All Questions
Tagged with reference-request nt.number-theory
1,408 questions
4
votes
2
answers
288
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Best known bounds for $\left|\sum_{n<x}\mu(nk)\right|$ (Reference request)
What is the best known bound for the Mertens function along arithmetic progressions? More specifically, what is the best bound known for
$$\sum_{n<x}\mu(kn)$$
as $k,x\to\infty$. This paper of ...
- 2,902
6
votes
2
answers
546
views
On circles and ellipses drawn on an infinite planar square lattice
Consider a plane with a square lattice formed by all points with both coordinates as integers. As can be easily seen, a simple parabola can be found that passes through infinitely many of the square ...
- 5,979
4
votes
1
answer
476
views
What are the p-adic algebraic numbers?
"Given $p$, what are the elements of $\mathbb{Q}_p$ algebraic over $\mathbb{Q}$?"
I periodically wonder this and come across this mathoverflow question which seems to be asking the same ...
- 609
8
votes
0
answers
240
views
Question on calculating character sums
I am wondering if there are any references that would help me with the following problem:
Let $p>2$ be a prime number, $n \in \mathbb{Z}^+$ odd such that $(n,p)=1$, $\chi$ an imaginary quadratic ...
- 707
3
votes
1
answer
174
views
Sequences generated by sum & product of terms (with rotating indices): combinatorial?
Fix an integer $t\geq0$ and consider the sequence $T_{0,t}=1$ and for $n\geq1$, with $k*=n-1-k+t\mod n$,
$$T_{n,t}=\sum_{k=0}^{n-1}T_{k,t}T_{k*,t}.$$
EXAMPLES. Some initial terms:
(a) the case $t=0$: $...
- 43.2k
7
votes
0
answers
427
views
Is there a name for these kinds of polynomials?
I've come across the following polynomials in my research and I am wondering if they have a name or if there is very much known about them:
\begin{equation}
F_{\chi}(T) = \sum_{a = 1}^{n-1} \chi(a)T^a
...
- 707
1
vote
2
answers
632
views
Is there a database about the particular values of $j$-invariant?
Is there a database that has all the known particular values of the $j$-invariant?
- 2,971
5
votes
1
answer
478
views
Problem related to divisibility of even power sum
The question was posted in MSE(12/19/20)link, but gets no answer. Hence I'm posting in MO
Define $S_m(n)=1^m+2^m+\cdots+n^m$
Can it be shown that
$S_{2m}(uv)\equiv0\pmod{uv}\iff S_{2m}(u)\equiv0\pmod{...
- 599
3
votes
1
answer
758
views
Looking for a paper of Lagarias and Odlyzko
I have been studying about the Chebotarev Density Theorem and have been hunting for the following paper of Lagarias and Odlyzko for quite a while:
Effective versions of the Chebotarev density theorem, ...
- 739
1
vote
1
answer
247
views
Gauss sum of imprimitive characters
On Wikipedia they state the following identity for the Gauss sum of an imprimitive character: suppose that $\chi: (\mathbb{Z}/m\mathbb{Z})^\times \rightarrow \mathbb{C}$ is a Dirichlet character with ...
2
votes
0
answers
98
views
Sublattices in the standard integral symplectic lattice
Let $V$ denote $\mathbb{Z}^{2g}$ with its standard integral symplectic form $\omega = \sum_{i=0}^{g-1}dx_{2i} \wedge dx_{2i+1}$ (or, the homology lattice of a genus $g$ surface with its intersection ...
- 427
15
votes
0
answers
673
views
Exposition of Drinfeld's proof of function field Langlands for GL(2)
I know, or think I know, the vague outline of the proof: the Galois-to-automorphic direction is "classical," i.e. follows from converse theorems due to Grothendieck et al., and for the ...
- 311
8
votes
1
answer
855
views
What is the motivation for excellent rings?
First of all I am not formally educated in mathematics so pardon my ignorance if this is obvious and I am skipping something vital, but I am interested nonetheless in what the original motivation and ...
- 81
14
votes
0
answers
358
views
How do we deduce the Jacquet-Langlands correspondence from Fargues' two towers?
In trying to understand the geometric proof of the local-Langlands and Jacquet-Langlands correspondence which uses Fargues's two tower theorem, I am having trouble finding a nice source on this, and I ...
- 3,539
10
votes
1
answer
591
views
Are there numbers whose binary and ternary representations simultaneously have few digit transitions? How frequent are those numbers?
For a natural number $n$, let $c_b(n)$ denote the number of digit transitions in the representation of $n$ in base $b$. By a digit transition, we mean a pair of successive, unequal digits: for ...
- 183
2
votes
0
answers
145
views
On some rational points on an elliptic curve over finite field
Let $p\equiv3\pmod4$ be a prime. We consider the elliptic curve $E$ over the finite field $\mathbb{F}_p$
(in affine coordinates) defined by
$$y^2=x^3+x.$$
Clearly the discriminant of $E$ is $-2^6$. ...
user125345
42
votes
5
answers
14k
views
The unproved formulas of Ramanujan
Are there any formulas due to Ramanujan that have still not been proved—or disproved?
