Questions tagged [nt.number-theory]
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
15,955
questions
11
votes
2
answers
1k
views
Does this product have analytic continuation?
The product
$$
F(s)=\prod_{p}\frac1{(1-p^{-s})^p},
$$
converges for $\mathrm{Re}(s)>2$, when $p$ runs over all primes. Does it admit analytic continuation beyond the line $\mathrm{Re}(s)=2$? Any ...
user1688
3
votes
1
answer
345
views
Polynomials splitting into linear factors modulo certain primes
Given integers $a,b$, we say a polynomial $f(x) \in \mathbb{Z}[x]$ is an $(a,b)$-filter, if $f(x)$ splits completely into linear factors modulo an odd prime $p$ only if $p=a \pmod b$. For example $x^2+...
- 537
2
votes
1
answer
209
views
When will the value of automorphic function $f(x)$ satisify an algebraic equation?
When will the value of automorphic function $f(x)$ satisfy an algebraic equation? Or what is the value of $x$ such that the value of automorphic function $f(x)$ is algebraic?
If the question is too ...
- 3,000
2
votes
0
answers
153
views
Sums of nonzero triangular numbers
OEIS A002097 gives numbers that are not the sum of 3 nonzero triangular numbers. They are just seven numbers: 1, 2, 4, 6, 11, 20, 29.
The site says that the seven numbers are all without references. ...
- 483
6
votes
1
answer
419
views
Complete residue system modulo n (permutation of numbers 0 to n-1) such that
I have a task:
Find all $n\ \epsilon \ N, \ n > 1$ for which a permutation $a_1,\ a_2,\ ...,\ an\ $ of numbers $ 0,1, ..., n - 1$ exists such that $a_1,\ a_1+a_2,\ ...,\ a_1+a_2+\ ...\ +an\ $ form ...
- 63
0
votes
1
answer
75
views
On symmetric linear diophantine equations
Given $k\in\Bbb N$ is there coprime $1<a,1<b$ with $(k,a^2-b^2)=1$ and coprime $1<c,1<d$ such that $k|(ac-bd)$ and $k|(ad-bc)$?
What is the smallest $\max(|a|,|b|,|c|,|d|)=\max(a,b,c,d)$ ...
- 13.7k
3
votes
1
answer
221
views
Selmer $p$-Groups
I searched so many articles about Bloch-Kato $p$-selmer Groups defined by $p$-adic representation but it seems that $p$ is not necessary to be odd.
Hence, I am wondering if it is worth considering ...
- 41
10
votes
0
answers
381
views
Residue of Eisenstein Series on GL(n)
Reference: Mœglin, C.; Waldspurger, J.-L.: Le spectre residuel de GL(n)
On GL($n$), the result of Waldspurger shows that if an automorphic representation $\pi$ is non-cuspidal and in the discrete ...
- 3,764
13
votes
1
answer
795
views
What kind of non-cuspidal automorphic representation are not isobaric sums?
Let's say $\pi$ is an automorphic representation on $GL_3(A_{\mathbb Q})$ (or $GL_n(A_{\mathbb Q})$).
If $\pi$ is not cuspidal, what $\pi$ can be other than isobaric sums?
If there is such a thing, ...
- 3,764
0
votes
1
answer
145
views
Solving integer equations
Let $n$ be a positive integer. Let $t_1 \le t_2 \le\cdots \le t_n$ be integers and let $k$ be an integer with $0\le k\le n$. Suppose that $\sum_{i=1}^{n}t_i=-k$ and $\sum_{i=1}^{n}t_{i}^{2}=k$. ...
1
vote
1
answer
547
views
On Robin's criterion for the Riemann Hypothesis
Statement 1 : (Robin) proved that if the R.H. is false then there exist constants $0<\beta <\frac{1}{2}$ and $c>0$ small , such that $\sum \limits_{d|n} d \geq e^\gamma n \ln \ln n+ n\frac{ c ...
- 185
9
votes
4
answers
743
views
Divisor sum estimate
What kind of upper bound can one get for
$$
\sum_{d|n}\frac{1}{d^{\sigma}},
$$
where $0<\sigma<1$? The best I can do is $\ll n^{(1-\sigma)/2}$ by breaking the sum at $\sqrt{n}$, using the ...
