**2**

votes

**1**answer

624 views

### Best upper bound on the number of divisors of $n$ that are larger than $N$.

I am looking for the best upper bound on $$\sum_{\substack{d | n\\ d \geq N}} 1.$$
I know that
$$
d(n) = \sum_{\substack{d | n}} 1 \leq e^{O(\frac{\log n}{\log \log n})}.
$$
For my application, I ...

**1**

vote

**1**answer

164 views

### Behavior of a quantity related to Fermat's 4n + 1 Theorem

One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem and I would like to ...

**3**

votes

**1**answer

388 views

### Heuristic for Montgomery's conjecture

This is my third question on this site regarding Montgomery's conjecture -- and I apologize
if this is too much -- but I am still not understanding well why this conjecture is believed to be true.
...

**65**

votes

**6**answers

7k views

### Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?

Let $p_n$ be the $n$-th prime number, as usual:
$p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.
For $k=1,2,3,\ldots$, define
$$
g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n).
$$
Thus the twin ...

**17**

votes

**1**answer

4k views

### Tightening Zhang's bound [closed]

Inspired by a blogpost by Scott Morrison and ongoing discussion there I decided to create this community wiki to track progress on the original bound of Yitan Zhang.
The original bound was ...

**2**

votes

**1**answer

226 views

### An estimate of an integral

On the bottom of the page 399. of Iwaniec and Kowalski's Analytic Number Theory, the authors claim that
$$h(t)=\int_{\mathbb H}k(i,z)y^s d\mu (z)$$ yields ...

**0**

votes

**0**answers

36 views

### Family of random sets represent all integers a.s.

Construct a family of sets $A_n$ such that $$|A_n|=\Theta\left((\log n)^2\right)$$
and the elements of $A_n$ are chosen uniformly at random mod $n$.
Say that a set $S$ represents $m\mod{n}$ if there ...

**12**

votes

**1**answer

2k views

### A technical question related to Zhang's result of bounded prime gaps

Here is a link on the internet: https://www.dropbox.com/s/su3uak2a057yrqv/YitangZhang.pdf
Can someone teach me how to use trivial estimation to reach (6.1) on page 24? Namely, how to impose ...

**4**

votes

**5**answers

778 views

### Spinoffs of analytic number theory

What are some techniques and theorems of analytic number theory that have proved useful outside of number theory?

**4**

votes

**1**answer

248 views

### Estimate on the prime-counting function $\psi(x)$.

There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I ...

**10**

votes

**2**answers

726 views

### Effective Chebotarev without Artin's conjecture

Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form
of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Artin's $L$-function and ...

**6**

votes

**6**answers

884 views

### Sequences equidistributed modulo 1

Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results.
H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1.
...

**0**

votes

**0**answers

177 views

### What is most current greatest lower bound on gaps between P2 almost primes

What is the current best result on the greatest lower bound on gaps between P2 almost primes where P2 represents a prime or the product of two semi-primes?

**1**

vote

**0**answers

89 views

### estimate for i-th smooth number, gap between consecutive smooth numbers

Does anyone know of the best estimates for $n_i$ and $n_{i+1}-n_i$ where $n_i$
is the $i-$th $y-$smooth number?
The best I could find was Tijdemann's estimate for the gap in terms of
...

**4**

votes

**2**answers

407 views

### Average involving the Euler phi function

Does
$$\frac{1}{N^2}\sum _{d=1}^N \log d \sum _{n=1}^{N/d} \frac{\phi(n)}{\log (dn)},$$
converges or not when $N$ goes to infinity?

**2**

votes

**1**answer

131 views

### A question about the second Chebyshev function $\psi(x) = \sum_{m=1}^{\infty}\vartheta(\sqrt[m]{x})$

Using a simple java application, I have noticed that for $x > 25$:
$$\psi\left(\frac{x}{5}\right) \ge \psi\left(\frac{x}{3}\right) - \psi\left(\frac{x}{4}\right)$$
where:
$$\psi\left(x\right) = ...

**7**

votes

**2**answers

366 views

### short character sums averaged on the character

Let $a$ be an integer, $p$ a prime (much) greater than $a$, and $\chi$ a Dirichlet character.
There is an abundant literature on the sums
$$S(\chi,a)=\sum_{i=1}^a \chi(i),$$
called short (or ...

**1**

vote

**0**answers

162 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 ...

**3**

votes

**1**answer

282 views

### Least non primitive root

There is an abundant literature, and even here on MO no shortage of questions, on the question of the smallest prime primitivee root modulo $q$ (where $q$ is a prime, or more generally
an odd prime ...

**1**

vote

**1**answer

200 views

### Bounds for the largest divisor of n less than n^0.5

Let $d(n)$ denote the largest divisor of $n$ less than $\sqrt{n}$. Are there good lower bounds for $d$ that hold for almost all natural numbers?
More precisely, is there a function $f$, say ...

**4**

votes

**3**answers

273 views

### A divergent series related to the number of divisors of of p-1

Let $d(n)$ denote the number of divisors of $n$. Is it known that the series
$$\sum_{p \text{ prime}} \frac{1}{d(p-1)}$$
diverges?
This would follow immediately from the Sophie Germain Conjecture. ...

**1**

vote

**1**answer

233 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 = ...

**1**

vote

**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 ...

**1**

vote

**0**answers

266 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

306 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 ...

**5**

votes

**1**answer

296 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 ...

**9**

votes

**2**answers

609 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 ...

**7**

votes

**0**answers

301 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 ...

**17**

votes

**0**answers

433 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**

votes

**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 ...

**7**

votes

**2**answers

338 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 ...

**5**

votes

**2**answers

650 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 ?

**10**

votes

**3**answers

770 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 ...

**8**

votes

**0**answers

685 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:
...

**1**

vote

**1**answer

611 views

**7**

votes

**0**answers

990 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

72 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

692 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 ...

**4**

votes

**1**answer

281 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

937 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

547 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

209 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, ...

**7**

votes

**2**answers

588 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

832 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 ...

**6**

votes

**2**answers

247 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

316 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

344 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)
...