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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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        2
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        63 views

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

        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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        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, ...
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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 ...
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        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 ...
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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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        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 $h(P)=\min(a_1, a_2,\ldots,a_k)$...
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        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{...
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        1answer
        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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        1answer
        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
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        1answer
        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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        1answer
        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 ...

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