All Questions
Tagged with reference-request nt.number-theory
1,409 questions
4
votes
0
answers
261
views
Reference request for some result of de Bruijn on zeros of some holomorphic function
In a video lecture on Youtube, ''Vaporizing and freezing the Riemann zeta function'', Terry Tao states that ''de Bruijn proved that if for some $t_0$ the zeros of $H_{t_0}$ are contained in the strip $...
- 57
4
votes
2
answers
784
views
asymptotic for restricted partitions
Let $m$ and $n$ be two positive integers and denote by $P(n,m)$ the number of partitions of $n$ into $m$ non-negative integers.
Is there an asymptotic formula for $P(n,m)$ ?? Any reference is welcome....
- 2,384
2
votes
3
answers
467
views
Concise introduction to Beta transformations
I'm looking for a concise introductory on the subject of beta transformations on the circle. I've found things that are related to its applications in the computational field or in number theory. A ...
- 3,574
7
votes
1
answer
382
views
$\log \log p / \log \log n$, where $p|n$, gets equidistributed in [0,1] (for almost all $n$)
According to Hardy-Ramanujan/Erdős-Kac we know that usually there are $\sim\log\log n$ prime numbers in a factorization. But if you pick up a natural number at random, and you factor it, what is the ...
- 1,384
0
votes
1
answer
132
views
Different solution of power Diophantine equation based on constant term
Let us define a power Diophantine equation by 2 algebraic functions $f,g$ (having different degree) and by integers $k, l >0$ where, there are finite solutions for $f(x)+k=g(y)$, but there exists $...
- 267
7
votes
1
answer
745
views
Is Квант the actual source of this problem?
If I recall correctly, Andreescu & Andrica attributed the olympiad problem which prompted this question by S. Pek to the Russian magazine Kvant. Does anybody here know if the problem actually ...
- 87
7
votes
1
answer
1k
views
Szpiro's conjecture for function fields and Mochizuki's approach to the number field case
Where can I find more details on the proof of Szpiro's conjecture for function fields, as mentioned in Minhyong Kim's answer to this MO question?
I am looking at this in the context of Mochizuki's ...
- 3,309
1
vote
0
answers
100
views
Divisibility properties of linear combinations of binomial coefficients [closed]
Let $p$ be a prime and $a_0,\ldots,a_n\geq 0$ be integers. Define
$$
S(a_0,\ldots,a_n)=\sum_{k=0}^n a_k\binom{n}{k}.
$$
I am trying to find out how much we know about
$$
v_p(S(a_0,\ldots,a_n)),
$$
...
- 81
5
votes
2
answers
957
views
Dirichlet's approximation only using prime power as denominator
I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if $x$ is a real number and $Q>0$ there exist $p,q\in \mathbb{Z}$ ...
- 1,692
16
votes
1
answer
1k
views
Has this strong number theoretic conjecture of Euler been proved, and where could I find such a proof?
Polya cites this work of Euler as an example of a conjecture which Euler considered impossible to doubt, and yet still needing a demonstration.
It is on pages 90-98 of "Induction and Analogy in ...
- 179
4
votes
1
answer
524
views
Siegel-Walfisz Theorem with smooth weights
Let
$$\psi(x;q,a)=\sum_{n\leq x\atop n\equiv a\pmod q}\Lambda(n)$$
where $\Lambda$ denotes the von Mangoldt function and $\phi$ to be Euler's totient function.
Then the Siegel-Wafisz theorem states ...
- 3,625
4
votes
0
answers
273
views
Kaczorowski's Paper on Distribution of Primes
I am looking for a digital copy of the following paper by Jerzy Kaczorowski: ON THE DISTRIBUTION OF PRIMES (mod4)
https://www.degruyter.com/view/j/anly.1995.15.issue-2/anly.1995.15.2.159/anly.1995.15....
- 41
3
votes
1
answer
320
views
Papers containing Ihara avoidance arguments
I am trying to understand some of the recent research in number theory. There is apparently a certain lemma, called Ihara's lemma, which can be established in some contexts and is unknown in other ...
user140765
2
votes
1
answer
192
views
A Vandermonde-type system
For a prime $p$ and $a_1,\dotsc,a_n\in\mathbb F_p^\times$, consider the system of equations
$$ \begin{cases}
\begin{align}
a_1 + \dotsb + a_n &= 0 \\
a_1x_1 + \dotsb + a_nx_n &...
- 23k
5
votes
1
answer
613
views
generating $q$-Catalan numbers
An $n$-Dyck path (or a Catalan path) is a lattice path $P$, unit East and North steps, in an $n\times n$ square grid which stays (weakly) above the main diagonal. Let $\square_n$ denote all such paths....
- 43.2k
10
votes
1
answer
314
views
Coefficient bounds on cusp forms, half-integer weight
Let $f(\tau) = \sum_{n=1}^{\infty} a(n) q^n$ be a cusp form on $\Gamma_0(4N)$ of half-integer weight $k \ge 5/2.$ The Ramanujan-Petersson conjecture in this case is that $$a(n) \ll n^{(k-1)/2 + \...
