All Questions
Tagged with prime-number-theorem analytic-number-theory
93 questions
4
votes
1
answer
1k
views
Difficulty with "A new elementary proof of the Prime Number Theorem" by Richter
I'm studying Richter's "A new elementary proof of the Prime Number Theorem" paper, and I'm finding some problems understanding some parts of it. For example, I don't see how to get, in Lemma ...
- 43
3
votes
0
answers
191
views
What smoothing to use for PNT-like results?
Consider a Dirichlet series $\sum_n a_n n^{-s}$ with desirable analytic properties (e.g., analytic extension to $\Re s>0$); one example would be $a_n=\mu(n)$. Say we want to estimate $\sum_{n\leq x}...
- 20.2k
4
votes
0
answers
150
views
Identities to go from $\sum_{n\leq x} \mu(n) \log \frac{x}{n}$ to $M(x) = \sum_{n\leq x} \mu(n)$?
Let $\mu$ be the Möbius function. Say we have a bound on $\check{M}(x) = \sum_{n\leq x} \mu(n) \log \frac{x}{n}$ of the form $|\check{M}(x)|\leq \epsilon x$ for all $x\geq x_0$.
It is then easy to ...
- 20.2k
4
votes
1
answer
255
views
First occurrence of formula for $\sum_{n\leq x} \mu(n) \log n$ in terms of $\psi(y)-\lfloor y\rfloor$?
The identity contained in the last two displayed equations in the following passage (from page 110 in Ayoub's An Introduction to the Analytic Theory of Numbers, 1963) gives us right away a simple ...
- 20.2k
11
votes
3
answers
1k
views
What is the intuition behind applying the Mellin Transform to prime distribution?
I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem.
I understand that applying the Mellin Transform to the partial sum of the van ...
- 113
5
votes
1
answer
736
views
Smallest prime factor of numbers
The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
user344548
2
votes
2
answers
424
views
"Squeezing" the primes?
The logical idea here is to map a curve that encodes the primes into the region $(0,1)^2$ and analyze the distribution there more easily and achieve tight bounds.
To assess the distribution of primes, ...
- 383
3
votes
1
answer
540
views
Prime number theorem via the explicit formula
Can the prime number theorem be obtained from the explicit formula,
$\psi(x)=x-\sum_{\zeta(\rho)=0}\frac{x^\rho}{\rho}+O(1)$?
Here, $\psi(x)=\sum_{k=1}^\infty\sum_{p^k<x}\log p$
- 3,699
0
votes
0
answers
90
views
Generalizations for the PNT to a subset of Dedekind domains?
The classical prime number theorem states that the prime counting function
$$\pi(X) := \# \{ p \leq X \ | \ \text{$p$ prime} \}$$
is asymptotically equal to $X/\log(X)$.
It is also known (and much ...
- 131
2
votes
1
answer
152
views
Estimating the minimum number of distinct least prime factors found in range of $c$ consecutive integers
When I look at the count of distinct least prime factors for a range of consecutive integers, I am seeing the same minimum number appear again and again. I am wondering if this number represents the ...
- 1,049
10
votes
1
answer
398
views
Vinogradov-Korobov prime number theorem for number fields
Without assuming the Riemann hypothesis, the traditional error bound of the prime-counting function $\pi(x)$ is $O(x\exp(-c(\log(x))^{1/2}))$. As shown by the Wikipedia page for the Landau prime ...
- 211
11
votes
2
answers
1k
views
Mertens-like theorem
Mertens' first theorem states that
$$
\sum_{p \leq n} \frac{\log p}{p} = \log n + O(1).
$$
I read in this paper that the following variant is "classical":
$$
\sum_{p \leq n} \frac{\log p}{p -...
1
vote
1
answer
122
views
Best possible unconditional partial sum estimate of $\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$:
Consider the following partial sum:
$$S(x,n)=\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$$
Here p runs through primes and $n$ is constant
What is the best possible unconditional( using best known version ...
