Questions tagged [analytic-number-theory]
On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.
3,066 questions
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Chen's Theorem with congruence conditions.
I would like to revisit a closed question of asterios in a more MO kind of way,
because it cuts quite close to a related question about sieving that might be of general interest.
The original ...
user631
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distribution of $\{na\}$ when $a$ is irrational number
(by $\{x\}$ I mean the fraction part of the real number $x$)
If $a$ is an irrational number and $n$ is a integral number, what is the distribution of $\{na\}$? I'm asking for some continuous function $...
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Analytic lower bounds on the first sign change of pi(x) - li(x)?
There have been many results on the first sign change of $\pi(x)-{\mathrm{li}}(x)$: among others, Lehman, te Riele, Bays & Hudson, Demichael, Chao & Plymen, and most recently Saouter & ...
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Exact formulas for the partition function?
I am curious, what kind of exact formulas exist for the partition function $p(n)$?
I seem to remember an exact formula along the lines $p(n) = \sum_k f(n, k)$, where $f(n, k)$ was some extremely ...
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Function zeros in strip 0 < Re < 1 [closed]
Hi everyone.
Could you plz tell me where the zeros of $f(s)$ in the strip $\{0 < \Re s < 1 \}$ are ?
Do they all have $\Re s= 1/2$ ?
$$f(s) = 1 - 2^{-s} - 3^{-s} + 4^{-s} - 5^{-s} + ...$$
$$=...
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Dirichlet's divisor problem via Lambert series
In Über die Bestimmung asymptotischer Gesetze in der
Zahlentheorie, Dirichlet proved his theorem on the asymptotic
behaviour of the divisor function using a Lambert series: let
$d_n = d(n)$ ...
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Pierpont primes
A Pierpont prime is a prime $p$ that can be written as $$p=2^u 3^v + 1.$$
What is known about Pierpont primes? I'm not a number theorist, and the best I can find is
http://en.wikipedia.org/wiki/...
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Is there any sense in which Dirichlet density is "optimal?"
A philosopher asked me an interesting math question today! We know that there are sets S of integers which don't have a "natural" or "naive" density -- that is, the quantity (1/n)|S intersect [1..n]| ...
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Asymptotics of Product of consecutive primes
I am looking for the asymptotic growth of product of consecutive primes. Is there anything that is known about this growth?
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Curious about a question on zeta zeros? [closed]
I have Edwards and Titmarsch books on Riemann zeta function with me. I could not find (maybe I did not read through that carefully), but are there results similar to the form like the one given below:
...
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Consequence of equidistribution or not?
Let $\theta \not\in \mathbb{Q}$. We know that $(n\theta)_{n \geq 1}$ is equidistributed modulo 1.
Let $\epsilon_n = \mathrm{sign}\bigl(\sin(n\pi \theta)\bigr)$ and $S_N= \sum_{n=1}^N \epsilon_n$.
I'...
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Upper bound for real part of Riemann Zeta function zeros
I have been reading about Riemann Zeta function $\zeta(s)$ and have been thinking about it for some time. I did some calculations, and reached a conclusion where $\Re(\rho) \le \log_2(3) - 1$ as $\Im(\...
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Asociated sum series of the Euler Product over the Twin Primes?
Please consider the (presumably infinite) Euler product over the twin primes:
$$ f(z) = \prod_{p\in\mathbb{P}}^{\infty} \Big( 1 - \frac{1}{(p(p+2))^ z} \Big) $$ (in which $p(p+2)$ is a divisor of $4((...
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sums of digits of powers of integers
It is known (Senge and Straus, 1971, see also C.L.Stewart, 1980) that for every natural $a $, not a power of 10, and every natural $s$, there are only finitely many $k$ such that the sum of decimal ...
user6976
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"probabilistic" density of primes?
A certain set $\cal P$ of primes is defined by two assumedly independent conditions:
The first condition on a prime $p$ can be characterized in terms of the type of splitting of $p$ in certain Galois ...
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A question on the prime divisors of p-1
For each positive integer n we may define the convergent sum $$ s(n)=\sum_{p}\frac{(n,p-1)}{p^2} $$
where the summation is over primes p and $(a,b)$ denotes the greatest common divisor of a,b.
It is ...
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Do the roots of R(x) have any significance for the prime counting function?
