# Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

11,224
questions

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### $B(\chi), L'(1,\chi)/L(1,\chi),\dotsc$

Let $\chi$ be a primitive Dirichlet character of modulus $q>1$. Write, as is customary, $B(\chi)$ for the constant in the expression
$$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} = B(\chi) + \sum_\rho ...

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### Other examples of irreducible similarities over the natural numbers

For a natural number $a$ define $X_a := \{a/k| 1\le k \le a \}$.
Then it is not difficult to show that $|X_a \cap X_b| = \gcd(a,b)$.
Using this one can define similarities over the natural numbers, ...

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63 views

### trying to prove that if the sum of two natural numbers is equal to 0, then both addends are 0 [on hold]

I'm stuck in a sense, that I'm trying to prove the question by contradiction and the case a != 0 and b != 0 isn't immediately clear to me and I tried using induction on a, keeping b fixed and the base ...

**5**

votes

**1**answer

89 views

### Maximum size of a critical set that sums to $n$

Say that a set $S \subset \mathbb Z^+$ can express $n$ if there is some way to add elements of $S$ (possibly more than once) to equal $n$. Call $S$ critical if moreover no proper subset of $S$ can ...

**2**

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60 views

### Geometry of the section $X_0(N) \to X_0(pN)$ given by the canonical subgroup

Goren and Kassaei's paper "The Canonical Subgroup: a Subgroup-Free Approach" takes the position that the canonical subgroup of order $p$ for elliptic curves over $\mathbb{Z}_p$ with $\Gamma$-level ...

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54 views

### Interesting problem over reverse function

First Let's define a function we will need
$F(x) = \sum_{k = 0}^{n-1} 10^{n-1-k} * d_{k}$ where $d_i$ are the digits of $x$ in such order $d_{n-1},d_{n-2},...,d_{1},d_{0}$ where $d_{n-1}$ is most ...

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129 views

### Are there at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$?

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$
Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$...

**1**

vote

**1**answer

113 views

### A question about a sum that involves gaps between twin primes, on assumption of the First Hardy–Littlewood conjecture

I wondered, inspired in a result mentioned from [1] (page 45), what should be the asymptotic behaviour of the sequence on assumption of the First Hardy–Littlewood conjecture
$$\sum_{\substack{\text{...

**8**

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**1**answer

161 views

### Universal property of $\mathbb S[z]$ and $E_\infty$-ring maps

Let $\mathbb S[z]$ be the free $E_\infty$-ring spectrum generated by the commutative monoid $\mathbb N$. That is, $\mathbb S[z] = \Sigma^\infty_+ \mathbb N$.
In Bhatt-Morrrow-Scholze II (https://...

**3**

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**1**answer

172 views

### Primes from arithmetic and geometric progressions

The five primes, 131, 157, 211, 349, 739, are neither in arithmetic or geometric progression, but are instead the sum of the five corresponding terms of an arithmetic and geometric progression.
Are ...

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161 views

### Convergence of $\sum_{n=1}^\infty x_n^k$

I thought that this question is more suitable for MSE, and asked it there. (Link to the MSE question) However, it does not get any answer despite the upvotes. It appears that I might have ...

**4**

votes

**1**answer

138 views

### Difference of two integer sequences: all zeros and ones?

Suppose that $c$ is a nonnegative integer and $A_c = (a_n)$ and $B_c = (b_n)$ are strictly increasing complementary sequences satisfying
$$a_n = b_{2n} + b_{4n} + c,$$
where $b_0 = 1.$ Can someone ...

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78 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 ...

**2**

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**1**answer

102 views

### Find integer $k$ such that $k \alpha_i \bmod{1}$ are simultaneously small for all $i$

A classical result shows that if $\alpha$ is irrational, then $\{k \alpha \bmod{1}\}_{k \in \mathbb{Z}}$ is dense over $[0,1]$.
Can we extend this result as follows?
Suppose $\alpha_1,\dots,\...

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47 views

### Reference request for a formula of the abscissa of convergence for Dirichlet integerals [on hold]

Following On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$, define $$I_s=\int_{1}^{\infty}(\pi(x)-Li(x))x^{-s-1} \mathrm{d}x,$$ where $\pi$ and $Li$ have their usual meanings. Define ...