- 149
13
votes
4
answers
2k
views
Proving Mertens' theorem using the prime number theorem
Mertens' Theorem states that
$$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + O(1/\log x).$$
This is weaker than the prime number theorem; in fact according to the Wikipedia page, the prime number ...
- 21.1k
2
votes
1
answer
283
views
Explicit bounds on number of primes of given size
How many prime numbers of $b$ bits are there?
Beyond the prime number theorem, one can give explicit bounds on the number of primes below some integer $n$, or in a given interval. For instance, Rosser ...
- 456
10
votes
0
answers
416
views
Are prime numbers among sums of prime numbers distributed as $\frac n{2\ln(n)}$?
Let $(s_n)_{n\in\mathbb N}$ be defined as follows:
For $n\in\mathbb N$, $s_n:=2+3+5+\cdots+p_n$ is the sum of the first $n$ prime numbers (e.g.: $s_1=2$, $s_2=5$, $s_3=10$, $s_4=17$, $\ldots$).
Let $\...
0
votes
0
answers
136
views
Bounded sums involving primes
I'm trying to generalize the Theorem 2.7.1 in [1] where they prove:
$$\sum_{p \leq x} f(p) = \int_{2}^{x} \frac{f(t)}{\log{t}} dt + \epsilon(x)f(x) - \int_{2}^{x} \epsilon(t) f^{'}(t) dt $$
where $\...
- 109
0
votes
0
answers
91
views
How to use prime number theorem In such cases?
Let,
$$A(x)=\sum_{p\leq x}f(p)$$
Where $p$ is a prime number.
Under the Prime Number theorem we have that,
$$\pi(x)=Li(x)+O\left(\frac{x}{e^{a\sqrt{\ln(x)}}}\right) $$
as $x$ approach infinity.
Now,
$$...
- 111
2
votes
1
answer
300
views
Averages of Möbius function in arithmetic progressions
It is mentioned in multiple occasions here that the bound
$$
\mathop{\sum_{n=1}^{N}}_{n\equiv a\mod l} \mu(n) = o(N)
$$
is equivalent to the prime number theorem in arithmetic progressions. But I am ...
- 589
1
vote
1
answer
310
views
Asymptotic lower bound for the number of square free with at least two prime factors
In one of Soundararajan's papers, he claims without proof that it is a standard exercise to show that the number $N(X)$ of positive square-free integers $d \equiv 1 \; \bmod \; 8$ less than $X$, with ...
- 577
2
votes
1
answer
460
views
How essential is the vanishing of the Dirichlet $L$-functions to Dirichlet's theorem on primes in arithmetic progressions?
I seem to recall that the prime number theorem (PNT) is equivalent to the fact that the Riemann zeta function $\zeta(s)$ is non-zero on all of $\text{Re}(s) = 1$ (see https://math.stackexchange.com/...
- 781
4
votes
1
answer
246
views
Short proof of the error bound in PNT assuming a zero-free strip?
I am looking for a short proof of the fact that $\zeta(z)\neq 0$ for $\Re z>a$ implies the prime number theorem with an error bound $O(x^{a+\varepsilon})$ for any $\varepsilon>0$, which would be ...
- 8,992
6
votes
2
answers
390
views
A lower-bound for the square-mean of Fourier coefficients of cusp forms at primes argument
There is a basis question which puzzles me for a while. The question is the following:
Let $X\ge 2,$ and $\lambda(n)$ be the $n$-th Fourier coefficient of a $GL(2)$ newform of prime level $N>1$, ...
- 563
11
votes
4
answers
707
views
Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?
Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...
- 20.2k
3
votes
0
answers
1k
views
On new (purely analytic) perspective towards theory of prime numbers
[I'm going to ask this question very carefully as a question similar to this received a critical response on this platform.
I myself am very skeptical about this but I want to know, from the experts' ...
- 375
2
votes
0
answers
311
views
Proving that the Riemann zeta function is zero free on Re=1 using the prime number theorem
Is $\frac{-\zeta'(s)}{\zeta(s)}+\frac{-s}{s-1}$ an analytic continuation, holomorphic for $Re\ s > 0,\ s\neq 1$, of $f(s)=s\int_{1}^{\infty}\frac{\psi(x)-x}{x^{s+1}}\mathrm{d}x$?
