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

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        0answers
        68 views

        Does following number theoretic assumption seem legitimate?

        Given $a_1,\dots,a_m\in\mathbb Z$ with $m>1$ and $n$ an integer. Define $$Q(\{a_1,\dots,a_m\},n,t)=r_1^2+\dots+r_m^2\mbox{ where }\forall i\in\{1,\dots,m\},\quad t\cdot a_i\equiv r_i\bmod n$$ $$Q(\{...
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        votes
        1answer
        123 views

        Inverse Problem for Iwasawa Modules

        Let $\Lambda$ denote the Iwasawa algebra and $M$ a finitely generated torsion $\Lambda$ module. Does there exist a number field $K$ and a $\mathbb{Z}_p$-extension $K_{\infty}/K$ such that the $p$-...
        1
        vote
        1answer
        151 views

        Definition of Haar integral in Bushnell and Henniart

        In Bushnell and Henniart's The Local Langland's Conjecture for GL(2) they define a right Haar integral on a locally profinite group $G$ as being a non-zero linear functional $$ I: C^{\infty}_{c}(G) \...
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        votes
        0answers
        123 views

        According to Legendre's conjecture, I draw the following conjecture: [on hold]

        Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between $n^{2}$ and $(n+1)^{2}$ for every positive integer $n$ .\ Let:$x\in \mathrm{positive~real~number}...
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        votes
        1answer
        232 views

        Is $j(\tau)^{1/3}$ the hauptmodul for the congruence subgroup generated by $\tau\rightarrow\tau+3, \tau\rightarrow-1/\tau$?

        The 3rd root of the modular invariant $j$ is $$ j(\tau)^{1/3}=q^{-1/3}(1+ 248q+ 4124q^2+ 34752q^3+\cdots),$$ where $q=e^{2\pi i \tau}$. I was wondering if $j(\tau)^{1/3}$ the hauptmodul for the ...
        1
        vote
        1answer
        173 views

        A generalization of Landau's function

        For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$ the least common multiple of all ...
        3
        votes
        0answers
        74 views

        Counting lattice points in adelic spaces

        Let $\mathbb{A}$ denote the ring of adeles of $\mathbb{Q}$, let $\mu$ be the Haar measure of $\mathbb{A}$, and let $\|\cdot\|_{\infty}$ denote the sup-norm of the components in the Archimedean of $\...
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        votes
        0answers
        34 views

        Number Theory (modular equivalence and multiplicative inverse) [on hold]

        i. Apply the definition of modular equivalence and write down what $ p^{-1}p ≡ 1 (mod \, q)$ means ii. Rearrange what you get and apply Bezout’s identity to conclude that if $gcd(p, q) = 1$ then $p?1 ...
        7
        votes
        0answers
        93 views

        Integers with fixed number of prime factors in arithmetic progression

        Famously, there are arbitrarily long arithmetic progressions $x$, $x+y$, ..., $x+ky$ consisting of primes, by Green-Tao. I was wondering whether the following generalization is also known (by the same ...
        2
        votes
        0answers
        55 views

        Congruence of normalized eigenforms at two primes

        Let $f_i\in S_{k_i}(\Gamma_0(N_i))$ be normalized cuspidal eigenforms for $i=1,2$ and let $K$ be the composite of the fields of Fourier coefficients generated by $f_1$ and $f_2$ and let $\mathfrak{p}...
        6
        votes
        1answer
        191 views

        On $\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i+j}(2i+j)$ and $\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i-j}(2i-j)$ modulo a prime $p>3$

        QUESTION: Is my following conjecture true? Conjecture. Let $p>3$ be a prime and let $h(-p)$ be the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$. Then $$\frac{p-1}2!!\prod^{...
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        vote
        0answers
        217 views

        On a certain representation of the Riemann zeta function in Montgomery-Vaughan

        I (TK. Isaac) recently saw some elegant representation of the Riemann zeta function in Montgomery-Vaughan's ''Multiplicative Number Theory'' (p.338), which however, seems untrue unless i'm missing ...
        2
        votes
        1answer
        168 views

        idempotents of a character

        Let $K$ be a number field and $\Delta = Gal (K/\mathbb{Q})$ and $\chi: \Delta \rightarrow \mathbb{Z}_p^*$ be a non-trivial Dirichlet character, $e_{\chi} = (1/\mid \Delta \mid) \sum_{\sigma \in \Delta}...
        5
        votes
        0answers
        150 views

        Legendre's three-square theorem and squared norm of integer matrices

        Let $\mathbb{N}$ be the set of non-negative integers. Let $E_n$ be the set of integers which are the sum of $n$ squares. Let $F_n$ be the set of integers of the form $\Vert A \Vert^2$ with $A \in M_n(...
        1
        vote
        0answers
        55 views

        Is the number of newforms in a fixed maximal ideal bounded?

        Let $N\in \mathbb{Z}_{\geq 1}$. Enumerate the set of primes $p_i$ which do not divide $Np$ in ascending order. Let $\mathbb{F}$ be a finite field of characteristic $p$, $\bar{\chi}$ denote the mod $p$...

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