# Tagged Questions

**6**

votes

**0**answers

173 views

### Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$

If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log ...

**10**

votes

**2**answers

460 views

### Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line ...

**0**

votes

**1**answer

124 views

### Estimating the height required to find a given small value of $|\zeta(s)|$ near the line $\sigma=1$

There are some qualitative theorems of Bohr, Jessen and Titchmarsh (e.g. The Theory of the Riemann zeta function, E.C. Titchmarsh, pages 306-308) proving that there is a $K=K(a,\alpha,\beta)$ such ...

**6**

votes

**0**answers

173 views

### Non-trivial bounds for polynomials at a fixed point

Let $f$ be a polynomial of degree $d$. Of course $|f(z)|\sim C|z|^d$ as $|z|\rightarrow\infty$ but also, since any polynomial is completely determined by its values at any $d+1$ points, we may ask how ...

**1**

vote

**0**answers

72 views

### Growth of sums of multiplicative functions over Squarefrees

When one looks at the quotient of Euler products $$\prod_p\frac{\sum_{\alpha=0}^{\infty}f(p^{\alpha})p^{-\alpha s}}{1+f(p)p^{-s}}$$
with $|f|\leq 1$, it is observed that the resulting expression ...

**1**

vote

**0**answers

74 views

### analytic continuation related to Chebyshev functions

Let $\psi$ be the Chebyshev function. I would like to prove that the function
$\sum_{n\ge0}(\sum_{k=0}^n\binom{n}{k}e^{\psi(k)})w^n$ can be analyticaly continued on an open set of $\mathbb C$ ...

**5**

votes

**1**answer

206 views

### Fourier expansion of Takagi-function (everywhere non differentiable function).

Let us consider Takagi-function defined by
$T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$,
where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$.
$T(x)$ has its period $1$, so ...

**7**

votes

**3**answers

845 views

### Objections to and arguments for the simplicity of all Riemann zeros

It seems to be that the simplicity of all the zeros is quite widely accepted as a working hypotheses, and it is known that a positive proportion are as such.
Titchmarsh explains in the last chapter ...

**3**

votes

**1**answer

298 views

### Automorphisms of complete discrete valuation ring

Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb ...

**0**

votes

**0**answers

104 views

### On modes of convergence of the Dirichlet series for reciprocal powers of zeta

Let $\mu_k(n)$ denote the nth coefficient in the Dirichlet series for $\zeta^{-k}(s)$. It can be shown that if there exists a $u=u(x,s)$ such that
$$|\sum_{n\leq x}\frac{\mu_k(n)}{n^s} ...

**3**

votes

**2**answers

243 views

### j-invariant duplication, triplication and quintuplication formulae… how?

I am interested in finding the derivation of the duplication, triplication and quintuplication formulae for Klein’s j-invariant, which are equations (13) – (24) of the corresponding page (Klein’s ...

**3**

votes

**3**answers

804 views

### Can the sum of two roots of unity be a root of unity?

Let $p$ be prime, and $z_0, z_1, ..., z_{p-1}$ be all the $p$-th roots of unity, i.e. solutions of the equation $z^p = 1.$
Is it true or false that a combination of two (or more, in general) of the ...

**3**

votes

**0**answers

249 views

### A Dedekind Eta trajectory / horocyclic flow (Reference request)

I've been exploring the composition of essentially the Dedekind $\eta$-function with
parabolic Möbius transformations,
...

**7**

votes

**0**answers

384 views

### The natural generalization of Euler's derivation of the Basel sum

Euler proved that $$\sum_{n=0}^\infty \frac{1}{n^2} = \frac{{\pi}^2}{6}$$ by comparing the $z^3$ term in the power series expression of $\sin(z)$ given by
$$\sin(z) = z - \frac{z^3}{3!} + ...

**1**

vote

**0**answers

86 views

### Isometric automorphisms of $\tilde{S}$

Hello,
this question is a follow-up from Structure groups and a special class of L-functions
Let $\tilde{\phi}:\tilde{\mathcal{S}}\to\tilde{\mathcal{S}}$ be an automorphism of $\tilde{\mathcal{S}}$, ...

**1**

vote

**1**answer

290 views

### A more direct proof of Dedekind Reciprocity

Question: Let $s(p,q)=\sum_{i=1}^{q-1}((i/q))((pi/q))$ where (p,q)=1 and $((x))=x-[x]-1/2$ for $x\notin Z$. I want to prove that $s(p,q)+s(q,p)=(p/q+\frac{1}{pq}+q/p)/12-1/4$ using at least one of ...