- 101
26
votes
2
answers
3k
views
Was Vinogradov's 1937 proof of the three-prime theorem effective?
Was Vinogradov's first proof of the three-prime theorem effective?
Reasons for my question: Vinogradov presented his proof in 1937 in a monograph; the English translation by K.F. Roth and A. ...
- 20.2k
12
votes
1
answer
307
views
Partition of [3n] into summoids
Let $ [n] $ be the set $ \{1,2,\ldots n\}$.
A summoid is a subset $ A \subset [n] $ of the form $ \{a,b,a+b\} $ (you can choose a better name, if it doesn't exist already).
Now, I developed by ...
5
votes
1
answer
259
views
Central binomial coefficients deprived of $2$'s: not radicals?
In the paper, P Erdos, R Graham, I Ruzsa, E Straus, On the prime factors of $\binom{2n}n$, Math. Comp., 29:83–92, 1975, it was conjectured that the central binomials are never square-free for $n>4$....
- 43.2k
7
votes
2
answers
929
views
English translation of Voronoi's dissertation
I am looking for an English translation of Voronoi's doctoral dissertation, "On a generalization of the Algorithm of Continued Fractions." I can only find it in the original Russian.
- 173
12
votes
1
answer
1k
views
Do Shintani zeta functions satisfy a functional equation?
Probably my questions are known or evident to the experts but I'm a bit puzzled. First of all there seem to be two kinds of zeta functions that go under the name of Shintani zeta functions.
First, ...
- 2,029
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
4
votes
1
answer
170
views
The number of solutions of the equation $ax_1x_2+by_1y_2=n$
The equation $x_1x_2+y_1y_2=n$ is well-studied (Ingham, Heath-Brown, Deshouillers & Iwaniec, Ismoilov) because it arises in an additive divisor problem. The number of solutions in positive ...
- 12.3k
6
votes
2
answers
2k
views
Variation on the Subset Sum Problem
Given a nonempty set of integers, and given that there exists a subset of this set whose elements sum to zero, is finding the smallest such subset NP-complete?
Disclaimer: The above question ...
- 259
20
votes
6
answers
4k
views
Erik Westzynthius's cool upper bound argument: update?
Version 2 of this writeup is
available, and includes a newer and simple upper bound thanks to
MathOverflow 88777 as
well as indirect references to future writeups. Details of further work
...
- 13k
10
votes
2
answers
5k
views
Cohen-Lenstra Heuristics reference
I am looking for good references (preferably, books) on Cohen-Lenstra Heuristics (on Real Quadratic fields) which explain in detail the reasons behind its fundamental assumption (higher the ...
- 1,305
5
votes
2
answers
274
views
The Heegner hypothesis for a mean value result of Murty-Murty/Bump-Friedberg-Hoffstein
The classical mean value result of Murty and Murty (1991) and Bump, Friedberg, and Hoffstein (1990) on derivatives of modular form L-functions $L(s,f)$ proves (roughly speaking) the existence of ...
- 3,799
1
vote
1
answer
218
views
Chinese Remainder Theorem for Remainder Intervals
Given $n$ natural numbers $m_1,\dots,m_n$ and $n$ remainder intervals $[a_1,b_1],\dots,[a_n,b_n]$ holding $a_i < b_i$ for all $i\leq n$ the task is to search for the smallest natural number $x$ ...
- 11
6
votes
0
answers
410
views
Efficient solutions to general Bézout’s identity $a_1 b_1 + \dots + a_n b_n = 1$
Suppose I have integers $a_1, \dots, a_n$ which are coprime, meaning that
$$a_1 b_1 + \dots + a_n b_n = 1$$
has a solution in integers $b_1, \dots, b_n$.
I would like to get a bound saying ...
- 4,164
4
votes
1
answer
288
views
Is total degree version and $x,y$ degree version of Coppersmith's theorem correct?
The notes here https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a ...
- 13.9k
7
votes
4
answers
1k
views
Consequences of the Inverse Galois Problem
Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false?
We know a lot of things that would be true if the Riemann Hypothesis holds. ...
9
votes
1
answer
682
views
On the exact reference of a cute Diophantine problem
The problem asks to prove that the Diophantine equation $x^{3}+y^{3} = (x+y)^{2}+(xy)^{2}$ does not have any solutions in natural numbers $x, y$.
I believe that this problem appeared in the section ...
- 8,842
9
votes
2
answers
1k
views
The p-adic valuation of a linear recurrence
Let $(u_n)_{n \geq 0}$ be an integer-valued linear recurrence of order $k \geq 1$. Precisely,
$$u_n = a_1 u_{n-1} + \cdots + a_k u_{n - k} \quad \forall n \geq k ,$$
for some $a_1, \ldots, a_k \in \...
user40023
2
votes
2
answers
257
views
Reference request for function by which to compute coefficients of continued fraction of algebaic number
The simple continued fraction is in the form
$$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance. Obviously,the coefficients $x_i$can be computed by computable function $x_i=f(i),...
