0
votes
0answers
55 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 ...
0
votes
0answers
58 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
1answer
192 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 ...
6
votes
3answers
801 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
1answer
286 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
0answers
98 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
2answers
236 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
3answers
685 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
0answers
229 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
0answers
379 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
0answers
85 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
1answer
274 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
2answers
374 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
1answer
380 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 that the remaining numbers ...
1
vote
1answer
251 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 ...
3
votes
1answer
447 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
1answer
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
1answer
241 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
1answer
348 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
0answers
272 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
2answers
345 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
1answer
543 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
1answer
910 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
2answers
438 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
1answer
419 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
1answer
424 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
2answers
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
1answer
550 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
0answers
262 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
3answers
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
1answer
425 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
3answers
690 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
3answers
436 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
1answer
885 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. ...
25
votes
1answer
5k 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] ...
3
votes
3answers
617 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
0answers
120 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
1answer
749 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
2answers
179 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
1answer
291 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 ...
22
votes
6answers
3k 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 ...
16
votes
3answers
457 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 ...