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# Questions tagged [prime-numbers]

Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.

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Let $\pi(x)$ be the number of primes $p$ not exceeding $x, \theta(x) = \sum_{p\leq x} \log p$ and $Li(x)$ be the logarithmic integral. Is it true that $$\pi(x)-Li(x) = \theta(x) - x + O(x^{1/2}\log^{... 0answers 66 views ### Convergence with the recurrence T_{n+1}=T_n^2-T_n+\frac{n}{p_n} For each integer n\geq 1 I define the recurrence$$T_{n+1}=T_n^2-T_n+\frac{n}{p_n}, with $T_1=1$, where $p_k$ denotes the $k-th$ prime. So multiplying by $(-1)^n$ and telecoping gives that for ...
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### On the integral $I_s = \int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$-follow up question

This is a follow up on On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$ According to the answer that i got, $I_s$ is not known to converge for any real $s<1$. But suppose $I_s$ ...
144 views

### On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$

Define $\pi(x)$ to be the prime counting function and Li(x) the logarithmic integral. Let $I_s$ be defined as above. Is $I_s$ known to be convergent for any real number $s<1$ ?
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### Is it possible to get a conjecture similar to Mandl's conjecture for a different arithmetic function of number theory, mainly related to primes?

I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number ...
381 views

### On the product $\prod_{k=1}^{(p-1)/2}(x-e^{2\pi i k^2/p})$ with $x$ a root of unity

Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...
1k views

### Is there a Kolmogorov complexity proof of the prime number theorem?

Lance Fortnow uses Kolmorogov complexity to prove an Almost Prime Number Theorem (https://lance.fortnow.com/papers/files/kaikoura.pdf, after theorem $2.1$): the $i$th prime is at most $i(\log i)^2$. ...
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### What about series involving strong primes?

I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' ...
316 views

### A conjectural formula for the class number of the field $\mathbb Q(\sqrt{-p})$ with $p\equiv3\pmod8$

Question. Is my following conjecture new? How to prove it? Conjecture. Let $p>3$ be a prime with $p\equiv3\pmod 8$, and let $h(-p)$ denote the class number of the imaginary quadratic field \$\...
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### An attempt to get a variant of Agoh–Giuga conjecture

The idea of this post is an attempt to explore a variant of the so-called Agoh–Giuga conjecture. In past days, and today, I tried to think about variants of this conjecture exploring congruences about ...

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