# Tagged Questions

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vote

**1**answer

202 views

### What are the best known lower and upper bounds for the second Chebyshev function $\psi(x)$

I was reading through Jitsuro Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$ when $x \ge 25$.
In the paper, he uses the following bounds for the second Chebyshev function ...

**1**

vote

**0**answers

154 views

### Estimates for the size of the product set [n].[n] [duplicate]

Possible Duplicate:
Number of elements in the set {1,…,n}*{1,..,n}
Writing $[n]$ for the set $\lbrace1,2,...,n\rbrace$, let $P_n$ denote the product set $[n].[n]$, i.e.
$$ P_n = ...

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**0**answers

113 views

### Off critical line zeros for half integer weight $L$-functions

Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put
$$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$
to be the $L$-function.
Further ...

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vote

**0**answers

256 views

### Poles of products of Gamma functions

I want to know if there can be a general statement about the poles (Laurent expansion) of such products of Gamma functions as a function of $p \in \mathbb{R}$ in the limit $\epsilon \rightarrow 0$,
...

**6**

votes

**3**answers

304 views

### Jacobi sums on tori

The Jacobi sum of $n$ multiplicative character $\chi_1,\dots,\chi_n$ on a finite field
$\mathbb F_q$ is defined as
$$J(\chi_1,\dots,\chi_n) = \sum_{x_1,\dots,x_n \in \mathbb F_q, x_1+\dots+x_n=1} ...

**0**

votes

**1**answer

162 views

### This might be a trivial question on Hurwitz's zeta function.

In the book I am reading they write that for Hurwitz zeta function, $\zeta(x,s)=\sum_{n=0}^{\infty} \frac{1}{(x+n)^s}$, the next sum in the RHS converges for $\Re(s)>-1$, and I don't see how ...

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**1**answer

292 views

### What analytic tools can provide a lower bound for this Diophantine equation?

The resolution of the Diophantine equation $$m! = n(n+1)$$ was asked on M.SE. My intuition says that this cannot be solved by elementary means - apologies if I am mistaken.
I felt that the following ...

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**2**answers

583 views

### Ordinary Generating Function for Mobius

Is there any information known for the Ordinary Generating Function for Mobius?
$$
\sum_{n=1}^{\infty} {\mu(n)}x^n
$$
I know that
It has radius of convergence 1.
Does not have limit as ...

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**0**answers

288 views

### Montgomery's conjecture and lower bound on certain Fourier transform.

Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...

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**0**answers

414 views

### On Determinants of Laplacians on Riemann Surfaces

History of the Formula: In their famous paper "On Determinants of Laplacians on Riemann Surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the space $T^n$ of ...

**18**

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**4**answers

1k views

### Good uses of Siegel zeros?

The short version of my question goes: What is known to follow from the existence of Siegel zeros?
A longer version to give an idea of what I have in mind: The "expectional zeros" of course first ...

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**2**answers

332 views

### Recovering $\sum_{n \leq x} a(n)$ from $\sum_{n \leq x} a(n)e^{-n/x}$

In the theory of automorphic forms and multiple Dirichlet series, we often take inverse Mellin transforms of Dirichlet series to come up with Tauberian theorems, like the Ikehara Tauberian method. In ...

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**2**answers

647 views

### Is there an explicit expression for the imaginary part of some non-trivial zero of zeta,

Is there an explicit expression for the imaginary part of some non-trivial zero of zeta, in terms of well-known constants, such as say $\gamma$ or $\pi$ say ?

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**3**answers

747 views

### Deducing BSD from Gross-Zagier and Kolyvagin

Does anyone know which papers deduce BSD for elliptic curves $E/\mathbb{Q}$ of rank 0 or 1 from the papers by Gross and Zagier and Kolyvagin? If I understand these theories right, there is still a ...

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**0**answers

674 views

### Analytic continuation of the Dirichlet generating series of the multiplicative partition function

Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series:
...

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**1**answer

584 views

**5**

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**0**answers

945 views

### “Must read ”papers on analytic number theory

Question: What would be some must-read
papers for an aspiring analytic number
theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: ...

**1**

vote

**0**answers

71 views

### linear independence of values of the polylogarithm at different roots of unity

I am interested in the real and imaginary part of the complex polylogarithm
$$L_{k+1}(\zeta):=Re(\frac 1 {i^k}\sum_{m=1}^\infty \frac{\zeta^m}{m^{k+1}}),$$
where $\zeta$ is a primitive $n$-th root ...