If so, what are they?
I believe this conjecture is due to Ramanujan and still open: if $x$ is a real number and $2^...
- 22.3k
0
votes
0
answers
87
views
Reference request for additive persistence of a number
It is well known fact that each natural number can be represented uniquely in any base. So we can define digit sum function whose value is sum of digits of the natural number in given base.
Let $f(n,b)...
- 249
2
votes
0
answers
491
views
Examples of almost Dedekind domains that are not Dedekind
All I know about almost Dedekind domains (which I have come to learn about only recently) is that they are integral domains whose localization at every prime is a discrete valuation ring. In other ...
- 739
0
votes
1
answer
192
views
English translation of Hasse's "Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage"
I want to read through Hasse's paper about cubic number fields: Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage, Mathematische Zeitschrift 31 (1930) pages 565-...
- 577
2
votes
0
answers
186
views
Dyadic models in number theory and "spillover"
In a classic blog post, Tao discusses the appearance of "dyadic models" in various guises in various areas of math. The number-theoretic version of the idea is to study polynomials over a ...
3
votes
0
answers
152
views
Finiteness of points over the cyclotomic extension for modular forms
Let $\rho(f):G_\mathbb{Q} \rightarrow GL_2(K_f)$ be the Galois representation attached to some cuspidal modular form $f$ where $K_f$ is a finite extension of $\mathbb{Q}_p$.
Let $V_f$ be the vector ...
- 313
1
vote
0
answers
213
views
Attempts to prove the Cohen - Lenstra heuristics based conjecture
In the well known Cohen - Lenstra paper published in 1983, the authors present an experimentally well-supported conjecture on computing certain asymptotics of class groups of real abelian and complex ...
- 577
4
votes
1
answer
307
views
English references to Cohen - Martinet Heuristics
I read through the celebrated paper of Cohen - Lenstra heuristics. But unfortunately, the Cohen - Martinet paper is originally written in French, which I do not understand. So I would like to know if ...
- 577
1
vote
0
answers
253
views
Who formulated the conjecture that the set of real parts of zeros of the Riemann zeta function is dense in $[0,1]$?
Does anyone know who formulated this conjecture related to Riemann's zeta function?
Conjecture. The set $$\{ x : \exists y \space \space \zeta (x+iy) = 0\}$$ is dense in $[0, 1]$.
In ...
19
votes
0
answers
523
views
univariate integer version of Hilbert's 17th problem
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
- 109k
4
votes
1
answer
264
views
Reference request: Long exact sequence in profinite Galois cohomology up through $H^2$
I'm looking for a reference of the following statement. Let $G$ be the Galois group of a Galois extension $L/K$, not necessarily finite. Let $A,B,C$ be groups with a continuous $G$-action, and let
$$1\...
- 7,527
4
votes
0
answers
506
views
Collatz conjecture and a diophantine equation
Let $M \ge 2$. Inspired by the Collatz iteration / algorithm ($M=2$), I tried the following function:
$$C_M(n) = n/M, \text{ if } n \equiv 0 \mod M, \text{ otherwise } (M+1)n+\{(M-n) \mod M \}$$
We ...
- 3,064
3
votes
0
answers
73
views
Reference: Asymptotic bit-complexity of algebraic operations and transcendental functions
This question is a reference request. Does anyone know of a reference that lists the asymptotic bit-complexity of algebraic operations and transcendental functions implemented on a Turing machine that ...
- 131
1
vote
0
answers
37
views
Raggedness measure of a sequence
This surely has been done, maybe I googled the wrong adjective...
Define a raggedness measure $r$ of a sequence $S$ in this way:
Two members $S_i,S_j$ of the sequence (who don't have to be adjacent!) ...
- 4,829
0
votes
1
answer
365
views
Where can I find the problem by Lagarias?
Jeffrey Lagarias proved, unconditionally, that:
$$
\sigma(n)<H_n+2\exp(H_n)\log(H_n)\qquad n>1
$$
This was posed as a problem in:
J. C. Lagarias, Problem 10949: A generous bound for divisor ...
- 53
14
votes
1
answer
749
views
reference for: no finite set of positive (integer) binary quadratic forms represents all primes
This recent question asks for a set of forms (binary quadratic) representing all primes.
Set of quadratic forms that represents all primes
When the question was asked on MSE last month
https://math....
- 25.7k
1
vote
0
answers
108
views
Question related to sequence of recurrence relation $a_k=\operatorname{rad}(a_{k-1}+a_{k-2})$ for $k\ge 2$ where $a_0=0,a_1=1$
Define radical of an integer Wiki
$$\displaystyle{\mathrm{rad}}(n)=\prod_{{\scriptstyle p\mid n\atop p\:{\text{prime}}}}p$$
Example $n=504=2^3\cdot3^2\cdot7$ therefore ${\displaystyle \operatorname{...
