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6 votes
1 answer
2k 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 ...
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 ...
4 votes
0 answers
151 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 ...
3 votes
0 answers
192 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}...
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 ...
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=...
5 votes
1 answer
737 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 $\...
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, ...
3 votes
1 answer
541 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, $\psi(x)=\sum_{k=1}^\infty\sum_{p^k<x}\log p$
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 ...
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 -...
21 votes
3 answers
6k views

Why is the Chebyshev function relevant to the Prime Number Theorem

Why is the Chebyshev function $\theta(x) = \sum_{p\le x}\log p$ useful in the proof of the prime number theorem. Does anyone have a conceptual argument to motivate why looking at $\sum_{p\le x} \log ...
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 ...
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 ...
30 votes
3 answers
4k views

Heuristic argument for the prime number theorem?

Here is a bad heuristic argument for the prime number theorem. Let $n$ be a positive integer and assume that PNT holds up to $n$. Then $n$ itself is prime if and only if for each prime $p<n$ the ...
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 ...
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 $\...
1 vote
1 answer
317 views

An explicit value for a bound proof

I saw a proof that $|p_n - li^{-1}(n)| \leq n e^{-c \sqrt{\ln(n)}} $, without saying anything about $c$ ! My questions is, what the explicit value of $c$ ?? It just says for some number $c$ without ...
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 ...
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 $\theta$. ...
2 votes
1 answer
461 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/...
6 votes
2 answers
392 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 $N>1$, ...
2 votes
0 answers
313 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 $s_{0}$ ...
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{\...
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 ...
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}\,:\...
-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 ...
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 ...
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 ...
15 votes
5 answers
9k views

Upper bounds for the sum of primes up to $n$

Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ ...
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 ...
4 votes
2 answers
674 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 $\...
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 ...
1 vote
1 answer
867 views

$n$th prime: a better approximation

Let $p_n$ be the $n$-th prime, then from Wikipedia I got that $p_n \approx n \left(\ln n + \ln \ln n -1 + \frac{\ln \ln n-2}{\ln n}+\frac{6\ln \ln n-( \ln \ln n)^2-11}{\ln^2 n} \right)$. What is a ...
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 ...
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 ...
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
194 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\...
36 votes
2 answers
7k views

Why do primes dislike dividing the sum of all the preceding primes?

I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...
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 $\...
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\...
50 votes
5 answers
3k views

Motivated account of the prime number theorem and related topics

Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often ...
6 votes
2 answers
411 views

A simultaneous generalization of the Grunwald-Wang and Dirichlet Theorems on primes

By Grunwald-Wang Theorem, if for some odd number $n$ the equation $x^n=a$ has no solutions in $\mathbb Z$, then the equation $x^n=a\mod p$ has no solutions for some prime number $p$. I am interested ...
6 votes
2 answers
389 views

asymptotic for li(x)-Ri(x)

Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$ where $$\operatorname{Ri}(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \...
3 votes
0 answers
206 views

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.
13 votes
2 answers
2k views

Asymptotics of the n-th prime using the gamma function

In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that. $$ p_n = n \...