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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

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Let $\chi$ be a primitive Dirichlet character of modulus $q>1$. Write, as is customary, $B(\chi)$ for the constant in the expression $$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} = B(\chi) + \sum_\rho ... 0answers 23 views Other examples of irreducible similarities over the natural numbers For a natural number a define X_a := \{a/k| 1\le k \le a \}. Then it is not difficult to show that |X_a \cap X_b| = \gcd(a,b). Using this one can define similarities over the natural numbers, ... 0answers 63 views trying to prove that if the sum of two natural numbers is equal to 0, then both addends are 0 [on hold] I'm stuck in a sense, that I'm trying to prove the question by contradiction and the case a != 0 and b != 0 isn't immediately clear to me and I tried using induction on a, keeping b fixed and the base ... 1answer 89 views Maximum size of a critical set that sums to n Say that a set S \subset \mathbb Z^+ can express n if there is some way to add elements of S (possibly more than once) to equal n. Call S critical if moreover no proper subset of S can ... 0answers 60 views Geometry of the section X_0(N) \to X_0(pN) given by the canonical subgroup Goren and Kassaei's paper "The Canonical Subgroup: a Subgroup-Free Approach" takes the position that the canonical subgroup of order p for elliptic curves over \mathbb{Z}_p with \Gamma-level ... 0answers 54 views Interesting problem over reverse function First Let's define a function we will need F(x) = \sum_{k = 0}^{n-1} 10^{n-1-k} * d_{k} where d_i are the digits of x in such order d_{n-1},d_{n-2},...,d_{1},d_{0} where d_{n-1} is most ... 0answers 129 views Are there at least one of a, b, c, a+b, b+c, c+a have h(P) \le 3? Given a positive integer P>1, let its prime factorization be written$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$Define the functions h(P) by h(1)=1 and h(P)=\min(a_1, a_2,\ldots,a_k)... 1answer 113 views A question about a sum that involves gaps between twin primes, on assumption of the First Hardy–Littlewood conjecture I wondered, inspired in a result mentioned from  (page 45), what should be the asymptotic behaviour of the sequence on assumption of the First Hardy–Littlewood conjecture$$\sum_{\substack{\text{...
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://...
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 ...
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 ...
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 ...
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 ...
102 views

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 $\alpha_1,\dots,\... 0answers 47 views Reference request for a formula of the abscissa of convergence for Dirichlet integerals [on hold] Following On the integral$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 ...

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