All Questions
Tagged with reference-request nt.number-theory
1,409 questions
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Jacobi's two-square theorem
Jacobi's theorem is: the number of ways of representing $N$ as a sum of two squares is $4(d_1(N)-d_3(N))$ where $d_i(N)$ is the number of divisors of $N$ that are of the form $4k+i$. I was wondering ...
- 221
3
votes
0
answers
85
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Kronecker limit formula for antiperiodic boundary conditions
The celebrated Kronecker limit formula gives the $\zeta$-reguralized determinant of the Laplacian on the torus $\mathbb{R}/(\mathbb{Z}\omega_1+\mathbb{Z}\omega_2)$ in terms of Dedekind eta function of ...
- 8,992
2
votes
0
answers
325
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Reference/PDF request for paper by Sathe and related note
I am looking for a PDF version of the following articles:
Sathe, L. G. - On a problem of Hardy on the distribution of
integers having a given number of prime factors, (I. - IV.) J. Indian Math. Soc. ...
- 593
9
votes
2
answers
3k
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Transformation formulae for classical theta functions
I am looking for a reference for the transformation formulae
for the classical theta-functions
$$\theta_4(\tau)=\sum_{n=-\infty}^\infty (-1)^n q^{n^2}$$
and
$$\theta_2(\tau)=\sum_{n=-\infty}^\infty q^{...
- 20.8k
5
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2
answers
403
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Is Howe's construction of tame supercuspidal representations independent of additive character?
Let $F$ be a $p$-adic field.
In "Tamely ramified supercuspidal representations of $Gl_n$" (Am. J. Math 73 (1977)), Howe constructs a supercuspidal representation $\pi_{\psi}$ of $GL_n(F)$ from the ...
- 1,453
18
votes
3
answers
6k
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The multiplicative order of 2 modulo primes
Artin's Conjecture says that any positive integer, which is not a square, is a primitive root modulo infinitely many primes. Christopher Hooley gave in
Hooley, Christopher (1967). "On Artin's ...
- 25.5k
8
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2
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852
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Does anyone have access to a copy of Yury G. Teterin's 1984 (Russian) preprint "Representation of numbers by spinor genera"
Encouraged by
Does anyone have an electronic copy of Waldspurger's "Sur les coefficients de Fourier des formes modulaires de poids demi-entier"?
I realized I could ask for this rare item ...
- 25.7k
2
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0
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125
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How many integers below $n$ can be expressed as a sum of $k$ $m$th powers?
For $m,k \geq 2$, let $C_{m,k}(n)$ denote the number of positive integers less than or equal to $n$ which can be expressed as a sum of $k$ $m$th powers.
I am interested in the asymptotic behavior of ...
- 163
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2
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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$$
Where trivial solutions are $x_i=y_j$.
Can you send me any ...
- 477
1
vote
1
answer
307
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Basic question regarding notation of summation over primitive characters
This seems like a very standard notation in analytic number theory, and I see it a lot. But I was confused with it and I would greatly appreciate any clarification.
When one writes sum of the shape
$$...
- 3,625
8
votes
2
answers
537
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Famous results about the value of a given limit assuming it exists
Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...
- 6,982
4
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1
answer
491
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Continued fraction of Liouville's constant
A friend and I were discussing the properties of continued fractions (as "best" approximations). For fun, we checked the continued fractions of Liouville's constant. The terms in the sequence fit a ...
- 4,432
6
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1
answer
740
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Numbers divisible only by primes of the form 4k+1
Let $A(N)$ denote the number of positive integers $n\le N$ composed of prime numbers $p\equiv 1\pmod 4$ only. Is there an asymptotic formula for $A(N)$ (as $N$ tends to infinity)?
- 63
5
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2
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3n+1 problem and cycles
Just to make sure I am up to date with this problem. I know (or I think I do) that it is not yet proven that there are no non-trivial cycles for the collatz sequence (please correct me if I am wrong). ...
- 2,275
3
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2
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513
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Asymptotic formula for the average number of zeros of a polynomial modulo p
Let $f$ be a non-constant polynomial with integers coefficients, and for each prime number $p$ let $\eta_f(p)$ be the number of zeros of $f$ modulo $p$. It is known that the average (in the natural ...
user40023
9
votes
1
answer
419
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Infinitely many solutions to $a^4+b^4+c^4=18$ over $\mathbb{Z}[i]$
We got infinitely many solutions to $a^4+b^4+c^4=18$ over
$\mathbb{Z}[i],i^2=-1$. Probably we can get infinitely many solutions
to $a^5+b^5+c^5=N$ over $\mathbb{Z}[\alpha]$ for algebraic $\alpha$.
...
- 25.4k
15
votes
2
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1k
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Evaluating the integral $\int_{1}^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$
I am trying to find a formula for the following integral for non-negative integer $k$:
$$\int_1^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$$
My first thought was to use the formula $$\...
