# Tagged Questions

**0**

votes

**1**answer

141 views

### On complex exponential sum estimation

Let $c>0$ a real number, let $N$ a large natural number and let $e\left(x\right):=e^{2\pi ix}$. Is it true that $\forall k\in\left[1,\,2N\right]$, $k$ natural number, that ...

**2**

votes

**0**answers

208 views

### Convergence of certain L-series

Suppose $|a_{n}| \leq 1$ completely multiplicative function assuming real values.
Suppose further that,
$
L(s)=\sum_{n} \frac{a_{n}}{n^s}
$
may be continued analytically to the left of $s=1$ a bit ...

**7**

votes

**1**answer

283 views

### Abscissa of convergence of Dirichlet series

Let $D(s)$ be a Dirichlet series with abscissa of convergence $\sigma_c=\sigma_a$. Does it follow that the Dirichlet series defined by $P(s)=D(s)\bar D(\bar s)$ has the same abscissa of convergence?
...

**0**

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**0**answers

85 views

### Derivative of a function related to Dedekind zeta function

Lef $K$ be an algebraic number field of degree $[K:\mathbb{Q}]=n$. For simplicity suppose $K$ is totally real. Define $f(s) = \zeta_K(s) \zeta(1-s)^{n-1}$ where $\zeta = \zeta_{\mathbb{Q}}$.
From the ...

**1**

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**1**answer

93 views

### Has a universality theorem been proved for the Davenport-Heilbronn L function?

The question is in the title: has a universality theorem in the sense of Voronin been proved for the Davenport-Heilbronn function, or do we expect such a theorem to hold true only for L functions that ...

**2**

votes

**0**answers

192 views

### computing a certain contour integral [closed]

I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side ...

**0**

votes

**2**answers

251 views

### Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$?

Any closed form for series like $$F(x)=\Sigma_{i=2}^{\infty}x^p,\text{p is prime}$$ or $$F(x)=\Sigma_{i=0}^{\infty}x^{i!}$$?
More generally,we can obtain a power series from decimal expansion of a ...

**1**

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**0**answers

150 views

### Convergent series, asymptotics and truncation

In regard to the characteristics of certain "explicit formulae" arising in number theory, I am pondering the connection between the rate of convergence of series and the asymptotic order of the ...

**18**

votes

**0**answers

825 views

### When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...

**1**

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**0**answers

159 views

### Euler constant transcendality. [closed]

What attempts have been made to prove that $\gamma := \lim_{k\rightarrow \infty} \sum_{k=1}^{n} \frac{1}{k} -\log n$ is transcendental?
Any reference for stuff that has been proved on this constant?
...

**0**

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**1**answer

545 views

### Can infinite polynomials be expressed as a product of its linear factors?

Background:
In the 1700s, Euler solved the Basel Problem, which was to solve $\sum_{n=1}^\infty\frac{1}{n^2}$ in closed-form. Euler showed that it was equal to $\frac{\pi^2}{6}$ by first expressing ...

**4**

votes

**1**answer

543 views

### Bernoulli number formula involving roots of Taylor polynomial of $\exp-1$

Please prove, give more symbolic or numeric support (counterexample!?), simplify or drop me a reference (or some vague hunch).
We have
$$B_n = n!\sum_{\lambda} \frac{\lambda^{2n}}{p'_n(\lambda)}$$
...

**1**

vote

**1**answer

301 views

### The Dirichlet series of the Hasse–Weil L-function

I have the following question:
Is there is a paper claiming that the Dirichlet series of the Hasse–Weil $L$-function (associated with an elliptic curve over rationals) is of finite order.
Thank you in ...

**20**

votes

**1**answer

973 views

### Provable zero-free region for any entire function that analytically is similar to zeta(s)

Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$:
$f(z)$ is bounded when $\Re z>1+\delta$
$f(z)$ is unbounded when $\Re z=1$
$f(z)$ grows ...

