# Tagged Questions

**8**

votes

**2**answers

473 views

### Prove that the Dirichlet eta function is monotonic

Let us consider $\eta(p):= \sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n^p}$ for $p>0$. Has anyone come along with an elementary proof that $\eta(x)$ is monotonically increasing on this set? By ...

**12**

votes

**1**answer

285 views

### Cesaro(?)/Euler(?) - summation of the $s(p)=\sum_{k=0}^\infty (-1)^{H(k)} (1+k)^p$ for $p=1,2,3,…$ (where $H(k)$ is the Hamming-weight)

In another thread (in MO) there was a question about a series where the signs at the terms alternate with the "Hamming-weight", that means according to the number of bits in the binary representation ...

**4**

votes

**1**answer

153 views

### Asymptotic expansion of the Mordell integral

my question concerns the Mordell integral
$$h(z;\tau):=\int_{-\infty}^\infty \frac{e^{\pi i\tau w^2-2\pi zw}}{\cosh(\pi w)}dw,\qquad \Im(\tau)>0,\quad z\in\mathbb{C},$$
which frequently occurs in ...

**27**

votes

**3**answers

1k views

### A translation of the Cantor set contained in the irrationals

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here.
What would ...

**6**

votes

**0**answers

842 views

### Questions on de Branges' work on the Riemann hypothesis

According to Wikipedia, Louis de Branges de Bourcia has obtained some notable
results, such as a proof of the Bieberbach conjecture in 1985, which is now
known as de Branges' theorem. Initially, his ...

**7**

votes

**1**answer

411 views

### Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?

Is the mapping
$$
f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}
$$
surjective?
If not, what is its image?
If yes, what can be said about ...

**11**

votes

**1**answer

986 views

### Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer?

Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$
runs over the integers?
The existence of the limes inferior follows from Dirichlet's approximation theorem,
but the ...

**8**

votes

**0**answers

144 views

### Inertia group vs. differential equations

The tame quotient of the inertia group of $\mathbf Q_p$, say, is the profinite group generated by the Frobenius $\sigma$ and the monodromy $\tau$, subject to the relation $\tau^{p-1} [\tau, \sigma] = ...

**4**

votes

**2**answers

269 views

### Bruhat-Schwartz functions and derivatives in p-adic numbers

First of all, I am not an expert in neither classical, nor $p$-adic functional analysis, but anyway, I stumbled over the following lately:
Let $\varphi:\mathbb{Q}_p\rightarrow\mathbb{C}$.
...

**2**

votes

**0**answers

328 views

### How well can you approximate a function by a band-restricted function?

Say I have a compactly supported $C^1$ function $f:\mathbb{R} \to \mathbb{R}$.
Let $R>0$. Let $\nu$ be some reasonable measure on $\mathbb{R}$ -- take, for instance, (a) $d\nu(t)=dt$ or (b) ...

**20**

votes

**1**answer

994 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 ...

**0**

votes

**1**answer

242 views

### Growth rate of a sum

Consider a positive sequence $x_n >0$ that satisfy the condition that there exists a constant $0<\alpha<1$ such that $x_{n+1} \geq \alpha (x_1+\ldots{} +x_{n})$.
What can be said about the ...

**28**

votes

**7**answers

2k views

### How should an analytic number theorist look at Bessel functions?

(And a related question: Where should an analytic number theorist learn about Bessel functions?)
Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...

**1**

vote

**2**answers

817 views

### How to compute \prod_{n=1}^{\infty} (1-p^{-n})

We know it converges for any prime p. I just want to know how to compute its exact value:
$$\prod_{n=1}^{\infty} (1-p^{-n})$$

**1**

vote

**0**answers

181 views

### Euler divergent series $(-1)^nn!$ in $\mathbb{R}$ and $\mathbb{Q}_p$

Consider the series:
$$\sum\limits_{n=0}^{\infty}(-1)^nn!$$
Well know that if p is prime, this series convergence in $\mathbb{Q}_p$. Let $s_p\in\mathbb{Q}_p$ is sum of this series.
Also let
...

**3**

votes

**3**answers

219 views

### Limit connected with a periodic function

I am posting the following question from Math.Stackexchange:
Let $f$ be a $1$-periodic function, i.e., $f(x+1)=f(x)$, defined on the interval $(0, 1)$ by the formula
$$
f(x)=2x-1.
$$
For a real ...

