Questions tagged [prime-number-theorem]
The Prime Number Theorem is a theorem that describes the distribution of the primes. It says that the number of primes less than or equal to a real number $x$ is asymptotic to $\frac{x}{\ln x}$.
42 questions with no upvoted or accepted answers
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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 $\...
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Implications of divergence of $1/\zeta(s) $ at 1/2
$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function.
This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$.
Its convergence is unknown if $1/2< s&...
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Montgomery's conjecture and lower bound on certain Fourier transform.
Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...
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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
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A variant of the equidistribution of primes in an imaginary quadratic number ring
It is known that the arguments of prime elements of $\mathbb{Z}[i]$ are equidistributed in $(0,2π)$ (by Theorem 5.36 of Iwaniec and Kowalski, or one of Kubilius' papers cited below). This theorem ...
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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 $x$, $$\frac{1}{x}\sum_{n\...
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Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?
This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO.
For every ...
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Asymptotic expansion for the average of $\omega(n)^2$
Let $\omega(n)$ be the prime factors counting function. I computed that for any $k\geq 0$, there exist certain constants $c_{-1},c_0,c_1,c_2,...c_k$ such that
$$\sum_{n\leq x}\omega(n)^2=x(\log\log x)...
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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 ...
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Euclides' sieve
This is probably a well-known problem. Given a set or multiset of natural numbers let us construct its "Euclides" closure: on each step we take all possible products $P_i$ of the elements in the set, ...
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Does $\pi(x) \geq \mathrm{Li}(x)$ imply that $\vartheta(x) \geq x$
As the question in the title asks, does $\pi(x) \geq \mathrm{Li}(x)$ imply that $\vartheta(x) \geq x$? Here $\pi(x) = \#\{p \leq x\}$, $\vartheta(x) = \sum_{p \leq x} \log p$ and $\mathrm{Li}(x) = \...
user95470
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Prime powers between $x$ and $x+x^\theta$
By the result of Baker, Harman, Pintz (http://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf), for any sufficiently large $x$ the interval $[x-x^{21/40},x]$ contains a prime number. This ...
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Effective prime number theorem
The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...
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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}...
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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' ...
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Cancellation in this exponential sum?
I would like to know whether it is possible to obtain cancellation in the sum
$$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$
where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.
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Does data suggest $| \pi_2 (n) - 2\Pi \int_2^n \frac{dx}{\ln(x)^2} | < \ln(n+2)^2 \sqrt (n+2) $?
Let $\Pi$ be the twin prime constant and $\pi_2(n)$ the twin prime counting function.
Define
$$ t(n) = \left| \pi_2(n) - 2 \Pi \int_2^n \frac{dx}{\ln(x)^2} \right| $$
Is it consistent with current ...
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An approach to the prime number theorem with Rademacher variables and a recursive formula for the prime pi function?
Consider the bipartite graphs defined here:
Why is this bipartite graph a partial cube, if it is?
We do random walks on them with equal propability and since the graphs are finite and connected the ...
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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 $s_{0}$ ...
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A question on $a_i(n) = a_i(\pi(n)) + a_i(n-\pi(n))$ with $a_i(n) = 1$ for $n \le i$
Let $a_i(n) = a_i(\pi(n)) + a_i(n-\pi(n))$ with $a_i(n) = 1$ for $n \le i$ where $\pi(n)$ is the prime-counting function.
By definition, it is obvious that $a_1(n) = n$ and $a_2(n)$ is https://oeis....
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Bombieri-Vinogradov up to smaller moduli?
Bombieri-Vinogradov theorem (taken from Wikipedia) states:
Let $x$ and $Q$ be any two positive real numbers with
$x^{1/2}\log^{-A}x\leq Q\leq x^{1/2}.$
Then
$$\sum_{q\leq Q}\max_{y<x}\max_{1\le a\...
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Wiener-Ikehara Theorem and Signal Processing
I am trying to understand the Wiener-Ikehara Tauberian theorem which can be a step to understanding the prime number theorem. Let
$$ \hat{a}(s) = \int_0^\infty e^{-us}\, da(u) $$
with $a(u)$ some ...
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Nontrivial nonrandom properties of prime numbers
What are some nontrivial nonrandom properties of prime numbers. Consider the simple model where each number is prime with probability 1/log(n) by Montgomery and extensions of it. Once you add some ...
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Primality testing by reversible computation using the prime number theorem
Suppose we want to build a primality testing algorithm for the numbers limited to the set $A =\{1, ..., 2^n\}$ and $n$ is reasonably large. The prime-number theorem tells us that there are ...