**4**

votes

**2**answers

397 views

### Zeros of incomplete exponential functions

Let
$$ f_N(z) = \sum_{n=N}^\infty \frac{z^n}{n!},$$
where $N$ is a positive integer.
Where are the (complex) zeros of those functions located?
It would be sufficient for me to know what the ...

**11**

votes

**4**answers

563 views

### non-trivial zeros of partial zeta functions

Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\omega$ be either ...

**14**

votes

**3**answers

1k views

### Perron, Fourier

Perron´s formula is in some sense just Fourier inversion, but I have never seen proven it that way in a textbook. I take this must be because the conditions for the Fourier inversion formula to hold ...

**4**

votes

**1**answer

307 views

### Need there be infinitely many Gaussian primes along lines that contain at least one?

Greetings from EuroCG 2012, from which I post via iPod, so apologies for lack of problem motivation, background research and mathematical formatting.
Question:Suppose L is a horizontal or vertical ...

**5**

votes

**2**answers

394 views

### Help with a mellin-type integral

greetings . i've been trying to do this integral for many days now, with no clue on how to attack it . the integral is a mellin inverse of some kind, and appears in analytic number theory .
...

**1**

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**0**answers

233 views

### Are there any known bounds on this function?

For a sequence of functions $f_{k}(z,s)=\frac{1}{k} \sqrt[s]{Li_s(z^k)}$ with $s>2$ and $Li_{s}(s)$ is the Polylogarithm, I am trying to show
If $\Re f_{1}(e^{\frac{2\pi i}{3}},s) > \Re ...

**1**

vote

**1**answer

579 views

### Convergence of Dirichlet series

I have arrived at an elementary-looking "result" via a sketchy argument. Having unsuccessfully searched for the statement and its "proof" in the literature, I would like to hear if anyone knows ...

**8**

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**0**answers

318 views

### $C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$

Suppose that $f:{\bf C}\mapsto {\bf C}$ is a $C^\infty$ function such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$, ie $f(z)$ is algebraic over the field ${\bf Q}(z)$ generated by $z$ ...

**1**

vote

**1**answer

541 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 ...

**4**

votes

**2**answers

784 views

### Evaluating the integral $\int_0^\infty \frac{\psi(x)-x}{x^2}dx.$

Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$ be the weighted prime counting function. I am trying to evaluate the integral $$\kappa:=\int_{1}^{\infty}\frac{\psi(x)-x}{x^{2}}dx$$ in several different ways. ...

**1**

vote

**1**answer

402 views

### Relation between partially computable function and complex function

Given a partially computable function, is there an analytic complex function which is equal to it at every point of it's domain? Or under what condition does a partially computable function correspond ...

**2**

votes

**2**answers

1k views

### Does the Euler product formula diverge for any zero of the Riemann zeta function?

Simple question (but not for me):
Does the Euler product formula diverge for any zero of the Riemann zeta function?
The reason why I ask this is that I heard we should not use the Euler product ...

**12**

votes

**3**answers

1k views

### Is the Euler product formula always divergent for 0<Re(s)<1?

It is known that the Euler product formula converges for Re(s)>1.
(which represents the Riemann zeta function.)
My question: Is the Euler product formula always divergent for
0 < Re(s) < 1 ?
...

**4**

votes

**1**answer

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 ...

**52**

votes

**1**answer

3k views

### Is there a “classical” proof of this $j$-value congruence?

Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve ...

**9**

votes

**5**answers

2k views

### Complex Analysis applications toward Number Theory

I'm an undergrad who is taking a Complex Analysis Course mainly for its applications in number theory.
So I would like to ask some guidelines about which theorems/concepts should I focus on in order ...

**4**

votes

**0**answers

318 views

### Adeles of Holomorphic Functions

In number theory, an adele is a kind of product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ...

**1**

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**0**answers

385 views

### Has the Weierstass transform been used to give Hermite series representations of the Riemann zeta function?

The inverse of the Weierstrass transform
expands a function as a series of Hermite polynomials $H_{n}$. There are several ways to invert the Weierstrass transform which led me to the following ...