**6**

votes

**2**answers

677 views

### Sums of two squares: What is known about the distribution of r(n)?

The distribution of sums of two squares has been studied by Landau. What is known about the distribution of the function $r(n)$, the number of representations of $n$ as the sum of two squares? Some ...

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votes

**1**answer

278 views

### Power free values of reducible polynomials

Browning in this paper proves that if $f \in \mathbb{Z}[x]$ is an irreducible polynomial of degree $d \geq 3$ and $k$ is an integer $\geq 3d/4 + 1/4$, then we have $$\#\{n \in \mathbb{Z} \cap [1, X] ...

**28**

votes

**1**answer

922 views

### Prime Number Races in 2 Dimensions

Is the mapping $$f: \ \mathbb{N} \rightarrow \mathbb{Z}[i], \ \ \ n \ \mapsto
\sum_{2 < p \leq n \ {\rm prime}} e^{\frac{p-1}{4} \pi i}$$ surjective?
In 1999, when I was an undergraduate student, ...

**5**

votes

**6**answers

535 views

### Representations with Triangular Numbers

A well known theorem of Gauss says that any natural number $n$ may
be written as the sum of three triangular numbers -
$$
n={a_{1} \choose 2}+{a_{2} \choose 2}+{a_{3} \choose 2}
$$
The following ...

**4**

votes

**0**answers

205 views

### Translation of an article by Wolfgang Schmidt on normality for real numbers in different bases.

I would greatly appreciate a pointer to a translation from German into English of the article by Wolfgang Schmidt, Über die Normalität von Zahlen zu verschiedenen Basen, from Acta Arithmetica VII, ...

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votes

**2**answers

571 views

### Question about the Hardy-Littlewood method (quite basic)

Hi, I have a question about the Hardy-Littlewood method.
Writing $R_s(n)$ for the number of ways to write $n$ as a sum of $s$ $k$-th powers and $f(\alpha )$ for the sum $\sum _{m=1}^Ne(\alpha m^k)$, ...

**19**

votes

**4**answers

2k views

### The Riemann Hypothesis and the Langlands program

On page 263 of this book review appears the following:
Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic ...

**0**

votes

**1**answer

766 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 ...

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**2**answers

245 views

### Minimal period of arithmetic progressions occurring in sets of positive density.

Let $A$ be a subset of ${\mathbb N}$ with positive upper-Banach density, and for each integer $k\geq3$, define $R_k=R_k(A)$ to be the smallest positive integer $r$ such that $A$ contains a length $k$ ...

**2**

votes

**1**answer

303 views

### Asymptotic formula in Analytic Number Theory

Hi,
Could anyone tell me in what sense the following is an "asymptotic formula":
Theorem 1 from
...

**2**

votes

**1**answer

326 views

### Siegel-Walfisz Theorem

Hi,
Could anyone explain to me how A and B are the same/different/equivalent?
A = The Siegel-Walfisz Theorem as stated in Wikipedia (this is the statement in Davenport)
...

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votes

**4**answers

711 views

### Using Quotient of Prime Numbers to Approximation Reals

We know a positive rational number can be uniquely written as $m/n$ where $m$ and $n$ are coprime positive integers. Particularly, we can pick out those numbers with $m$ and $n$ both prime.
Question ...

**7**

votes

**1**answer

395 views

### Lower bound for exponential sums.

Let $D$ be a subset of $\mathbb Z/n \mathbb Z$ containing $0$. For $m$ an integer, set $$\alpha(m,D)=\sum_{d \in D} e\left (\frac{m d }{n}\right ),$$
where as usual $e(x) = e^{2 i \pi x}$ This is an ...

**3**

votes

**1**answer

235 views

### Ratio of consecutive divisors and average

Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard:
(1) ...

**4**

votes

**1**answer

562 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
$$B_n = n!\sum_{\lambda} \frac{\lambda^{2n}}{p'_n(\lambda)}$$
...

**4**

votes

**0**answers

135 views

### Estimate needed on a sum involving the prime counting function

Let $F(x) = \sum_{1 < n \leq x} (-1)^{\pi (n)}$ where $\pi (n) $ is the prime counting function.
I am trying to understand how $ F(x) $ behaves as $ x \to \infty$. In particular, what are the ...