**2**

votes

**0**answers

731 views

### 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 ...

**6**

votes

**2**answers

411 views

### Alternating sums of GCDs

The exciting question on alternating sums of binomial coefficients triggered me to ask the following much simpler question (sorry if it is too simple, but I'm not a number theorist, so I must be ...

**1**

vote

**1**answer

158 views

### Local nonarchimedean Sobolev inequality

Let $B$ be the unit ball in $\mathbb{R}^n$. A local version of the Sobolev inequality on $\mathbb{R}^n$ says that for any $p\in[1,\infty]$ there exist constants $C >0$ and $k \in \mathbb{N}$ such ...

**3**

votes

**4**answers

2k views

### Non trivial zeros of the Zeta function

The Zeta-function can be written as the following infinite Hadamard product of its non-trivial zeroes:
$\zeta(s) = \pi^{\frac{s}{2}} \dfrac{\prod_\rho \left(1- \frac{s}{\rho} ...

**7**

votes

**2**answers

357 views

### Asymptotics of the $q$-harmonic series as $q\to1$

The following (very simply looking!) problem occurs in regularization
of the harmonic series
which can be formally thought of as the limit as $q\to1$, $|q|<1$, of
$$
...

**12**

votes

**1**answer

970 views

### Why is the functional equation of the Riemann zeta function equivalent to the Poisson summation formula?

We can derive from the Poisson summation formula the modularity of the Theta function, which results in the functional equation. In his book on the Riemann Zeta function, Patterson mentions also that ...

**5**

votes

**1**answer

1k views

### The Guinand-Weil explicit formula without entire function theory

I'll admit from the outset that this question is slightly vague. The actual question appears at the end of the post.
The explicit formula of Guinand and Weil can be written in the following way:
For ...

**10**

votes

**2**answers

577 views

### Order of $\zeta(1+it)$

What is known about the order of $\zeta(1+it)$?
Iwaniec-Kowalski gives (pp. 226 citing a result of Vinogradov-Korobov)
$|\zeta(1+it)| \lesssim (\log t)^{2/3},$
and oppositely Titchmarsh gives ...

**5**

votes

**1**answer

1k views

### Do the tails of the decimal expansion of pi form a dense set in [0,1]?

Let $a_n=10^n \cdot \pi$. Is the set of numbers $\{a_n-\lfloor a_n \rfloor : n \in \mathbb{N}\}$ dense in [0,1]?
What is the best known result near this question?
Apparently John Nash asked this on ...

**8**

votes

**2**answers

909 views

### On rational functions with rational power series

Let $f(z)=\sum_{n\geq 0}a_n z^n$ be a Taylor series with rational coefficients with infinitely non-zero $a_n$ which converges
in a small neighboorhood around $0$. Furthermore, assume that
...

**9**

votes

**0**answers

236 views

### Positivity of polynomial sequences via generating series

In this question I address
the problem of proving the nonnegativity of a numerical sequence
$a_0,a_1,a_2,\dots$ via generating series technique. In the notation
$A(x)=\sum_{n=0}^\infty a_nx^n\ge0$ ...

**0**

votes

**1**answer

782 views

### Generalizations of a product formula for the gamma function

Hello and Happy holidays.
I am interested in generalizations of the following product formula for the gamma function
$\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$:
\begin{align}
...

**7**

votes

**3**answers

925 views

### A Question concerning the Fourier Transform of $\mathbb{R}$

Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace ...

**1**

vote

**1**answer

325 views

### Upper bound for the quality of an $abc$-triple

A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the quality or power of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log ...

**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} ...

**3**

votes

**0**answers

367 views

### What are the consequences of allowing the ABC-conjecture $\kappa_{\epsilon}$ to also vary with $\omega(abc)$?

A commonly encountered form of the ABC-conjecture is the following:
For all $\epsilon > 0$, there is a constant $\kappa_{\epsilon} > 0$ (depending only on $\epsilon$) such that for all coprime ...

**1**

vote

**2**answers

494 views

### Sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$

Are there known sharp upper bounds (in terms of $k$ or $\omega(k)$, the number of distinct prime divisors of $k$) for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the ...

**7**

votes

**0**answers

181 views

### Semi-norms for Schwartz-Bruhat space over Q_p

I'm an analyst beginning to do some work over the p-adics, in particular work with spaces of functions from $\mathbb{Q}_p$ to $\mathbb{C}$. The Schwartz-Bruhat space in this case is given by the space ...