- 11.4k
11
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1
answer
622
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Gauss, Jacobi, Kloosterman sums and representation theory in the $\mathbb F_1$-world
This question is inspired by Why are Bessel function and Kloosterman sum similar? - it developed in me desire to understand Kloosterman sums better.
There seems to be common knowledge that Gauss, ...
- 18.4k
12
votes
2
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499
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"Pythagoras number" for integral matrices
It is classically known that every positive integer is a sum of at most four squares of integers, i.e. every sum of squares of integers is a sum of four squares of integers. Now consider a symmetric $...
- 3,031
3
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1
answer
481
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Reference for explicit formula for $\sum_n \Lambda(n) \chi(n)$ with smooth weights
Let $\Lambda$ be the von Mangoldt function and $\chi$ a primitive character mod $q$, then we have the explicit formula
$$
\sum_{n \leq X} \Lambda(n) \chi(n) = \delta_{\chi} X - \sum_{ |Im \ \rho| \leq ...
- 3,625
6
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0
answers
467
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Eisenstein series of Hilbert modular forms
I am reading Shimura's paper "The Special Values of the Zeta Functions Associated With Hilbert Modular Forms" and I do not exactly understand his definition of the Eisenstein series in section 3.
...
- 123
24
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0
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1k
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Exotic 4-spheres and the Tate-Shafarevich Group
The title is a talk given by Sir M. Atiyah in a conference with the following abstract:
I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory)...
- 1,629
6
votes
2
answers
1k
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Does anyone have an electronic copy of Waldspurger's "Sur les coefficients de Fourier des formes modulaires de poids demi-entier"?
Is there an electronic copy of Waldspurger's paper "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" floating around the internet somewhere? This appeared in J. Pures Math. ...
- 13.1k
14
votes
0
answers
481
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If $ab^2$ is a sum of three squares, then so is $a$. How to see it quickly?
Here $a, b$ are positive integers, and the squares are the squares of integers. This follows from Legendre's three squares theorem, but is there a direct way?
- 109k
2
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147
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Well-known estimate for $L(s,\chi)$ for $\sigma=\text{Re}s\geq 1/2$
This is a very short question.
Let $s=\sigma+it$ be a complex number with $\sigma \geq 1/2$.
In the paper 'Jutila, Matti. "On the Mean Value of $L(1/2, \chi)$ FW Real Characters." Analysis 1.2 (...
- 663
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0
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333
views
Explicit bounds for the Mertens function
It is a consequence of some forms of the prime number theorem that with $\mu$ the Möbius function, for all $A > 0$, there exists $c_A$ such that for all sufficiently large $x$, $$\frac{1}{x}\sum_{n\...
- 1,890
2
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2
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484
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Growth of $r_{2}(n)$
Let $n$ be a positive integer. From Jacobi's two-square theorem we know that the number $r_{2}(n)$ of representations of $n$ as a sum of two squares is given by
$$
r_{2}(n)=4(d_{1}(n)-d_{3}(n)),
$$
...
- 433
3
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0
answers
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views
Hermitian sublattices of a given type
Consider an unramified quadratic extension $E/F$ of non-archimedean local fields, and suppose that $\langle\cdot,\cdot\rangle$ is a fixed Hermitian form on $E^d$ such that $\mathcal{O}_E^d$ is self-...
- 2,516
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1
answer
467
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Analytic properties of Eisenstein series
Let $\Gamma$ be a discrete subgroup of $SL_2(\mathbb{R})$ which has a cusp at $\infty.$ suppose that $\mu(\Gamma\setminus\mathbb{H})<\infty,$ consider the Eisenstein series :$$E(z,s,\Gamma)=\sum_{\...
- 400
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3
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What is the etymology for the term conductor?
This is related to the previous question of how to define a conductor of an elliptic curve or a Galois representation.
What motivated the use of the word "conductor" in the first place?
A friend ...
- 3,268
3
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1
answer
171
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Congruence of normalized eigenforms at two primes
Let $f_i\in S_{k_i}(\Gamma_0(N_i))$ be normalized cuspidal eigenforms for $i=1,2$ and let $K$ be the composite of the fields of Fourier coefficients generated by $f_1$ and $f_2$ and let $\mathfrak{p}...
user130124
8
votes
0
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Why is the set of lifts of a p-divisible group canonically the same as the set of lines that span $M(G)/FM(G)$?
Let $M$ be the Dieudonne module of a p-divisible group $G_0$ over $k$, and let a lift of $G_0$ to $A$ be a p-divisible group $G$ over $A$ such that $G \otimes_A k \simeq G_0$. Let $\omega_G$ be the ...
- 3,539
21
votes
3
answers
3k
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Twin Prime Conjecture Reference
I'm looking for a reference which has the first statement of the twin prime conjecture. According to wikipedia, nova, and several other quasi-reputable resources it is Euclid who first stated it, but ...