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Do $k$ specific prime factors uniquely determine the continuous composite sequence of length $k$?
猜想:不存在两个长度为k且一共含有k个不同素因子的连续合数序列的素因子集合相等。例如
24 25 26 27 (2 3 5 13)
其余长度为4的连续合数序列要么素因子个数大于4 要么素因子集合不等于{2 3 5 13} 这个连续合数猜想的重大特定情况下的猜想。难度可与哥德巴赫猜想匹敌，非天才误入。再比如2 3 5 唯一决定了8 9 10 除此之外再无其他。总之经过计算机验证没有找到反例。
...
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R.H. equivalent statement condition
Is the inequality $\prod \limits_{p \leq \sqrt{x}} (1+\frac{1}{p^2-1}) \prod \limits_{p \leq x} (1+\frac{1}{p}) \leq e^\gamma \ln(\theta(\sqrt{x})+\theta(x))$ where $\theta(x)$ is the Chebyshev's ...
user95470
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A prime number simplex
Let $\ n\in\mathbf N:= \{1\ 2\ \ldots\}\ $ be a natural number.
Let $\ K\ $ be a non-empty finite set of primes. Let $\ \kappa:=|K|.\ $ Consider a $\kappa$-dimensional simplex $\ S_K\subseteq \mathbf ...
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Distribution of primes and near-primes among $\prod p_k \pm 1$
For $n\in \Bbb{Z}^+$ define the statement "$n$ is $k$-social" to mean that
$$
\prod_{i=1}^n p_i +1 \mbox{ has exactly } k \mbox{ prime factors}
$$
where $p_i$ is the $i$-th prime.
So for example $5$ ...
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Density of ratios of an arbitrary increasing sequence of prime numbers
It is well known that the set $\left\{ \frac{p}{q} : p,q \textrm{ prime numbers }\right\}$ is dense in the positive real numbers $\mathbb{R}_{>0}$. Not having a background in number theory, I ask ...
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Siegel Walfisz Theorem for algebraic number fields
Is there a generalization of the Siegel Walfisz to algebraic number fields? This has been done for the prime number theorem in the prime ideal theorem.
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Limit of Sequence of unusual Prime Product
Let $p_n$ be the nth prime and $p_L$ be closest to its square root:
\begin{equation}
p_L^2 \approx p_n \approx x
\end{equation}
Let $\sigma \in Z^+$ be a positive integer constant. Define the ...
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Explicit upper bounds on the number of primes up to the square of the $n^\text{th}$ prime number $p_n$
I'm looking for explicit upper bounds on the number of primes up to the square $m=p_n^2$ of the $n^\text{th}$ prime number.
Such estimates can rely on the knowledge of the exact number of primes up to ...
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Using Ehrhart polynomials to count primes?
As indicated below, one could use the Ehrhart polynomials of the simplex in number theory.
Here are the questions without context first:
Questions:
The sum $$\sum_{k=0}^t (-1)^k ( \operatorname{...
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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 ...
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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 $\...
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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,
$$...
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better estimates than the prime number Theorem in Euclidean domains
For a unique factorization domain we know that we have some the analogues of fundamental theorem of arithmetic, and can build elements by using 'building blocks'. For me the easiest examples are ...
- 1,561
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Repetitions of residua following a prime
NOTATION $\ p(0)\!=\!2\quad p(1)\!=\!3\quad\ldots\ $ -- the strictly increasing sequence $\ \mathbb P\ $ of all primes.
Conjecture
$$\forall_{k\in\mathbb Z_{>4}}\,\exists_{m\,n\in\mathbb P}\quad (\...
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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 ...
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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 $S_2(x) = \sum_{n \le x}r_2^2(n)$. ...
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Rate of growth of ratio of n-th prime to im part of n-th zero of $\zeta(z)$
Let $P(n)$ be the $n$-th prime, and let $Z(n) = -i (\rho_n - \frac12) $ where $\rho_n $ is the n-th zero of he zeta function (that is, the zero with n-th smallest positive imaginary part). Let $R(N) =...
- 1,247
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asymptotics of primes in arithmetic progressions
If $a$ and $q$ are given coprime positive integers, what is the best known error term for
$$
\sum_{p<x,\,p\,\text{is prime},\,p\equiv a \pmod q} \frac{\log p}p-\frac{\log x}{\varphi(q)}?
$$
Is it, ...
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