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

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0
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13 views

Elementary analytic number theory question

Let $a_x$ refer to remainder of $a$ divided by $x$. Fix an $\epsilon\in(0,\frac{1}{100})$, $r\in(0,\frac{1}2)$. Is there an $N_\epsilon\in\Bbb N$ such that $\forall N>N_\epsilon$, we could find ...
3
votes
1answer
88 views

Relations between modular functions of certain $q$-continued fractions

Given the j-function $j:=j(\tau)$, and $q=e^{2\pi i\tau} = \exp(2\pi i\tau)$ where, for convenience, we set $\tau=\sqrt{-n}$. I. $\frac{A_2(q)}{A_1(q)} = \text{q-cfrac}:\;$ Icosahedral group ...
2
votes
2answers
289 views

$t$-analogue of the symmetric power of an additive character over $\Bbb{F}_q^*$

Let $G$ be a finite group and let $f: G \longrightarrow \Bbb{C}$ be any complex-valued function. For integers $k, n \geq 0$, an indeterminant $t$, and $x \in G$ let $f_k(x) := f \big( x^k \big)$ and ...
2
votes
0answers
78 views

Ramanujan conjecture and covariance of Kloosterman sums

There has been interest in moments and covariances/correlations of Kloosterman sums $S(m,n,c)=\sum_{ad=1 \mod c} e(\frac{ma+nd}{c})$ like $\sum_{m\in\mathbb F_c} S(m,n,c)^k$, ...
11
votes
1answer
314 views

Why is there a $\sqrt{5}$ in Hurwitz's Theorem?

Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states: For every irrational number $\alpha$, there are infinitely ...
3
votes
1answer
149 views

Least prime for which a square-free integer is a non-residue

Suppose $a$ is a square-free integer and $\left(\frac{a}{p}\right)=1$ for the primes $p\leq k$. I'll call $a$ a quasi-square of order $k$. What I am interested in is the maximum value of $k$ in terms ...
4
votes
0answers
75 views

Explicit bounds for exceptional zeros and/or $L(1,\chi)$ for real $\chi$

I would like to have an explicit upper bound (that is, one with explicit constants) for a possible real zero $\beta$ for $L(1,\chi)$ for real Dirichlet characters $\chi$. I need such a bound for real ...
8
votes
2answers
542 views

Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?

Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)} $$ Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?
0
votes
0answers
11 views

lattice basis reduction of the orbit of a rational vector on the torus

LEt $v=(p_1/q,...,p_n/q)$ be a vector of the torus $\mathbb{T}^n$, such that for any $i$, $p_i$ and $q$ are relatively prime. Let $L= \{ kv \mod \mathbb{T}^n , k=0,...,q-1 \}$. What is the lattice ...
23
votes
1answer
2k views

Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mathematica knows that: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$ The von Mangoldt function should then be: ...
2
votes
0answers
778 views
+50

Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta

The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation. In the discrete calculus there is ...
2
votes
0answers
60 views

Representations of $\mathbb{H}^{\times}$ and $\mathbb{H}^{\times}/\mathbb{R}^{\times}$

In an attempt to recapture Eichler's theta correspondence I have hit a stumbling block. Let $D$ be a quaternion algebra over $\mathbb{Q}$, ramified at $p,\infty$. Also let $V_j = ...
-4
votes
0answers
69 views

Is Zsigmondy's Theorem utilized in Sociology to the extent that a lay person might be familiar with its use? [on hold]

Is Zsigmondy's theorem somehow useful in sociological studies? Has it in some way been co-opted as a sociological term with a corresponding sociological theory, no matter how far off base from the ...
2
votes
1answer
107 views

Proving inequation with ceilings in Finite Field of characteristic $p$

Take $ui = pt_i +j_i$ where $p$ is a prime number and $u(p-r) \equiv 1 $ $(\mbox{mod p})$ for positive integers $1 \le i, r, j_i\le p-1$ and $t_i \ge 0$. How can I prove that: \begin{equation} ...
2
votes
0answers
94 views

Are these two $q$-continued fractions equivalent?