- 1,588
7
votes
1
answer
232
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Is anything known about this class of series involving the divisor function?
I hope it is OK to ask the following reference request. If my question is not suitable, please let me know and I will do my best to modify it!
Let $N\in\mathbb{N}$, let $q$ be a point in the open ...
- 661
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Published reference on the automorphism group of modular curves $X_1(N)$?
I wish to cite that the automorphism groups of $X_1(N)$ have already been completely calculated, and what they are, but I am having difficulty finding this calculation in the literature.
I have ...
- 3,539
3
votes
1
answer
108
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Is coprimality in $NC$?
Following reference https://pdfs.semanticscholar.org/e86e/8d7a267a29b9ad4ca112828109adfec55e8b.pdf claims integer coprimality is in $NC$ and it also has one citation. Is this claim valid?
- 13.9k
5
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1
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680
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When does this interesting sum diverge?
For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$
I don't know of any references or methods for this -- not even for $x=1$, for which the ...
- 3,443
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1
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714
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What is wrong with this counterexample to primality test assuming GRH? [closed]
From SMOOTH NUMBERS: COMPUTATIONAL NUMBER THEORY AND BEYOND Andrew Granville pp.13-14:
2j. Lenstra’s polynomial time test as to whether an integer that is conjecturally prime, is rigorously ...
- 25.4k
0
votes
1
answer
308
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Have you seen this prime distribution before?
The basic question is : has this system been considered before, and how do I find it? References to the literature would be most welcome, but I am asking for reasonable search terms. I will try the ...
- 13k
47
votes
1
answer
3k
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Which small finite simple groups are not yet known to be Galois groups over Q?
The subject line pretty much says it all. To expand just a little bit:
1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known ...
- 65.4k
6
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1
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413
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Looking for a copy of Algebraic Number Theory in honor of Iwasawa
I am looking for an electronic copy of this volume:
Advanced studies in Pure Mathematics, Volume 17
Algebraic Number Theory - in honor of K. Iwasawa
Edited by J. Coates, R. Greenberg, B. Mazur and I. ...
- 621
21
votes
2
answers
1k
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Most squares in the first half-interval
It is well known that if $p$ is an odd prime, exactly one half of the numbers $1, \dots, p-1$ are squares in $\mathbb{F}_p$. What is less obvious is that among these $(p-1)/2$ squares, at least one ...
- 14.7k
0
votes
1
answer
109
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Reference request: Markoff type equations
Consider the equation $x^2+ay^2+bz^2=(1+a+b)xyz.$ If there are infinitely many integral solutions, then up to permutations $(a,b)=(1,1),(1,2),(2,3).$ I have found a presentation by Waldschmidt (https:/...
- 298
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2
answers
334
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(Reference) A Shimura operator acting on Hermitian modular forms
In his book Arithmeticity in the theory of automorphic forms (http://bookstore.ams.org/surv-82-s) Shimura introduces at page 146 an operator $\Delta_p^q$ which should act on nearly holomorphic modular ...
- 253
1
vote
1
answer
990
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Is the existence of $\lim_{n\to\infty}\cos(n!\pi x)$ for given arbitrary irrational $x$ an open problem?
Motivated by a recent MSE question about the sequence of function $\cos(n!\pi x)$, I have read related several related questions:
On the behaviour of $\sin(n!\pi x)$ when $x$ is irrational.
Is there ...
user14319
4
votes
1
answer
601
views
Reference request, zeta function is rational function via Riemann-Roch?
I am looking for a reference to a proof that the zeta function of a function field in one variable over a finite field $\mathbb{F}_q$ is a rational function in $q^{-s}$ by using the Riemann-Roch ...
- 75
0
votes
0
answers
83
views
Is it possible to get a conjecture similar to Mandl's conjecture for a different arithmetic function of number theory, mainly related to primes?
I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number ...
19
votes
1
answer
2k
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Legendre and sums of three squares
The Three-Squares-Theorem was proved by Gauss in his Disquisitiones, and this proof was studied carefully by various number theorists. Three years before Gauss, Legendre claimed
to have given a proof ...
- 32.6k
3
votes
1
answer
330
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Growth Rate of Number of Divisors of Highly Composite Numbers
Let $H(k)$ be the $k^{th}$ highly composite number (HCN).
What is an asymptotic estimate for the size of $H(k)$ in terms of $k.$
What is an asymptotic estimate for the size of $d(H(k))$ where $d(n)$ ...
- 10.4k
5
votes
0
answers
268
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Reference Request on logarithm derivative of L-functions
I'm looking for references that show almost all Dirichlet characters $\chi \mod q$ satisfy
$$|\frac{L'}{L}(1+it, \chi)|=o(\log q)$$
where $t\in \mathbb{R}$ is fixed. I have been able to adapt a method ...
- 51