- 10.8k
0
votes
2
answers
192
views
Space of functions f such that the number of primes in $ [x, x+f(x)] $ remains bounded
Given a positive integer $ n $ , let $ S_{b}(n) $ the set of functions $ f $ fulfilling the following conditions :
1) $ f $ is continuous, positive and increasing on $(n,+\infty) $
2) for ...
- 6,932
8
votes
2
answers
2k
views
Fermat's Last Theorem in finite fields
Consider the finite field $\mathbb{F}_q$. Schur (1916) proved that, given $n$, when the field is sufficient large, this equation,
$$x^n+y^n= z^n$$
always has a nontrivial solution.
What conditions ...
- 719
6
votes
2
answers
313
views
Can a nontrivial $\phi \in Gal(\bar K /\bar k )$ preserve all algebraically closed subfields?
The answer is "yes" if $\bar K = \overline{k(x)}$ (in fact, every $\phi \in Gal(\overline{k(x)}/\bar k)$ preserves all algebraically closed subfields -- there's only two of them, so it's not that hard!...
- 61.6k
4
votes
0
answers
167
views
Siegel-Walfisz Theorem in Opera de Cribro
In Chapter 20 of the book Opera de Cribro, when proving there are infinitely many primes of the form $x^2+y^2$ where $y$ is also a prime, the authors used the Siegel-Walfisz Theorem in the following ...
- 211
2
votes
0
answers
377
views
unramified character
In a few papers on p-adic Galois representations, I've come across this statement:
Let unr(x) be the unramified character that sends a geometric Frobenius to x.
Could someone explain to me how this ...
- 141
2
votes
1
answer
555
views
Could Furstenberg's Argument Prove the Infinitude of Primes in Number Fields?
I have a somewhat unconventional view of the Prime Number Theorem as a "quantification" of the infinitude of primes. Here I recall the argument of Furstenberg. Define a topology $\mathcal{X}$ on $\...
- 22.6k
3
votes
0
answers
146
views
Given an embedding of $X$ into $\mathbb{P}^n_K$, do you get an induced embedding of any twist of it into $\mathbb{P}^n_K$?
Let $X$ be a projective algebraic curve over some number field $K$, and let $\varphi:X\hookrightarrow \mathbb{P}^n_K$ be an embedding of it (defined over $K$) into some projective space.
Now let $X'$ ...
- 460
16
votes
5
answers
678
views
Is every $GL_2(\mathbb{Z}/n\mathbb{Z})$-extension contained in some elliptic curve's torsion field?
I suppose this question could be phrased in terms of Galois representations, but I'm asking it this way.
Let $n>1$ be an integer. If $K$ is a number field with $\operatorname{Gal}(K/\mathbb{Q}) ...
- 1,499
1
vote
2
answers
450
views
A prime number determined by its congruence relation?
Let $p_i$ denote the $i$-th prime number. Is there any "good function" $k(n)$ such that when we know $x_i$ for which $p_n \equiv x_i \pmod{p_i}$ for all $i\leq k(n)$, it is possible to find a unique ...
- 1,169
11
votes
0
answers
189
views
Explicit L-factor for supercuspidals
I am interested in L-functions for the quasi-split unitary group $U$ in three variables, following the construction by zeta integrals of Gelbart-Piatetski-Shapiro. My aim is to understand the ...
- 5,314
9
votes
0
answers
411
views
Number of prime factors in a very short interval
Let $k \geq 3$ be a (large enough) integer, let $x \in \mathbb{R}$,
and set $I_x := [x, x + \log^k x]$.
Some believe that for $x$ large enough there exists a prime $n \in I_x$.
Equivalently, there ...
- 11.2k
32
votes
2
answers
2k
views
A Collatz-like problem on prime numbers
Consider the function $f$ on the prime numbers defined by $$ f(p):= \text{ the greatest prime factor of } 2p+1.$$ The iteration of $f$ from any prime $p<10^8$ converges to the cycle $$(3,7,5,11,23,...
- 25.9k
11
votes
4
answers
446
views
Sequential addition of points on a circle, optimizing asymptotic packing radius
Suppose I have to put $N$ points $x_1, x_2, \ldots, x_N$ on the circle $S^1$ of length 1 so as to achieve the largest minimum separation (packing radius). The optimal solution is the equally spaced ...