**1**

vote

**2**answers

378 views

### Analytical predicate for integers over complex numbers

A complex number $z$ is an integer if and only if $\sin(\pi z)=0$.
It follows that a complex number $z$ is an integer if and only $\sin^2(\pi z) = 0$. So for a real analytic function $f$ and any real ...

**5**

votes

**1**answer

499 views

### Extending an assignment property from Q to R (or C)

Property of any odd number of nonnegative integers:
Given $x_1 \leq \ldots \leq x_{2n + 1}$ with each $x_i \in \mathbb{Z}_{\geq 0}$, suppose that for any $x_i$ we remove, the remaining numbers can be ...

**1**

vote

**1**answer

255 views

### Automorphic and modular forms for subgroups of modular group and fuchsian groups

Is there a well-understood correspondence between subgroups G of $SL_2(\mathbb{Z})$ (not necessarily of finite index) and graded algebras of modular forms invariant under G?
Given an algebra of ...

**4**

votes

**2**answers

649 views

### Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard)

What are modular forms or cusps forms, resp. ?
We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The ...

**2**

votes

**1**answer

215 views

### $\sum \frac{n^2-1}{(n^2+1)^2}=\frac{1}{2}(1-\frac{\pi^2}{\sinh(\pi)^2})$

The following series evaluation
$\sum \frac{n^2-1}{(n^2+1)^2}=\frac{1}{2}(1-\frac{\pi^2}{\sinh(\pi)^2})$
seems attractive to me, and has a proof related to the evaluation of $\zeta(2)$.
Does this ...

**5**

votes

**1**answer

245 views

### information on an Euler product

The following Euler product came up in some sieving applications:
$f(z, s) = \prod_{\mbox{primes}} \left(1-\frac{z}{p^s}\right).$
What is known about this function? (Analytic continuation? ...

**4**

votes

**1**answer

356 views

### Viète's generalized infinite product yielding other converging values?

I took Viète's infinite product for $\frac{2}{\pi}$:
$\displaystyle \dfrac{2}{\pi} = \dfrac{\sqrt2}{2} . \dfrac{\sqrt{2+\sqrt2}}{2} . \dfrac{\sqrt{2+\sqrt{2+ \sqrt2}}}{2} \dots$
and made it generic:
...

**4**

votes

**0**answers

280 views

### Weight-2 modular forms under $\Gamma(N)$

I'm looking for explicit bases of weight 2 modular forms under $\Gamma(N)$, for small N (<16 would be enough). (Ideally in terms of Theta- or Eta-Functions)
It seems to me that this should ...

**2**

votes

**2**answers

376 views

### Converse to a theorem of Landau on Dirichlet series

Landau's Theorem for Dirichlet series with real coefficients ($c_n$) states that if the coefficients are of fixed sign for all sufficiently large $n$, then the point $\sigma_0$ on the abscissa of ...

**1**

vote

**1**answer

580 views

### A question about Ahlfors's proof of modular function being a covering space of the twice punctured plane

I have a question about Ahlfors's proof of modular function being a covering space of the twice punctured plane .See Ahlfors' complex analysis, second edition, page 272. You can either explain or ...

**16**

votes

**1**answer

957 views

### Factoring Integers using Complex Integrals

Suppose $n$ is an integer and we wish to factor it. As a special case we have $n = pq$ with $p,q$ distinct primes. The problem: factoring $n$ via complex analysis tools
Background
I have been ...

**5**

votes

**2**answers

446 views

### What monsters does the “growth condition” required of holomorphic modular functions bar?

Even though the title of this question pretty much captures what I'd like to know, I'll add
two side questions:
1) Is it difficult to get a handle on the totality of functions that arise if one ...

**4**

votes

**1**answer

430 views

### Polylogarithm inequality

Recall the polylogarithm $Li_s(z)=\sum_{n=1}^\infty \frac{z^n}{n^s}.$
For $|z|<1,$ is $\Re \left( \frac{Li_1(z)}{Li_2(z)} - \frac{2}{3}\frac{Li_2(z)}{Li_3(z)} \right)>0?$
The numerics suggest ...

