# Tagged Questions

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### Is it possible to sum the divergent series with prime coefficients?

It is known that the series $$P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime}$$ cannot be summed by means of (prime) zeta function regularization. (The result was ...
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### Metric on the set of subsets of the rational primes

I was thinking how to say that two sets of prime numbers are close to each other, and came up with this. NOTATION $\ \Delta(A\ B)\ :=\ (A\setminus B)\cup(B\setminus A)\ \$ is the symmetric ...
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### Repetend digit graphs for $1/n$ in base $b$

Here is a decimal expansion of $\frac{1}{34}$: $$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$ And here is a graphical representation of the 16-digit "repetend," as a directed ...
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### Lower bounds on the error term of the prime number theorem

Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t. $$f(x)\ll |\psi(x) - x|$$ where $\psi$ is the Chebyshev function.
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### Integers with a large prime divisor in short intervals

For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following: There exists some $c>0$, such that for all $x$ sufficiently large the number of integers ...
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### lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$?

Are there known any lower and upper bounds for $$\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k,$$ where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$? Or at least is it known ...
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### The conjecture of Montgomery and Soundararajan on primes in short intervals: Empirical inconsistencies?

Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...
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### Finding a suitable number

Let $n,m$ be two positive integers. By $r_n$ we denote the largest prime not exceeding $n$. If $r_n\leq m\leq n$ and $q$ is the largest prime factor of $n!/m!$ such that $q\geq 17$ and $q\geq n-m+3$, ...
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### Consecutive primes versus prime twins

First a warm-up. Let $\ V\$ be an arbitrary set of odd natural numbers. Let $\ S(V)\$ be the generated multiplicative semi-group. What are the necessary and/or sufficient conditions on $\ V\$ for ...
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### Optical methods for number theory?

I found a paper: 'A New Method of Finding the Distribution of Prime Number', saying We stack discs and annuluses with certain rules then turn on the light to illuminate. The projection of ...
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### Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?

OEIS A226181: 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ... Primes $p$ ...
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### Natural numbers that cannot be expressed as a difference between a square and a prime?

We wish to find the set of natural numbers that cannot be expressed as a difference between a square and a prime. e.g. $1 = 2^2 - 3$ $2 = 3^2 - 7$ $3 = 4^2 - 13$ and so on. The smallest such ...
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### Relation between Maier's theorem and a conjecture of Montgomery and Soundararajan

Let us consider the number of primes in the interval $[N,N+h]$, with $h\leq N$. According to the answer given by Lucia to a previous question on the distribution of primes, it is natural to consider ...
OEIS A167055 Numbers n such that $12n + 5$ is prime. $0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS $A167055$. I conjecture that the set of the sum of every two items of this ...
### n-ary quadratic forms with $S$-integer values
Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form. Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to ...