**4**

votes

**1**answer

277 views

### Distribution function for divisors of an Integer

For a fixed $n$, let $D_n(x) = \{ d|n : d \leq x \}$ . We assume here $p \leq x \leq n/p$,
where $p$ is the smallest prime factor of $n$.
For example if $n = p^i$ for some prime $p$ then $D_n(x) ...

**11**

votes

**2**answers

478 views

### Average orders of multiplicative functions

For a multiplicative function $f$ and $x>0$ let $$S_f(x)= \sum_{n \leq x} f(n).$$
Studying sums of this type is a favourite pastime of analytic number theorists. I'm trying to understand what kind ...

**14**

votes

**1**answer

908 views

### Is there a Montgomery's conjecture for Dirichlet characters and Artin representations ?

Edit: as GH noticed, the way I tried to state Montgomery's conjecture is wrong. There were some mistakes in the references I used, which compounded with some mistakes of mine, gave a very poor post. ...

**3**

votes

**0**answers

182 views

### Is this extension of the Selberg class trivial?

I came across the following modification of the Selberg class in some of my work (see below), and while I've moved on in some sense -- I submitted the paper in question -- I can't get it off of my ...

**7**

votes

**1**answer

661 views

### Least prime primitive root

For $p$ a prime number, let $G(p)$ be the least prime $q$ such that $q$ is a primitive root mod $p$, that is $q$ generates the multiplicative group $(\mathbb Z/p\mathbb Z$)* .
Is it known that ...

**2**

votes

**1**answer

383 views

### The tightest prime zipper

Define a prime zipper as an increasing function $f(n)$ mapping $\mathbb{N}$ into $\mathbb{N}$
with the property that, for every $n \ge 1$, there is at least one prime within the
inclusive interval $[ ...

**5**

votes

**2**answers

323 views

### Density of integers $n$ whose totient $\varphi(n)$ is larger than $\alpha n$

Fix $0 < \alpha < 1$ a real. Let $S_\alpha$ the set of integers $n \geq 1$ such that be $\phi(n)>\alpha n$. For $x>0$, let $S_\alpha(x)$ be the number of positive integers $n$ less han $x$ ...

**20**

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**1**answer

1k 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 ...

**9**

votes

**1**answer

572 views

### Prime Power Gaps

In 2000, Baker, Harman and Pintz proved that there is always a prime in
the interval $(n-n^{0.525}, n)$. There are also conditional results
implying smaller intervals. Nevertheless, I could not find ...

**1**

vote

**1**answer

249 views

### Does FE of Selberg Zeta function imply Trace formula?

Does the functional equation of the Selberg Zeta function imply the Selberg trace formula?
BTW, the trace formula implies the functional equation.

**13**

votes

**1**answer

839 views

### Small primes in arithmetic sequences

Fix an integer $a>1$. For $n \geq 1$ an integer, let $\pi_{n,1}(an)$ the number of primes
$p \leq an$ such that $p \equiv 1 \pmod{n}$, and $\pi(an)$ the number of all primes $p \leq an$. Let
...

**4**

votes

**1**answer

416 views

### An application of Mobius Inversion in a paper of Shintani

I've been reading about Shintani zeta functions and in particular with respect to finding the density of cubic discriminants as in the theorem of Davenport-Heilbronn. In Shintani's paper "On ...

**3**

votes

**1**answer

404 views

### About Theorem 3.2 in 'introduction to spectral theory of automorphic forms' by Iwaniec

In Theorem 3.2 of 'Introduction to spectral theory of automorphic forms' by Iwaniec,the first bound is about the coefficients of automorphic forms
$$\sum_{|n|\le ...

**25**

votes

**2**answers

2k views

### Number of elements in the set $\{1,\cdots,n\}\times\{1,\cdots,n\}$

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?

**14**

votes

**1**answer

827 views

### A question about Speiser's 1934 result on the Riemann hypothesis

A number of sources concerning Speiser's 1934 result state that the Riemann Hypothesis (RH) implies $\zeta'(s)\neq 0$ for all $0<\text{Re}(s)<1/2$. But I have seen some (possibly less reliable) ...

**1**

vote

**2**answers

379 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 ...