

**32**

votes

**8**answers

4k views

### Why does the Gamma-function complete the Riemann Zeta function?

Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or ...

**3**

votes

**1**answer

220 views

### Theorem 7b of Serre's “Propriétés galoisiennes des points d'ordre fini des courbes elliptiques”

Could someone please point me towards a proof of the statement in the second paragraph, in the proof of Theorem 7b of Serre's Propriétés galoisiennes...? The statement is as follows:
Let $F$ and $F'$ ...

**5**

votes

**1**answer

298 views

### Kernel of the character of congruence groups

Let $\Gamma$ be a congruence subgroup of $SL_2(\mathbb Z)$ and $\chi:\Gamma \to \mathbb{C}^*$ a character of $\Gamma$ with finite image. Is then $\ker(\chi)$ also a congruence group? If not, can ...

**2**

votes

**0**answers

214 views

### Periodicity with irrational numbers [migrated]

Recently, I invented the following theorem and found a proof,
it seems strange since it is very counter-intuitive to me.
The proof is long and non-conceptual.
Is there a place or a branch of math ...

**0**

votes

**1**answer

269 views

### Covering a finite subset of $\mathbb{N}$ with prime arithmetic progressions

Because of a problem I ran into I am trying to get a quick start in covering with arithmetic progressions.
First I want to say I am aware of this previously asked question:
Covering $\mathbb{N}$ with ...

**0**

votes

**1**answer

107 views

### sum over primes involving divisor function (variation of the Titchmarsh divisor problem)

This question was also asked on MSE.
Does there exist an asymptotic estimate for the following sum over primes
$$
\sum_{p\leq x} \frac{\tau(p-1)}{p}\;,
$$
where $\tau(n)=\sum_{d|n}1$ is the divisor ...

**10**

votes

**4**answers

2k views

### Can a sum of roots of unity be an integer?

Let $n \geq 2$, $H \lneq (\mathbb{Z}/n\mathbb{Z})^*$, $\zeta_k$ a primitive $k$-th root of unity. Is it possible that $$\sum_{h \in H} \zeta_k^{h} \in \mathbb{Z}$$ for every $k$ dividing $n$ such that ...

**2**

votes

**0**answers

103 views

### State-of-the-art for the descent principle in relation to surfaces over a number field

I'll start with some motivating remarks (edit: as pointed out in the comments, these motivational remarks do not hold for surfaces: there is an example of a conic bundle surface over a real quadratic ...

**9**

votes

**4**answers

3k views

### “Counter”-example for Gauss's Lemma on irreducible polynomials

Gauss's Lemma on irred. polynomial says,
Let R be a UFD and F its field of fractions. If a polynomial f(x) in R[x] is reducible in F[x], then it is reducible in R[x].
In particular, an integral ...

**3**

votes

**0**answers

102 views

### multiplicity of automorphic representation of unitary similitude group

Let $G$ be a unitary similitude group over $\mathbb{Q}$ (as in the book of Harris-Taylor), $\pi$ an irreducible automorphic representation of $G(\mathbb{A})$. I'm looking for some results on its ...

**-3**

votes

**0**answers

81 views

### Tripet prime reciprocals series [closed]

Does any body know if the series of reciprocals of triplet primes of form
p,p+2,p+6 or p, p+4,p+6 converges or diverges. Could this be used as a proof of infinity of twin primes

**1**

vote

**0**answers

32 views

### All all hypo-multiplicative functions linear combinations of quasi-multiplicative functions?

A function is called quasi-multiplicative by many authors if $f(m)f(n)=f(1)f(mn)$, a slight generalization of multiplicativity. (Basically a multiplicative function times a constant is a ...

**20**

votes

**6**answers

3k views

### Minimal prerequisite to reading Wiles' proof of Fermat's Last Theorem

May I respectfully ask what the minimal background needed to read Wiles' proof of Fermat's Last Theorem is?
I'm not an expert on number theory, but out of curiosity I wanted to understand - at a ...

**4**

votes

**0**answers

159 views

### What is the density of the reciprocal of the set of cubes?

In his MathOverflow question "How thick is the reciprocal of the squares?" Kevin O'Bryant asks if a certain set, the reciprocal of the set of squares (identifying sets with power series in ...

**6**

votes

**2**answers

257 views

### Field of definition of Galois representations of weight 1 modular forms

Let $f$ be a weight 1 modular form (let's say cuspidal, new, normalized, and a Hecke eigenform). Then there's an associated Artin representation $\rho_f: \operatorname{Gal}(\overline{\mathbf{Q}} / ...

**3**

votes

**0**answers

298 views

### Metric on the set of subsets of the rational primes

Note: this is a revision of an earlier post. It was kindly pointed out that my initial proposed metric was in fact not a metric, so this is a revised version.
I was thinking how to say that two sets ...

**2**

votes

**2**answers

320 views

### Looking for ways how to calculate $\Phi_n(i)$

$Φ_n(1)$ and $Φ_n(−1)$ for the cyclotomic polynomials are well-known.
I am now looking for
$$Φ_n(i)$$
and/or
$$Φ_n(−i)$$
with i the complex unit.
At this moments my endeavours result in intricate ...

**-1**

votes

**1**answer

161 views

### Squarefree numbers with all digits equal 1 [closed]

Is this true that numbers of the form $$\frac{10^k -1}{9}$$ are squarefree for all values of $k\in\mathbb{N}$?

**1**

vote

**2**answers

156 views

### Looking for a reference for a paper by Mordell

On page 384 of the book "Number Theory:Volume 1:tools and Diophantine Equations" by Henri Cohen there is reference to the fact that: "It has been proved by Schinzel, Mordell nd successors that such an ...

**47**

votes

**3**answers

4k views

### What is the status of the Gauss Circle Problem?

For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) ...

**-6**

votes

**0**answers

80 views

### Atleast one prime between [closed]

For Goldbach conjecture to be true, there must be at least one prime between p and 2p+2
GC states that any even number must be a sum of two primes
So there must be at least one prime between a prime ...

**1**

vote

**0**answers

144 views

### Is it trivial to get $200$ algebraic abc triples of equal quality $1.6978…$ over isomorphic number fields?

Got $200$ algebraic abc triples over distinct though isomorphic
number fields of equal quality $1.6978...$
Strongly suspect I can get as many as I like
(assuming the computations are correct).
Is ...

**8**

votes

**1**answer

229 views

### Regularity assumption in the simple trace formula

In the simple trace formula of Deligne Kazhdan one assumes that the test function is supported at the elliptic regular elements at one place and is a supercusp form at another place. Why can't one ...

**11**

votes

**2**answers

542 views

### Failing of heuristics from circle method

The heuristic from circle method for integral points on diagonal cubic surfaces $x^3+y^3+z^3=a$ ($a$ is a cubic-free integer) seems to fit well with numerical computations by ANDREAS-STEPHAN ELSENHANS ...