**14**

votes

**2**answers

4k views

### Irrationality of pi*e, pi^pi and e^(pi^2)

What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?

**7**

votes

**4**answers

1k views

### Non-vanishing of zeta(s), Re(s)=1, without complex analysis?

Say you are allowed to use Fourier analysis, complex variables, Euler-Maclaurin, etc., but no complex analysis - no holomorphic continuations, no definition of analytic function, and, in particular, ...

**17**

votes

**6**answers

1k views

### A gamma function identity

In some of my previous work on mean values of Dirichlet L-functions, I came upon the following identity for the Gamma function:
\begin{equation}
\frac{\Gamma(a) \Gamma(1-a-b)}{\Gamma(1-b)}
+ ...

**7**

votes

**3**answers

631 views

### Distribution of fractional parts of n^{3/2}

What can be said about the limiting distribution of the sequence of fractional parts of $\{n^{a},n>0\}$ for $a\in(1,2)$. I ran a computer experiment for $n\sqrt{n}$ and it looks like uniformly ...

**3**

votes

**2**answers

717 views

### Product over the primes

I'm trying to estimate the product
$$\prod_{p\lt q\lt r\lt s}1-\frac{24}{(pqrs)^2}$$
where $p,q,r,s$ are primes.
This is for the purpose of calculating the density of Sloane's A070284 [1]. The idea ...

**6**

votes

**2**answers

335 views

### How large (small) can be the measure of a set where a polynomial takes small values ?

A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other ...

**7**

votes

**2**answers

702 views

### Sum f(p) over all primes convergent with sum f(n) over all natural numbers divergent?

The sum $\sum_{n=1}^{\infty} 1/n^{s}$ is convergent for all real $s>1$ and diverges for all real $s \le 1$. The same holds for the sum $\sum_{p \ prime} 1/p^{s}$. Thus, for the functions $f(n)= ...

**10**

votes

**7**answers

5k views

### Greatest power of two dividing an integer

Does anyone know of a closed form for the function on $\mathbb{N}$ which returns the greatest power of two which divides a given integer?
To be more precise, any positive integer $n\in\mathbb{N}$ can ...

**19**

votes

**0**answers

1k views

### Trigonometry related to Rogers--Ramanujan identities

For integers $n\ge2$ and $k\ge2$, fix the notation
$$
[m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad
[m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}.
$$
Consider the following trigonometric ...

**13**

votes

**0**answers

634 views

### Polynomials with presumably positive coefficients

After seeing that some positivity problems get their solutions on MO,
I am quite enthusiastic of posing my (and not only) problem of positive flavour.
In order to state it, I have to introduce the ...

**19**

votes

**3**answers

2k views

### When is $n/\ln(n)$ close to an integer?

As usual I expect to be critisised for "duplicating"
this question. But I do not! As Gjergji immediately
notified, that question was from numerology. The one I ask you here
(after putting it in my ...

**20**

votes

**1**answer

2k views

### Polynomials with rational coefficients

Long time ago there was a question
on whether a polynomial bijection $\mathbb Q^2\to\mathbb Q$ exists. Only one attempt
of answering it has been given, highly downvoted by the way. But this answer ...

**25**

votes

**1**answer

776 views

### A question of Erdős on equidistribution

In his book Metric Number Theory, Glyn Harman mentions the following problem he attributes to Erdős:
Let $f(\alpha)$ be a bounded measurable function with period 1. Is it true that
...

**16**

votes

**3**answers

1k views

### Recursions which define polynomials

There are many examples (Somos sequences, special polynomials related to rational solutions of the Painleve equations) when a recurrence relation, which a priori produces a sequence of rational ...

**11**

votes

**2**answers

1k views

### Growth of the “cube of square root” function

Hello all, this question is a variant (and probably a more difficult one)
of a (promptly answered ) question that I asked here, at Is it true that all the "irrational power" functions are ...

**6**

votes

**1**answer

292 views

### at which rational points does the Hypergeometric function take rational values

A generic example is ${}_2 F_1(\frac{1}{3},\frac{2}{3},\frac{6}{5};\frac{27}{32})=\frac{8}{5}$. So my question: Is there any description of the set of rational points at which the hypergeometric ...