If so: Let $s_{0}$ ...
- 129
4
votes
1
answer
395
views
Mertens formulas aren't enough for prime number theorem
For the primes it's true that
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/\ln x)
$$
where, $M$ is suitable constant, and, moreover, the prime number theorem gives that
$$
\lim_{x\to\infty}\frac{\...
- 41
7
votes
2
answers
636
views
How to use the Prime Number Theorem in order to prove Selberg's Formula?
I`m reading Melvin B. Nathanson's "Elementary Methods in Number Theory"
and I can't think of a way of deducing Selberg's formula (9.3) from the prime number theorem.
This is one of the tasks ...
- 129
1
vote
1
answer
181
views
Density of gaussian primes inside consecutive disks centered along the real axis of complex plane
Let's define the family of consecutive subsets of $\mathbb{N}$:
$$S_n =\{x \in \mathbb{N}\,:\,|x-n^2|\le n\}$$
With the previous definition we have that
$$U_n=\bigcup_{k=1}^n S_k=\{x \in \mathbb{N}\,:\...
- 1,008
-2
votes
1
answer
181
views
Polynomials of minimum degree that interpolate primes in intervals
Given an interval $[a,b]$ what is the minimum degree of univariate polynomials in $\mathbb Q[x]$ that passes through all primes between $a$ and $b$ (denoted by $\mathbb P[a,b]$ with total number of ...
- 1,826
-3
votes
1
answer
237
views
L. Gegenbauer's proof of Infinitude of Primes [closed]
I was going through the paper 'Euclid’S theorem on the infinitude of primes: A historical survey of its proofs' by Romeo Mestrovic where he mentioned that
L. Gegenbauer proved Infinitude of Primes by ...
- 233
6
votes
1
answer
499
views
Understanding Sylvester' s $1871$ paper of primes in arithmetic progression of the forms $4n+3$ and $6n+5$
The following is the proof of infinitude of primes in arithmetic progression of the form $4n+3$ and $ 6n+5$ done by Sylvester in $1871$ in his paper "On the theorem that an arithmetical progression ...
- 233
5
votes
2
answers
435
views
Proving certain inequality related to Primes
I was reading the following paper. But I can't understand why the last line concerning $\frac{2}{\pi}$ is true. The proof is a work of Sylvester.
I would be happy if someone helps me in understanding ...
- 233
8
votes
1
answer
245
views
Asymptotic density of sums of consecutive primes
Call a positive integer respectable if it is a sum of consecutive prime numbers. For example, every prime numbers is respectable. So are $3+5=8$, $2+3+5=10$, $5+7=12$, $3+5+7=15$, $2+3+5+7=17$, $7+11=...
- 15.4k
14
votes
1
answer
1k
views
A naive question about the prime number theorem
Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$, where $\Lambda(n)$ is the von Mangoldt function.
Then as Chebyshev showed, the following equality holds
$$\sum_{n\leq x} \psi(x/n)=x\log(x)-x+O(\log(x)).$$
My ...
- 165
12
votes
1
answer
894
views
Newman's proof of the prime number theorem
I am teaching a graduate course in Complex Analysis and I am covering Newman's proof of the prime number theorem. I have been using the simplified version in the papers of
Zagier and Korevaar. However,...
- 411
4
votes
2
answers
673
views
Counting prime ideals and an explicit Landau prime ideal theorem
Let $K$ be a number field, $\mathcal O_K$ be its ring of integers, and $\mathfrak p$ be a prime ideal of $\mathcal O_K$. Let $x\in \mathbb R^+$, and $N(\mathfrak p)$ be the norm of the prime ideal $\...
- 403
0
votes
0
answers
167
views
On the difference $\operatorname{Li}(\theta(x))-\pi(x)$
In G. Robin's paper, more precisely in Lemme12, how does he use formula (39) to prove formula (36)?
[1] Robin, Guy, "Estimation de la fonction de Tchebychef θ sur le k -ième nombre premier et ...