- 3,104
9
votes
2
answers
2k
views
References on Taylor series expansion of Riemann xi function
I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$.
$$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$
where
$$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...
- 603
7
votes
1
answer
1k
views
What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?
The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...
- 6,982
3
votes
1
answer
194
views
Divergence of a series related to Schinzel's hypothesis H
The Series
Consider the series identity
$$\Phi(s) = \sum_{n=1}^\infty \frac{\mu(n) (\log n)^k}{n^s} \sum_{r \in R(n)} \zeta(s,r/n) = \sum_{n=1}^\infty \frac{\Lambda_k'(f(n))}{n^s}$$
$$R(n) = \left\...
- 33
1
vote
0
answers
77
views
Digit summation of squared numbers
In olympiad teaching period, we have a session that students must try to design a good problem for others. Many times we arrive to good questions but sometimes there are some challenges. In one of our ...
- 4,784
10
votes
2
answers
1k
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 ...
3
votes
2
answers
384
views
Poles of the Rankin-Selberg zeta function associated to Hilbert cusp forms
Let $K$ a totaly real number field, $\mathcal{O}_K$ its ring of integre and $h$ the narrow class number of $K$. Let $\mathbf{f}$ a collection $(f_1, ..., f_h)$ of Hilbert cusp forms $f_\lambda$
$(\...
- 400
5
votes
1
answer
235
views
Algebraic independence of $a^a$ and $b^b$ for algebraic irrationals $a,b$
Let $a,b$ be algebraic irrationals.
Are there conjectures or unconditional results about the algebraic
independence of $a^a$ and $b^b$?
Probably Schanuel's conjecture is related,
maybe only $\log{a},...
- 25.4k
4
votes
4
answers
561
views
An upperbound for divisor function squared on a short interval
Let $d(n)$ be the divisor function defined by $d(n) = \sum_{m|n} 1$. I am in need of estimate of the following type:
$$
\sum_{Q \leq n \leq Q + H} d^2(n) \ll H (\log (Q + H))^T
$$
where $T$ can be any ...
- 3,625
8
votes
2
answers
676
views
The number of solution of $x_1^2 + \cdots + x_k^2 \equiv \lambda \bmod q$
I'm playing with exponential sums...
If $q$ is an odd prime and $a$ an integer such that $q \nmid a$, then the following formula for the Gaussian sum is known
$$\sum_{x=0}^{q-1} e_q(ax^2) = \left(\...
user40023
7
votes
1
answer
675
views
Short divisor sum
Let $d(n)$ denote the number of positive divisors of the positive integer $n$.
Pick some positive $X,h \in \mathbb{R}$ and consider the sum
$$ S(X,h) := \sum_{X \leq n \leq X+ h} d(n).$$
In view of ...
- 11.3k
7
votes
2
answers
1k
views
Is there a von Koch-type theorem for the generalized Riemann hypothesis?
Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound
$$
\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).
$$
Q1: ...
- 965
3
votes
1
answer
441
views
What is the shortest length of an Egyptian fraction expansion for a given $p/q$?
An Egyptian fraction expansion is a sum of reciprocals of integers, for example:
$$\frac{4}{17} = \frac{1}{5} + \frac{1}{29} + \frac{1}{1233} + \frac{1}{3039345}$$
Every positive rational number $p/...
- 4,164
4
votes
1
answer
292
views
References on Erdos conjecture on arithmetic progressions
Erdos conjectured that any set $ A $ of positive integers such that $ \sum_{n\in A}\dfrac{1}{n} $ diverges contains arbitrary long arithmetic progressions. The celebrated Green-Tao theorem is a ...
- 6,982
2
votes
1
answer
242
views
Frequency of digits in powers of $2, 3, 5$ and $7$
For a fixed integer $N\in\mathbb{N}$ consider the multi-set $A_2(N)$ of decimal digits of $2^n$, for $n=1,2,\dots,N$. For example,
$$A_2(8)=\{2,4,8,1,6,3,2,6,4,1,2,8,2,5,6\}.$$
Similarly, define the ...
- 43.2k
6
votes
5
answers
2k
views
The missing Euler Idoneal numbers
It is known that if GRH holds there does not exist additional Idoneal numbers. (see www.mast.queensu.ca/~kani/papers/idoneal.pdf this paper puts on the question of correctnes for Wikipedia and Wolfram ...
- 3,463
20
votes
4
answers
2k
views
Does the set of happy numbers have a limiting density?
A positive integer $n$ is said to be happy if the sequence
$$n, s(n), s(s(n)), s(s(s(n))), \ldots$$
eventually reaches 1, where $s(n)$ denotes the sum of the squared digits of $n$.
For example, 7 is ...
- 856