- 5,926
7
votes
1
answer
470
views
Examples of the large sieve inequality where a constant larger than 1 is needed
Let $S(x) = \sum_{n=0}^{N-1} a_n e^{2 \pi i n x}$ be a trigonometric polynomial of length $N$. The analytic/harmonic large sieve inequality in its sharpest form states that
$$ \sum_{r=1}^R |S(x_r)|^2 ...
- 11.8k
5
votes
0
answers
183
views
Is $699 \ldots 998$ value of the Euler totient function?
Let $n_l = 7 \cdot 10^{l+1}-2 = 699\ldots 998$. I want to know if there is an $l$ such that $n_l$ can occur as the value of the Euler totient function $\varphi(n)$. For given $l$, it is easy to check ...
- 1,101
0
votes
2
answers
263
views
Is the number of real values of Zeta on the critical line up to some ordinate known?
The famous plot of $\zeta(1/2+it)$ for real $t$ seems to show that this function gets a non zero real value exactly once between two consecutive Riemann zeros. Moreover, letting $\rho_{i}$ the $i$-th ...
- 6,932
5
votes
1
answer
379
views
A Naive Question about Nekovar's Paper on Beilinson's Conjecture
I post this naive question on Math stackexchange, but have got no reply, so I decide to bother the mathoverflow community.
https://math.stackexchange.com/questions/2350436/rational-structures-of-...
- 2,961
18
votes
0
answers
2k
views
$G$ a group, with $p$ a prime number, and $|G|=2^p-1$, is it abelian?
During my research I came across this question, I proposed it in the chat, but nobody could find a counterexample, so I allow myself to ask you :
$G$ a group, with $p$ a prime number, and $|G|=2^p-1$, ...
- 3,801
5
votes
2
answers
215
views
Mod 2 eigensystems not defined over Z/2--looking for simple examples
Consider the weight 2 newform 67.2.1 b in the LMFDB table. It is defined over Q(root 5), and reducing modulo the inert prime (2) we get a mod 2 eigensystem defined over an extension of Z/2 but not ...
- 5,412
5
votes
0
answers
116
views
Arguments of Hecke-eigenvalues
Let $p$ be a prime and $\chi$ a primitive character mod $p$. Consider the orthonormal basis $B_k(q,\chi)$ of $S_k(q,\chi)$, the space of all weight $k$, level $p$ holomorphic forms with character $p$. ...
- 251
2
votes
0
answers
130
views
"Left over factors" of Fibonacci numbers squarefree?
One fact about Fibonacci numbers is they have the Mersenne-like property that $F_{mn}$ is divisible by $F_m$ and $F_n$ (but not necessarily $F_m F_n$, $F_3^2 \nmid F_9$ is a simple counterexample). ...
- 233
13
votes
2
answers
991
views
Action of SL(2,Z) on upper triangular primitive integer matrices of determinant N, from the right. Is it transitive?
I am porting this question across from StackExchange, since it has received no answers and perhaps is sufficiently deep to fit here.
I am considering the set of upper triangular matrices
$$D_N=\left\...
- 133
3
votes
2
answers
313
views
Is this BBP-type formula for $\ln 31$, $\ln 127$, and other Mersenne numbers also true?
In this post, a binary BBP-type formula for Fermat numbers $F_m$ was discussed as (with a small tweak),
$$\ln(2^b+1) = \frac{b}{2^{a-1}}\sum_{n=0}^\infty\frac{1}{(2^a)^n}\left(\sum_{j=1}^{a-1}\frac{...
- 12.1k
9
votes
2
answers
677
views
Oscillation of the summatory Möbius function
Let the Mertens function $$M(x) = \sum_{n \le x} \mu(n)$$ I assume (perhaps foolishly) that it is known that $M(x)$ changes sign infinitely often. If that's true, the question is a quantitative ...
- 95.6k
8
votes
2
answers
440
views
Number of points of elliptic curve over $\mathbb{F}_p$ with CM by $\sqrt{-2}$ when $p\equiv 1\bmod 8$
I was trying to calculate the number of points of the curve $E:y^2 = x^3 + 4x^2 + 2x$ over $\mathbb{F}_p$ for $p\equiv 1\bmod 8$ (In order to have $\sqrt{-2}\in\mathbb{F}_p$) but I did not succeed. ...