**5**

votes

**1**answer

461 views

### The identity $\sum_n \ln(n) x^n = \sum_p ln(p)(\sum_k\frac{x^{p^k}}{1-x^{p^k}})$

As in the famous Euler product identity, the primes occur on
only one side of the following:
$\sum_n \ln(n) x^n = \sum_p ln(p)(\sum_k\frac{x^{p^k}}{1-x^{p^k}})\ .$
My basic question: Does this ...

**1**

vote

**2**answers

245 views

### Finding representatives of PSL_2(Z) orbits

Given $\tau$ in the upper half plane, what is a good, systematic way to find a representative in the usual fundamental domain for the $PSL_2(Z)$-orbit of $\tau$? For example, let $\tau=\frac{2}{3} + ...

**10**

votes

**1**answer

560 views

### Non-standard enlargements, $\zeta(s)$ and analytic continuation

Consider an extension of the Riemann zeta function $\zeta(s)$ where $s$ now runs over a non-standard enlargement of the complex plane.
Observe that if $s=\sigma + it$ with $\sigma>1$ real and ...

**4**

votes

**0**answers

275 views

### Studying primes via the gamma function alone: $(x+1)\prod_n \Gamma(\frac{x}{n}+1)^{\mu(n)}$

Various questions on MO concerning the "surprise" occurrence of the gamma function in the functional equation of the Riemann zeta function got me wondering whether the Gamma function alone suffice for ...

**5**

votes

**3**answers

1k views

### Product of sine

For which $n\in \mathbb{N}$, can we find (reps. find explicitly) $n+1$ integers $0 < k_1 < k_2 <\cdots < k_n < q<2^{2n}$
such that
$$\prod_{i=1}^{n} \sin\left(\frac{k_i \pi}{q} ...

**12**

votes

**1**answer

443 views

### How can one “see” the Hopf fibration in the space of lattices in the plane?

This question is inspired from Etienne Ghys's talk on Knots and Dynamics from ICM 2006.
The map $L \mapsto (G_4(L), G_6(L))$ gives a bijection between all lattices $L\subset \mathbb{C}$ (including ...

**4**

votes

**3**answers

735 views

### How can one express the Dedekind eta function as a sum over the lattice?

The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. The algorithm is: rotate the lattice so that one of its basis vectors ...

**1**

vote

**3**answers

451 views

### structure of singular matrices whose entries have modulus one

Let $A$ be a $n \times n$ matrix all of whose entries has modulus 1.
Suppose the matrix $A$ is singular.
We will assume without loss of generality that all the entries in the first row and the ...

**3**

votes

**1**answer

1k views

### Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)

The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. ...

**26**

votes

**1**answer

6k views

### What, exactly, has Louis de Branges proved about the Riemann Hypothesis?

Hi,
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] ...

**4**

votes

**3**answers

653 views

### Coprimality and squarefree numbers

As observed on Mathworld, "Amazingly, the probabilities for random pairs of integers and Gaussian integers being relatively prime are the same as the asymptotic densities of squarefree integers of ...

**3**

votes

**0**answers

124 views

### When is the Locus of Equi-modular points of two monic polynomials with integer coefficients contained in the unit disk?

If $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are two monic polynomials (relatively prime) with integer coefficients and $$\Gamma=\lbrace z \rm{\ s.t.\ } |\lambda_{1}(z)|=|\lambda_{2}(z)|\rbrace,$$ when is ...

**16**

votes

**1**answer

757 views

### Irrational Numbers and the Riemann Surface of a Multi-Valued Function

Suppose a meromorphic function $f(z)$ has two poles, with residues $1$ and $\gamma$, respectively. Then the topology of the Riemann surface of the anti-derivative of $f(z)$ depends on whether or not ...

**1**

vote

**2**answers

180 views

### Simultaneous convergence of powers of unit complex numbers

Let $z_1,\ldots,z_n$ be complex numbers of modulus one. Does it exist an increasing sequence $k_j\in\mathbb{N}$ such that $\lim_{j\to\infty}z_i^{k_j}=1$ for all i?

**4**

votes

**1**answer

297 views

### Field of Definition of a Meromorphic Function

Question
Let X be a smooth, projective curve over the algebraic closure of ℚ. Let f:X->ℙ1 be a meromorphic function. Assume that the zeros and the poles are defined over some number ...

**33**

votes

**8**answers

4k views

### Why does the Gamma-function complete the Riemann Zeta function?

Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or ...

**18**

votes

**3**answers

553 views

### When are some products of gamma functions algebraic numbers?

I want to know when certain expressions of the form
$ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $
are algebraic numbers. These ...