Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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Existence of Hecke operators with distinct eigenvalues?

Consider the space of modular forms $M_k(N)$. Any modular form $f \in M_k(N)$ is determined by a finite number of Fourier coefficients (e.g., Sturm's bound), thus there is a finite set of Hecke ...
3
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96 views

Goldbach's problem in algebraic number fields [duplicate]

Are there any results on the representation of numbers in algebraic number fields as the sum of primes in the ring of integers in that field? There are some results for Waring's problem in other ...
4
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0answers
115 views

Easiest way to see that $\zeta_{\mathbb{Z}[i]}(s) = \zeta(s) L(s, \chi)$?

As the question suggests, what is the easiest way to see that$$\zeta_{\mathbb{Z}[i]}(s) = \zeta(s)L(s, \chi)?$$Here, $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...
2
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1answer
67 views

Upper bound for number of prime numbers in a range

Theorem 3.2 in http://arxiv.org/pdf/1405.2593.pdf shows that for any $x$ there are $\gg x\exp(-\sqrt{\log x})$ integers $x_0 \in [x; 2x]$ such that $\pi(x_0 + \log x) - \pi(x_o) \gg \log\log x$. Is ...
4
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1answer
53 views

Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$, does $L_c(s, \chi)$ necessarily equal $1$?

Consider an analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, let $c ...
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0answers
75 views

Twisting by a multiplicative Character in Katz, Perversity and Exponential sums

Let $C(x_1,\ldots,x_n)$ be a nonsigular cubic form with integral coefficients. In his Proof that $C$ fulfills the Hasse-Principle, if $n\geq 9$, Hooley used the following estimate that was provided ...
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73 views

$F[[T]] \times F[[1/T]]$ fundamental domain, show compactness

Let $p$ be a prime number. What is the easiest way to see that $(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$ is compact? Here $\mathbb{F}_p[T, 1/T]$ is embedded in ...
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161 views

Question about Fermat's Last Theorem [on hold]

Is there a way to prove that having $x \gt 0, z \gt 0, n \gt 2$ with $x, z, n \in \mathbb{Z}$, $$ \sum_{k = 0}^{n - 1}{\binom{n}{k} x ^k} = z ^ n $$ have no solution without using Fermat's Last ...
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63 views

Volume of arithmetic quotients of symmetric spaces

Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes_{\mathbb{Q}} F)$. Let $K_{\infty}$ be a maximal ...
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2answers
778 views

History of Geometric Analogies in Number Theory

My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens? For example, was it before Grothendieck introduced schemes to ...
2
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2answers
182 views

binomial/factorial identity mod p

In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result. Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...
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2answers
475 views

Number of prime numbers in a range

Denote by $A_n$ the number of prime numbers between $n$ and $n + \log n$. Is it true that $A_n < const$? UPD: Is it true that $A_n > \log \log n$ (or something another) for infinite number ...
3
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1answer
105 views

gamma-factor of a primitive element of the Selberg class

Suppose $F$ is a primitive element of the Selberg class and $\displaystyle{\prod_{j=1}^{r}\Gamma(\lambda_{j}s+\mu_{j})}$ with $r>1$ the product of Gamma functions appearing in the gamma factor ...
3
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1answer
152 views

When are the powers of 2 sum-free mod n?

I've encountered the following question in my research: Let $A$ be a subset of $\mathbb{Z}/n\mathbb{Z}$. Let me call $A$ "sum-free" if there is no solution to $x+y=z$ for $x,y,z \in A$ with distinct ...
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0answers
74 views

Bound on $g(n+1)/g(n)$ for Landau's function

I have read that for the Landau function $g(n)$ (http://mathworld.wolfram.com/LandausFunction.html), one knows that $\lim_{n \rightarrow \infty} g(n+1)/g(n) = 1$ (stated, but not proved in "On ...
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2answers
161 views

Simultaneous lcms

Suppose that we have some finite number of $k$-tuples then we define the lcm of two of these tuples to be the tuple of lcms of the co-ordinates. E.g. $[(9, 10), (5, 18)] = ([9, 5], [10, 18]) = (45, ...
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1answer
131 views

A conjecture on the prime counting function

I was thinking about the Second Hardy-Littlewood conjecture for quite sometime (some of my posts are related to this). In one of my earlier post I conjectured that the inequality ($\pi(x)$ denotes the ...
2
votes
1answer
121 views

isogeny clases of CM abelian varieties

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, ...
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1answer
78 views