- 599
4
votes
1
answer
127
views
Question about the notation $N_{\chi}(\alpha, T)$, the number of zeroes of the $L(s, \chi)$ in a rectangle
I am confused with what seems to be a standard notation in analytic number theory and I'd appreciate any clarification. I am interested in the zero density estimates, for example link.springer.com/...
- 3,625
7
votes
2
answers
948
views
Reference of J.L. Waldspurger's paper on Shimura correspondence
I want to find reference of Waldspurger's paper referred at "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (...
- 73
3
votes
1
answer
220
views
Reference request: relationship between discriminant and smoothness of a conic over arbitrary fields
I'm looking for an (ideally modern) reference of the relationship between the discriminant and smoothness of projective conics over arbitrary fields (including those of characteristic 2). Let $k$ be a ...
- 7,527
7
votes
0
answers
346
views
The space of $p$-adic norms
The 1963 paper by Goldman and Iwahori The space of $p$-adic norms deals with the space of norms on a finite dimensional vector space $E$ over a locally compact complete discrete valuation field $K$. I ...
- 433
19
votes
1
answer
1k
views
Deligne's letter to Bhargava from March 2004
I am quite interested in moduli spaces for Rings and Ideals, a letter from Deligne to Bhargava is cited in Melanie Wood's thesis Moduli spaces for Rings and Ideals (pdf), studying the minimal free ...
- 461
9
votes
1
answer
388
views
$π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?
(I posted this question on Math SE but it has had no answer for a year now so I would like to ask if anyone here can provide one.)
Thinking about the prime number theorem, I wondered whether it is ...
- 2,912
15
votes
1
answer
484
views
Looking for a paper on transfinite diameter by David Cantor
I have been reading about transfinite diameter and its applications to number theory and have been hunting for the following paper for quite a while:
Cantor D.: On an extension of the definition of ...
- 739
12
votes
1
answer
526
views
Equidistribution of $\{\alpha p\}$ for $p$ in an arithmetic progression
Let $\alpha$ be irrational. A famous theorem of Vinogradov says that $\{ \alpha p\}$ is equidistributed in $[0,1]$ as $p$ runs over all primes.
Let $a,q$ be natural numbers with $\gcd(a,q) = 1$. Then ...
- 21.3k
2
votes
0
answers
65
views
Request for resources or techniques for bounding the infinity norm of an infinite product convolved with a simple function
I'm attempting to bound an expression of the form.
$$
\lVert(\prod_{i=1}^{\infty} \phi_i) * s \rVert_{\infty}
$$
Where $\phi_i$ are bounded periodic step functions which can be replaced by smoothed ...
- 41
1
vote
0
answers
188
views
I'm looking for a proof of Polya-Bertrandias Theorem
I'm looking for a proof of Polya-Bertrandias rationality criterion in english (not the one from Amice).
- 11
12
votes
2
answers
1k
views
Short research articles
I am a masters student. I am interested in short articles which have counter examples and very few references. I want to write a short and interesting article.
For example; One of the best known ...
14
votes
3
answers
2k
views
Norms in quadratic fields
This should be well-known, but I can't find a reference (or a proof, or a counter-example...). Let $d$ be a positive square-free integer. Suppose that there is no element in the ring of integers of $\...
- 38k
10
votes
5
answers
771
views
Reference request: Diophantine equations
I am looking for a textbook, or preferably lectures, on the subject of Diophantine equations. I am familiar with the basic principles of modular arithmetic, conics and the Hasse Principle, and the ...
- 2,811
18
votes
1
answer
631
views
Best texts on Lie groups for number theorists
What are the most comprehensive textbooks on the structure of Lie groups and their infinite-dimensional representations if one is interested in their applications to number theory (so covering ...
- 221
7
votes
2
answers
788
views
Reference request for the explicit formula for $\sum_{n\leq x} \Lambda(n)n^{-s}$
Denote by $\Lambda(n)$ the von Mangoldt function, which is equal to $\log p$ if $p\geq 2$ is a prime, and $0$ otherwise. Let $\rho$ denote a complex zero of the Riemann $\zeta$-function. If I recall ...
- 1,019
2
votes
3
answers
281
views
Diophantine equation of a factorial type
I'm interested in nontrivial solutions of Diophantine equations of the type
$$a^2b^3 = \frac{c!}{(c-k)!} $$
For various values of k fixed, and of course $a,b,c \in \mathbb{Z^+}$
Does anyone have any ...
- 41
2
votes
0
answers
313
views
On the Chowla and twin prime conjectures
I'm reading https://terrytao.wordpress.com/tag/chowlas-conjecture/ and at some point it is mentioned that, the twin prime conjecture is a variant of Chowla's conjecture that $\sum_{n\leq x} \lambda(n)\...
- 1,019