2
votes
0
answers
144
views
A sum involving fractional part function
I was exploring some sum when I came across this sum which I have no idea the value, here is the sum
Let $ N $ be an integer with the prime decomposition $ N = p_1^{k_1} p_2^{k_2} ... p_m^{k_m} $.
...
- 21
-4
votes
2
answers
269
views
Has nontrivial solution in positive integers of a diophantine equation: $x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$ [closed]
Has nontrivial solution in positive integers of a diophantine equation as follows ?
$$x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$$
Where trivial solutions are $x_i=y_j$.
Can you send me any ...
- 477
29
votes
1
answer
941
views
Is this BBP-type formula for $\ln 257$ and $\ln 65537$ true?
We have the known BBP(Bailey–Borwein–Plouffe)-type formulas,
$$\ln3 = \sum_{n=0}^\infty\frac{1}{2^{2n}}\left(\frac{1}{2n+1}\right)$$
$$\ln5 = \frac{1}{2^2}\sum_{n=0}^\infty\frac{1}{2^{4n}}\left(\...
- 12.1k
1
vote
0
answers
124
views
Computing the successive minima of the following lattice in $\mathbb{R}^4$
Let us define the lattice $\Lambda$ in $\mathbb{R}^4$ defined by the matrix
$$
\Lambda = \begin{bmatrix}
A & 0 & 0 & 0 \\
0 & A & 0 & 0 \\
\gamma_1 & \gamma_2 &...
- 3,595
6
votes
2
answers
461
views
A naive diophantine approximation question
Let $\alpha$ be a positive real number (bigger than one, and irrational if it matters) (the one I am secretly thinking of is $\varphi,$ the golden ratio). I want to know, given an $\epsilon>0,$ ...
- 95.6k
4
votes
1
answer
573
views
Another conjecture on sum $A+B=C$
Could You give your ideas, your comment, or a referen for a conjecture as follows:
Consider $A, B, C$ be three positive integers numbers. By Fundamental theorem of arithmetic we write:
$A=a_1^{x_1}...
- 477
4
votes
0
answers
196
views
Linear combination of characters
For each $i \in \mathbb{N}$, let $G_{i}$ be a finite abelian group and $\widehat{G_{i}}$ the $\overline{\mathbb{Q}}$-valued character group of $G_{i}$. Suppose that $|G_{i}| \rightarrow \infty$ as $i \...
- 51
1
vote
2
answers
202
views
Galois stability of characters
Let $G$ be a finite cyclic group and $\widehat{G}$ the character group. Let $S \subset \widehat{G}$ be a Galois-stable subset i.e. if $\chi \in S$, then the Galois conjugates $\chi^{\sigma} \in S$ for ...
- 51
6
votes
2
answers
560
views
Counting points on lattices in inside a box- Geometry of numbers
Let $\Lambda$ be a lattice in $\mathbb{R}^n$, and let $|\mathbf{x}|$ denote the $L^2$ norm. There is a fairly standard argument involving successive minima to obtain an estimate on $N(R)$ which is the ...
- 3,595
1
vote
1
answer
431
views
Proof of prime gap bound? [closed]
In another question on mathoverflow (What is the best currently proven bounds on prime gaps?) the following bound on the prime gap was quoted:
$G(X)\ll \frac{X^{0.525}}{\log X}$
How do you prove this, ...
- 45
5
votes
0
answers
230
views
Lindelöf Hypothesis and the Karatsuba conjectures
I'm aware of Shao-Ji Feng's result that Karatsuba's weaker conjecture ("conjecture 1") is true conditionally on the Lindelöf Hypothesis.
Shao-Ji Feng, "On Karatsuba conjecture and the Lindelöf ...
- 17.5k
7
votes
2
answers
466
views
Rankin-Selberg integral for GL(3) form with Odd Maass form on GL(2)
Let $F$ be a Hecke-Maass cusp form for $SL_3(\mathbb Z)$.
Let $u$ be a Hecke-Maass cusp form for $SL_2(\mathbb Z)$.
The following integral
$$\mathcal L(s,F\times u)=\int_{{SL}(2,\mathbb{Z})\...
- 3,764
4
votes
1
answer
329
views
Another integral that has a closed form involving finite series of $\zeta(2k+1)$'s. Could it be reflexive?
In the context of a series of questions here, here and here, about closed form expressions involving finite series of $\zeta(2k+1)$'s for certain integrals, I would like to raise another one:
$$f(n):=...
- 4,169