Differences of consecutive ordered fractional parts

Let $r$ and $h$ be a real numbers and $n>0$. Write the fractional parts $\{k*r+h\}$, for $k = 1,2, . . . n$, in increasing order as $$ a_1 < a_2 < \cdots < a_n.$$ Let $D_n$ be the set of ...
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0answers
41 views

Automorphism group of the gamma factor of a certain type of L-function [on hold]

Let $F$ be an element of the Selberg class with polynomial Euler product, $\gamma_F$ its gamma factor appearing in the functional equation of $F$, which is defined up to a multiplicative factor. Is ...
5
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2answers
155 views

PSL(2,p) as quotient of triangle groups

As a by-product of some Magma computations, I've observed that, for each prime $p$ such that ${\rm PSL}(2,\mathbb{F}_p)$ can be a quotient of the $(3,3,4)$-triangle group (i.e. $p \equiv \pm 1 ...
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267 views

Need explicit formula for certain “$q$-numbers” involving gcd's

The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the ...
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2answers
259 views

When is a sequence the sum of two Beatty sequences?

In other words, given a sequence $(s_n)$, how can we tell if there exist irrationals $u>1$ and $v>1$ such that $$s_n = \lfloor un\rfloor + \lfloor vn\rfloor$$ for every positive integer $n$? ...
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391 views

One-to-one correspondance between zeta zeros and the prime powers? [on hold]

This question is highly speculative, but I would really appreciate some insight into the problem. Previously asked on MSE without answer here. I have noticed an interesting property related to the ...
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0answers
320 views

Fermat's Last Theorem in $\mathbb{Z}/n\mathbb{Z}$

Let $\mathbb{N}$ denote the set of positive integers. We define a relation $R\subseteq \mathbb{N}^3$ by $$ R = \{(x,y,z) \in \mathbb{N}^3: \exists n\in \mathbb{N}: 1< n \leq \max\{x,y,z\} \land ...
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2answers
148 views

Transitivity of discriminant for flat algebras

Sorry if the question doesn't feed this site, I'm reposting it from MSE. Nobody answered it there and I couldn't find the proof in general case(whenever it was mentioned the proof was referred to as a ...
3
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0answers
59 views

Searching information on a certain function with a fixed point property connecting Moebius $\mu$ and Fibonacci numbers

Let $\mu$ be the Moebius function and define for $1\leq n\in\mathbb{N}$ $$ f(n) = \left\{ \begin{array}{ll} \mu\left(\frac{n}{2}\right) + \mu\left(\frac{n}{4}\right), & n\equiv 0, 4, 8\mod 12, \\ ...
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167 views

Can estimate upper bound of $|p_{i}|$ or $|q_{i}|?$

when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2$$ I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that ...
3
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1answer
144 views

higher dimensional analogue of EGZ theorem

The EGZ theorem states that any multiset of $2n-1$ integers has a subset of size $n$ the sum of whose elements is a multiple of $n$. Kemnitz-Reiher theorem is a 2-dimensional analogue of EGZ. Here is ...
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1answer
269 views

On a claim of Zagier on extending a map to cocycle

Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$_2(\bf{ Z})$' (link, p. 6), studies the action of $\Gamma=PSL_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ ...
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square classes of quadratic extensions of 2-adic fialds [migrated]

I have a question about square classes of quadratic extensions of 2-adic fields. I appreciate anybody help me to understand. Why all elements of $1+\mathfrak{p}^5$ are square in ...
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1answer
175 views

Does unique factorization for automorphic L-functions imply a weakened form of Ramanujan conjecture?