- 13
0
votes
1
answer
169
views
Reference request for this equivalence of the prime number theorem
Let $\psi(x)=\sum_{p^{k}\leq x} \log p$, $k\in \mathbb{N}$. If i recall correctly, the convergence of the integral $s\int_{1}^{\infty} (\psi(x)-x)x^{-s-1} \mathrm{d}x$ at $s=1$ is equivalent to the ...
user146617
7
votes
0
answers
461
views
On a paper of Alain Connes entitled 'Around Wilson's Theorem '
A relatively recent paper Alain Connes - Around Wilson's theorem
introduced the function
$$
S(n,x ) = \sum_{i=1}^n \sin^2\Bigl(\frac{(i-1)! x}{i}\Bigr).
$$
In the same paper, he proved that the ...
user145059
15
votes
3
answers
2k
views
Elementary lower bounds for the number of primes in arithmetic progressions
Some version of the Prime Number Theorem provides the asymptotic behavior of the number of primes in arithmetic progression $qn+a$ with $(q,a)=1$, $n \ge 1$. I was wondering there are Chebyshev-type ...
- 6,214
0
votes
1
answer
474
views
An upper bound for $\sqrt{p_{n+1}}$
Let $C$ be a positive constant. Is it true that for all sufficiently large integers $n$ the inequality $$\prod_{i=1}^n (1+\frac{1}{\sqrt{p_i}})>C\sqrt{p_{n+1}}$$ holds? (Here with $p_k$ is denoted ...
0
votes
1
answer
256
views
Lower bound for $\prod_{p\equiv 3 \pmod 4} p^{v_p(n!)}$
What is the best lower bound known for $$\prod_{p\equiv 3 \pmod 4} p^{v_p(n!)},$$ where the product is taken over all the primes(congruent to $3$ modulo $4$) less than or equal to $n$.
3
votes
2
answers
386
views
Prime divisors of $\prod_{i=1}^n (i^2+1)$
Is it true that for every positive integer $n$ there is a prime $p>n,$ which divides $\prod_{i=1}^n (i^2+1)$ ?
5
votes
0
answers
193
views
Asymptotic expansion for the average of $\omega(n)^2$
Let $\omega(n)$ be the prime factors counting function. I computed that for any $k\geq 0$, there exist certain constants $c_{-1},c_0,c_1,c_2,...c_k$ such that
$$\sum_{n\leq x}\omega(n)^2=x(\log\log x)...
7
votes
1
answer
1k
views
A curious prime counting approximation or just data overfitting?
I am not sure, if this is a research problem. If not I will move this question to ME:
Let $\Omega(n) = \sum_{p|n} v_p(n)$, which we might view as a random variable.
Let $E_n = \frac{1}{n} \sum_{k=1}^n\...
user6671
0
votes
0
answers
114
views
The best error term for the second moment
Let $r_2(n)$ be the number of representations of a positive integer $n$ as a sum of two prime squares, i.e. $n=p^2+q^2$. Consider $S_1(x)= \sum_{n \le x} r_2(n)$ and $S_2(x) = \sum_{n \le x}r_2^2(n)$. ...
3
votes
3
answers
380
views
Oscillations of $\theta(x)-x$, for the Chebyshev $\theta$ function
Is anything known about the relative "periodicity" of the oscillations of $\theta(x)-x$, that is, how frequent, in general terms, are the sign changes? Here, $\theta(x)$ is the Chebyshev $\theta$. ...
- 1,018
5
votes
1
answer
297
views
Landau's theorem using nth roots
This question was asked earlier at MSE .
Let $\omega$(n) denote the number of distinct primes dividing $n$. The Mobius function is defined as $\mu(n) = (-1)^{\omega(n)}$ if $n$ is squarefree and $\...
- 1,411
6
votes
0
answers
333
views
Explicit bounds for the Mertens function
It is a consequence of some forms of the prime number theorem that with $\mu$ the Möbius function, for all $A > 0$, there exists $c_A$ such that for all sufficiently large $x$, $$\frac{1}{x}\sum_{n\...
- 1,890