I'm calculating the roots of the function
\begin{equation}
R(x) = \sum_{k=1}^{\infty}\frac{\mu(k)}{k}li(x^{1/k})
\end{equation}
This function seems to have a largest and smallest positive root. Can ...
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Proof in the literature of an equality involving the prime counting function
Let
\begin{equation}
R(x) = \sum_{k=1}^{\infty}\frac{\mu(k)}{k}li(x^{1/k})
\end{equation}
where $\mu$ is the Mobius function and
\begin{equation}
li(x) = \int_0^x \frac{dt}{\log t}
\end{equation}
Is ...
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A question about Mobius inversion
I don't know how precise I can make this question. I want to know whether there is a theorem that says that a certain phenomenon always happens, but I think the best I can do in order to pin down the ...
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An asymptotic expression for the solution to the squares problem suggested by statistical mechanics
The $s$ squares problem is to count the number $r_s (n)$ of integer solutions $(x_1,x_2,...,x_s)$ of the Diophantine equation $x_{1}^{2}+x_{2}^{2}+...+x_{s}^{2}=n$ in which changing the sign or ...
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Class Numbers and 163
This is a bit fluffier of a question than I usually aim for, so apologies in advance if this doesn't pass the smell test for suitability.
Likely my favorite fun fact in all of number theory is the ...
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How to find the almost period of an exponential polynomial
Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the ...
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Values of zeta at odd positive integers and Borel's computations
Someone recently quoted to me this recent article that claims to prove that $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$.
[Edit: published reference: Musha, Takaaki.
Negation of the conjecture for ...
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What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
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complete estimates of the error for a well-known asymptotic expression of partition p(n,m)
Let $p(n,m)$ be the number of partitions of an integer $n$
into integers $\le m$, we have a well-known asymptotic expression:
For a fixed $m$ and $n\to\infty$,
$$p(n,m)=\frac{n^{m-1}}{m!(m-1)!} (1+...
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What does the probabilistic model suggest the error term in the PNT should be?
Let $\Lambda(n)$ be the von Mangoldt function. The prime number theorem is equivalent to the statement that $\sum_{n \leq N} \Lambda(n) \approx N$. Defining $\lambda_{*}(n)= \Lambda(n)-1$ we may ...
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Complex and Elementary Proofs in Number Theory
The Prime Number Theorem was originally proved using methods in complex analysis. Erdos and Selberg gave an elementary proof of the Prime Number Theorem. Here, "elementary" means no use of complex ...
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The Wiener-Ikehara approach to the PNT
Was providing an alternative proof of the PNT one of the main impulses that led to the discovery of the Tauberian theorem of Wiener and Ikehara or the other way around?
In any case, do you know who ...
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Does Weyl's Inequality prove equidistribution?
Let $f(n) = \theta n^d + a_{d-1} n^{d-1} + \cdots a_1 n + a_0$ be a polynomial with real coefficients, and $\theta$ irrational. Let $S_N = \sum_{n=1}^N e^{2 \pi i f(n)}$. Weyl's Equidistribution ...
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What is the best known upper bound for the number of twin primes?
A quantitative form of the twin prime conjecture asserts that the the number of twin primes less than $n$ is asymptotically equal to $2\, C\, n/ \ln^2(n)$ where $C$ is the so-called twin prime ...
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Approaches to Riemann hypothesis using methods outside number theory [closed]
Background: Once an analytic number theorist remarked to me that all attempts to prove the Riemann hypothesis using number theoretic methods have failed. Since then that remark stuck in my mind.
The ...
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Dirichlet's Divisor Function
We know that by Dirichlet's formula for the Divisor function $ \displaystyle \sum\limits_{n \leq x} d(n) = x \log{x} + (2C-1)x + \mathcal{O}(\sqrt{x})$.
What is the best approximation available till ...
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The Riemann zeta function and Haar measure on the profinite integers
In an answer to a question on MU about the Riemann zeta function, I sketched a proof that the probability distribution on $\mathbb{N}$ which assigns $n$ the probability
$$\frac{ \frac{1}{n^s} }{\zeta(...
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Heuristic reason for Polya's conjecture
Let $\lambda(n)$ be Liouville's function, so that for each positive integer $n = p_1^{m_1}\cdots p_r^{m_r}$, we have that $\lambda(n) = (-1)^{\sum^{r}_{k=1}{m_k}}$. In 1919, Polya conjectured that $L(...
