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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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        What is easier to find, the next prime number or next zero of zeta function?

        I mean at a fairly large height. At what height does the difficulty, change sign? Let us give the number of the prime numbers, with 5 decimals accurate. (When we use the zeros of zeta function ...
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        0answers
        45 views

        How to choose a prime p s.t. n-th cyclotomic polynomial splits into as much as possible irreducible polynomials while p is almost constant size?

        The reason I ask this question is that cyclotomic polynomial is critical to the construction of lattice-based cryptography. In most of the existing lattice-based cryptographic schemes, $n$ is usually ...
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        1answer
        268 views

        Perfect Runs of Consecutive Integers

        A run of 2 or more consecutive integers is said to be perfect if the sum of its terms equals the sum of all the their proper divisors, adding common divisors as often as they occur. For example, 672, ...
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        2answers
        188 views

        Robin's inequality and the zeros of the Riemann zeta function

        Robin showed that if $a\in(1/2, 1]$ is the supremum of the real parts of the zeros of the Riemann zeta function $\zeta(s)$, then $f(x)=\Omega_{\pm} (x^{-b})$, where $b$ is some number on $(a-1/2, 1/2],...
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        142 views

        Weak k-Tuple conjecture form and what we should prove [closed]

        Let $d \in \mathbb{N}, d \geq 2$ and consider the tuple $G(d,c) = (c_1,c_2,\cdots,c_d)$, with $c_1<c_2<\cdots<c_d$ are evens positive integers. I am trying to found an asymptotique formula ...
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        1answer
        1k views

        Why shouldn't this prove the Prime Number Theorem?

        Denote by $\mu$ the Mobius function. It is known that for every integer $k>1$, the number $\sum_{n=1}^{\infty} \frac{\mu(n)}{n^k}$ can be interpreted as the probability that a randomly chosen ...
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        1answer
        273 views

        What was the first elementary proof that $\pi(x)=o(x)$?

        Denote by $\pi(x)$ the number of primes $\leq x$. I'm interested in knowing who came up with the first elementary proof that $\pi(x)=o(x)$. I know that Chebyshev demonstrated elementarily before ...
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        2answers
        141 views

        on generating prime numbers [closed]

        I apparently need to rephrase the question(s), so here goes: I'm an amateur mathematician, and I have been working for quite some time on finding a more efficient way of factoring large semiprime ...
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        0answers
        142 views

        Estimating integral of product of terms $\cos(t\log p)$

        I would like to prove the following proposition from A. Harper's paper "Sharp conditional upper bound for moments of the Riemann Zeta Function" Proposition. Let $T$ be large and let $n=p_1^{\...
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        1answer
        249 views

        Reversing the CRT: Is $5$ tough?

        Given odd primes $p\ne q$, by the CRT we can find an integer $x$ such that $x\equiv 2^{p-1}\pmod q$ and $x\equiv 2^{q-1}\pmod p$. Can this procedure be reversed? For which integers $x$ there exist ...
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        1answer
        150 views

        Order of magnitude of $\sum \frac{1}{\log^2{p}}$, or $\sum \frac{1}{\log^a{p}}$ for arbitrary power $a$ [closed]

        In this MO question, it says that we have $$ \sum_{p<n} \frac{1}{\log{p}} =\frac{n}{\log^2 n}+O\left(\frac{n\log\log n}{\log^3 n}\right).$$ where the sum is on all primes $p$, up to some max ...
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        4answers
        362 views

        Covering the primes with pairs of consecutive integers

        Is it true that for every sufficiently large positive integer $n$, one can always find at most $k=\lfloor\pi(n)/2\rfloor$ integers, $a_1,a_2,a_3,a_3,\dots a_k$, between $1$ and $n$, such that each of ...
        21
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        1answer
        2k views

        Does the average primeness of natural numbers tend to zero?

        This question was posted in MSE. It got many upvotes but no answer hence posting it in MO. A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value ...
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        0answers
        96 views

        Decoding non-trivial zeros of the Riemann zeta function from prime numbers

        It is well-known how to calculate the prime-counting function $\pi$ from with the help of the non-trivial zeros of the Riemann zeta function (see here). Assume now basically the opposite. If we are ...
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        0answers
        106 views

        Questions on Riemann's explicit formula

        If we consider this version of the prime-counting function $$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$ (with $\pi$ being the normal prime-counting function), then we can write $\...

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