In this MSE post, Nicco Mnisi defined a particular $q$-continued fraction of order $12$. More generally, define the cfrac found in Ramanujan's Notebooks, Vol III, Chap. 16, page 24, $$U(q) = ...
10
votes
1answer
646 views

The maximum of the preimage of [1,x] through Euler's totient function

A friend of mine and I have shown the following: "For each $x \geq 1$ let $m := m(x)$ be the greatest positive integer such that $\varphi(m) \leq x$, where $\varphi$ is the Euler's totient function. ...
18
votes
3answers
530 views

Variety acquiring rational point over any quadratic extension

Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$? If ...
7
votes
2answers
322 views
+50

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 $\pi(x_0 + \log x) - \pi(x_0) \gg \log\log x$. Is ...
2
votes
0answers
170 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
6
votes
1answer
272 views

Uniformly small sums of roots of unity

I have considerable numerical evidence that for all $0\leq k\leq{{n-1}\over 2}$ ($n$ odd) there exists a subset $ S_k$ of {1,2,...,n} of cardinality $k$ such that the modulus square of ...
0
votes
0answers
64 views

Bound of Chebyshev function and zeros of zeta function

It is an elementary argument (such as in Multiplicative Number Theory, section 18) that, if the Chebyshev's function $f(x) = \sum_{n \le x} \Lambda(x) = x + O(x^\alpha)$ for some $\alpha < 1$, then ...
-4
votes
0answers
118 views

Must a proof of the asymptotic Goldbach conjecture be effective to imply GRH?

It was shown by Hardy and Littlewood that GRH (i.e. the Generalized Riemann Hypothesis for Dirichlet L-functions) implies that every large enough odd number is the sum of three primes. Later on (circa ...
2
votes
1answer
192 views
+50

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 ...
11
votes
1answer
576 views

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 ...
1
vote
6answers
329 views

Isotropic ternary forms

It is well known that some questions about isotropic ternary forms reduces to the study of the special case $f_0(X)=xz-y^2, X=(x,y,z)$, see page 301 of Cassel's "Rational quadratic forms" (Dover, ...
39
votes
2answers
2k 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 ...
5
votes
0answers
113 views

Radical of a polynomial values

It has been observed by Langevin and Elkies that the following holds: Assume that the $ABC$- conjecture is true. Suppose that $f(x)\in\mathbb{Z}[x]$ has no repeated roots. Fix $\varepsilon >0.$ ...
-1
votes
0answers
21 views

Freedom of speech in scientific discussions - An invitation to more tolerance in Scientific debates [migrated]

I hope this post enjoys some tolerance, and don't get closed or put on hold immediately. I believe that freedom of speech in scientific discussions is one of the key values which enriches the debates ...
3
votes
0answers
336 views
+50

Best known bounds on certain exponential sums

What are the best bounds currently known for the following exponential sum: $$\sum_{x < p \le 2x} e(\alpha p^k)$$ for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...
2
votes
1answer
159 views

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
votes
0answers
119 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 ...
5
votes
3answers
503 views

Any rigorous way to claim that sums with repeat summands are few?

Let $B \subset \mathbb{Z}^+$. Define $r_{B,h}(n)$ to be the number of ways of writing $n$ as the sum of $h$ elements of $B$ and $R_{B,h}(n)$ the number of ways to write $n$ as the sum of $h$ DISTINCT ...
6
votes
0answers
188 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 $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...
4
votes
1answer
73 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 ...
2
votes
0answers
87 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 ...
6
votes
0answers
88 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 ...
19
votes
2answers
907 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
votes
2answers
196 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 ...
15
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0answers
278 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 ...
-2
votes
0answers
174 views

Question about Fermat's Last Theorem [closed]

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 ...
4
votes
2answers
493 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 ...
1
vote
1answer
79 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 ...
2
votes
2answers
170 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, ...
3
votes
1answer
113 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
votes
1answer
157 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 ...
1
vote
0answers
82 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 ...
0
votes
1answer
137 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 ...
34
votes
1answer
2k views

Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...
2
votes
1answer
127 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 $K$, ...
0
votes
0answers
45 views

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

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 ...