Selberg orthonormality conjecture for automorphic L-functions was proven under Ramanujan conjecture, and SOC itself implies unique factorization for those L-functions. My question is: does the ...
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96 views

field of constants of a curve [closed]

I'm trying to gain some intuition about the field of constants of a curve. If $C$ is over a field $k$, then it is defined as the set of elements of $k(C)$ algebraic over $k$. If I understood ...
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1answer
271 views

Computing an eigencuspform in $S_2(\Gamma_0(1776))$

Consider $$\bar{\rho}:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(\mathbb F_7)$$ the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then ...
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2answers
1k views

Arctangents of odd powers of the golden ratio

While trying to answer this MSE question, I found that arctangents of many odd powers of the golden ratio $\varphi=\frac{1+\sqrt5}2$ are expressible as rational linear combinations of arctangents of ...
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0answers
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Let p be prime, is there a divisor d of p-1 or p+1 with gcd(d!,p+d)=1 such that p+d is prime ? [closed]

My previous question was incomplete. Please accept my apologies.
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2answers
266 views

Logarithmic integral, $π(x)$ and $x/(\ln x)$

The function $\text{Li}$ (logarithmic integral) is defined for $x>0$ by $$ \text{Li}(x)=\int_2^{x}\frac{dt}{\ln t}. $$ The prime number theorem, proven by Hadamard and de la Vallée-Poussin in 1896 ...
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4answers
692 views

Number of $\mathbb F_p$ points constant mod $p$?

I have some affine varieties $X$ defined over $\mathbb Z$, and associated integers $c(X)$, with the property that $\# X_{\mathbb Z/p} \equiv c(X) \bmod p$ for all $p$. (In particular $c(X)$ is usually ...
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0answers
99 views

Number of primes one larger than divisors of a fixed number, which is LCM of 1,2,3,…,L

I am hoping someone can estimate the number of primes that come up this way: take a number $L,$ then let $$ C = \operatorname{lcm} (1,2,3,\ldots,L). $$ We know that $C$ has quite a lot of divisors; ...
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2answers
855 views

How to prove that this equation has only one solution?

I can't find a way to prove that the following equation has only one solution : $$ X = \frac{2^Q - 1}{2^{P+Q} - 3^P} $$ with $X,P,Q$ integers $> 0$. One trivial solution is $X = 1, P = 1, Q = ...
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Let p be a prime number. Then there (should) exist a number d, divisor of p-1 or p+1 such that p+d is prime. [closed]

This is a conjecture I am trying to prove. I cannot find any useful hints ; anyone ?
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1answer
91 views

Line bundle of Half integral weight modular forms

Let $\omega_{X}$ denote the line bundle of cusp forms of weight $\frac{1}{2}$ over $X$, where $X=\Gamma\backslash \mathbb{H}$ and $\Gamma$ is any arbitrary fuchsian subgroup. Is it true that the line ...
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1answer
125 views

Chinese remainder theorem for cyclic subfactor planar algebras

This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh. The chinese remainder theorem can be stated as follows: Let $n_1, \dots, n_r \ge 2$ be positive integers ...
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66 views

Approximation to a certain Weyl-sum

Let $ S(\alpha)=\sum_{x\leq X}\sum_{y\leq Y}e \left(\alpha x y^3 \right)$, for some $X,Y \geq 1$ and write $\alpha = a/q + \beta$ for $(a,q)=1$, as usual. For the 'classical' cubic Weyl-sum ...
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182 views

Hodge–Tate structures of modular forms

The title refers to the paper of Faltings: Hodge-Tate structures and modular forms. Math. Ann. 278 (1987), no. 1-4, 133–149. The main theorem in the paper says that the associated Galois rep to a ...
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1answer
491 views
+50

Subsequence and integers as a sum of $\frac{1}{n}$

For all $M \in \mathbb{Z}$, is there a finite sequence of positive integers (not necessarily distinct) $(n_i)_{i \in I}$, s.t. $\sum_{i \in I} \frac{1}{n_i} = M$, and there is no subsequence $(n_i)_{i ...
3
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2answers
223 views

Representations of complex semi-simple algebraic group “defined over $\mathbf{Z}$”?

If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over ...
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0answers
119 views

Conway's box function iterated to produce a hierarchy of nested sets of real numbers

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...
3
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1answer
141 views

Products of relative prime numbers with least sum

Let $P(n)$ be the set of subsets $P$ of $\mathbb{N}$ with the properties All elements of $P$ are relative prime to each other. The product of all $k \in P$ is greater or equal to $n$. Now let ...
5
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1answer
237 views

Order of vanishing of an integer polynomial at a point

Let $f(x,y)$ be a polynomial with integer coefficients, and let $\alpha=(\alpha_1,\alpha_2)\in \mathbb{C}^2$ be a complex point. I want to show that $f$ cannot vanish at $\alpha$ to high order unless ...