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Distribution of quadratic residues of a fixed number without using Dedekind zeta function
Let $n > 1$ be a square-free natural number, which is fixed. The assertion to be proved is the following:
Let $p$ run through primes. Then, $$\left( \frac{n}{p} \right)$$ is equally distributed ...
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Product over the primes
I'm trying to estimate the product
$$\prod_{p\lt q\lt r\lt s}1-\frac{24}{(pqrs)^2}$$
where $p,q,r,s$ are primes.
This is for the purpose of calculating the density of Sloane's A070284 [1]. The idea ...
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Counting points on lattices
I expect that the following is a standard problem from analytic number theory, but I don't know where exactly to look for an answer.
Let f: ℤr→ H be a surjective homomorphism into a ...
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Question on consecutive integers with similar prime factorizations
Suppose that $n=\prod_{i=1}^{k} p_i^{e_i}$ and $m=\prod_{i=1}^{l} q_i^{f_i}$ are prime factorizations of two positive integers $n$ and $m$, with the primes permuted so that $e_1 \le e_2 \cdots \le e_k$...
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2-adic valuation of the class number
I came across the following line and was wondering what it meant exactly and how you go about showing it.
Let d be a fundamental discriminant.
Let P(d) = the divisors of d except for the largest.
The ...
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Making a character small at a reciprocal
The following question emerged from thinking about the Erdős discrepancy problem. I don't know whether an answer would be directly helpful, but it might, and in any case I find the question quite ...
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Calculating the infinite product from the Hardy-Littlewood Conjecture F
The Hardy-Littlewood Conjecture F [1] involves the infinite product
$$\prod\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$$
where $\varpi$ ranges over the odd primes and $\left(\frac D\...
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Why Is $e^{\pi\sqrt{232}}$ an Almost Integer?
We have already discussed why $e^{(\pi\sqrt{163})}$ is an almost integer.
Why are powers of $\exp(\pi\sqrt{163})$ almost integers?
Basically $j(\frac{1+\sqrt{-163}}{2} ) \simeq 744 - e^{\pi\sqrt{163}...
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Is Li(x) the best possible approximation to the prime-counting function?
The Prime Number Theorem says that $\lim_{n \to \infty} \frac{\pi(n)}{\mathrm{Li}(n)} = 1$, where $\mathrm{Li}(x)$ is the Logarithm integral function $\mathrm{Li}(x) = \int_2^x \frac{1}{\log(x)}dx$. ...
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Primes as the first coefficient of a reduced indefinite quadratic form
Given a discriminant d>0 (make it fundamental if that is easier), when can a prime p be the the $x^2$ coefficient of a reduced indefinite quadratic form?
That is, for what p is there a reduced form $...
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Achieving consecutive integers as norms from a quadratic field
This question is inspired by my inability to make any progress on Will Jagy's question.
Giving a positive answer to this question should be strictly easier than proving Jagy's conjectures.
Suppose ...
- 156k
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votes
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answers
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How does one use the Poisson summation formula?
While reading the answer to another Mathoverflow question, which mentioned the Poisson summation formula, I felt a question of my own coming on. This is something I've wanted to know for a long time. ...
- 29k
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votes
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Uniform variant of Stirling's approximation
Stirling's formula is usually stated in the form $\log \Gamma(s) = (s-\frac12) \log{s} - s + \log\sqrt{2\pi} + E(s)$, where
$E(s) = c_1/s + c_2/s^2 + \dots + O(s^{-K})$ for certain absolute ...
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Variants of Grönwall's theorem
Except the original Grönwall's theorem that $$\limsup_{n \to \infty} \frac{\sigma(n)}{n \log \log n} = e^{\gamma},$$ and the two variants $$\limsup_{\begin{smallmatrix} n\to\infty\cr n\ \text{is ...
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Is there another proof for Dirichlet's theorem? [duplicate]
Possible Duplicate:
Is a “non-analytic” proof of Dirichlet’s theorem on primes known or possible?
Dirichlet's theorem on primes in arithmetic progression states that there are ...
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What are the analytic properties of Dirichlet Euler products restricted to arithmetic progressions?
There are (at least) two ways of writing down the Dirichlet L-function associated to a given character χ: as a Dirichlet series
$$\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$
or as an Euler product
$$\...
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