**13**

votes

**1**answer

391 views

### How fast can we numerically calculate Kloosterman sums?

Define the usual Kloosterman sum by $$S(m,n;c) = \sum_{\substack{x \pmod{c} \\ (x,c) = 1}} e\Big(\frac{mx + n\overline{x}}{c}\Big),$$
where $x \overline{x} \equiv 1 \pmod{c}$, and $e(x) = e^{2 \pi i ...

**3**

votes

**1**answer

297 views

### On link between Riemann hypothesis and partial GRH

Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of ...

**0**

votes

**0**answers

123 views

### Number of S-integer points inside high dimensional spheres

I am interested in the number of points inside high dimensional spheres, in fact for sets of the form $S(t,m)=\{x\in R^{n}:mx\in Z^{n}\ {\text{and}}\ ||x||^2\leq t(n)\}$ where $t(n)$ is a constant ...

**1**

vote

**2**answers

244 views

### A conjecture of Montgomery: reference request

In the answer to this question, engelbret mentions "a conjecture of H. L. Montgomery (not the one on pair correlations, another one), which implies both the GRH and the Elliott-Halberstam ...

**10**

votes

**1**answer

346 views

### Distribution mod 1 of exponential growth sequences

Let $t_n$ be a sequence of real numbers and $C,r>1.$ Suppose that for every $n\geq 1$ we have $\frac{1}{C}r^n\leq t_n \leq Cr^n.$ Does there exist a real number $\xi$ and an $\varepsilon>0$ ...

**1**

vote

**1**answer

511 views

### Exponential sums

I would like to estimate the following sum
$\sum_{N <n \leq 2N}e(vn^{l})$, $l \geq 1$ constant(not integer) and $v$ is a parameter(integer) that doesn't grow too fast(a small power of N).
The ...

**5**

votes

**1**answer

443 views

### Other implications of Zhang's method

I have been reading a bit about Zhang's proof and the associated Polymath8 project.
Though Tao's high level summary
...

**1**

vote

**1**answer

514 views

### An integral representation of the Riemann zeta function

I am referring to the equality in equation $3.29$ (page 12) and $4.20$ (page 17) in this paper.
I am unable to recognize where this comes from or what is the general expression for values other than ...

**0**

votes

**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 ...

**3**

votes

**0**answers

209 views

### Inequalities in paper by Jean Bourgain

The question refers to the following paper by Jean Bourgain: http://arxiv.org/abs/math-ph/0011053
Specifically, I can't derive the following inequality in (1.20):
\begin{equation}
...

**-3**

votes

**1**answer

174 views

### a question on Siegel Upper Half Space [closed]

Is it known whether siegel upper half plane is dense in the space of nonsingular matrices of same dimension .$.http://en.wikipedia.org/wiki/Siegel_upper_half-space.
Actually the question i have in my ...

**0**

votes

**0**answers

108 views

### Motivation behind the appearance of Bessel functions in partial trace formulas

Bessel functions occur naturally on the Kloosterman side (or geometric side) of Petersson's formula and Kuznetsov's formula. Is there an intuitive explanation for their appearance? For instance, is ...

**4**

votes

**1**answer

304 views

### subconvexity problem for $GL(3) × GL(2)$ $L$-function without involving in symmetric lift

A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated.
...

**1**

vote

**1**answer

214 views

### Trace formula for the Möbius function

I found this conjecture while working with the Möbius function
$$ \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi ...

**10**

votes

**1**answer

379 views

### Estimate term in Ramanujan Lost Notebook (classic analytic number theory)

This is the fault of Igor Rivin, who asked about sums of divisor functions. I will put in links eventually. What I would like to know is the size of the right hand side in Ramanujan's formula (381), ...

**0**

votes

**1**answer

340 views

### How to do such a partitioning?

Assume:
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K, \qquad x \in \mathbb{R}_+^K , \qquad w = e^{-j\frac{2\pi}N}
$$
and,
$$
f(l) = \sum_{i=1}^K \sum_{j=1}^K x_i x_j w^{(p_i-p_j)l}
$$
I am going to ...

**7**

votes

**1**answer

410 views

### The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$

The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
...

**2**

votes

**2**answers

311 views

### Summation of certain series

Suppose $f(n)$ is a periodic function with period $q$. Now from this paper we get that if $\displaystyle\sum_{n=1}^{q}f(n)=0$ then ...

**5**

votes

**0**answers

200 views

### Average of Fourier coefficients of a cusp form of half integral weight

Suppose $f$ is a cusp form of half integral weight $k$ w.r.t. the group $\Gamma_0(4)$ ($k$ is not very low, can assume $k \ge 11/2$), and $a_n$ is its Fourier coefficient. The Linnik bound says that ...

**8**

votes

**1**answer

332 views

### Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ...

**7**

votes

**1**answer

858 views

### A reformulation of the Riemann Hypothesis

I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes.
Let's define ...

**16**

votes

**1**answer

710 views

### Infinitely many primes, and Mobius randomness in sparse sets

Problem 1: Find a (not extremely artificial) set A of integers so that for every $n$, $|A\cap [n]| \le n^{0.499}$, ($[n]=\{1,2,...,n\}$,) where you can prove that $A$ contains infinitely many primes.
...

**2**

votes

**1**answer

82 views

### Question about BFI's “Primes in arithmetic progressions to large moduli”

Ref: http://link.springer.com/article/10.1007%2FBF02399204
My question is about the proof of Theorem 0(b). On p.213, we see the expression
...

**7**

votes

**1**answer

499 views

### Density of prime pairs whose gap is less than the average gap

By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...

**5**

votes

**0**answers

349 views

### a generalization of a formula of Shimura

Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$.
Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$.
Shimura have the following formula
$L(s, Ad\; ...

**4**

votes

**0**answers

158 views

### Maximal order of Hooley's Delta function?

There is a large literature on Hooley's
$$
\Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1
$$
giving its normal and average order. What is known of its maximal order?
Clearly $\Delta(n)\le d(n)$ ...

**2**

votes

**1**answer

353 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

147 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

346 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.
...

**61**

votes

**6**answers

6k 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 ...

**16**

votes

**1**answer

4k views

### Tightening Zhang's bound

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 ...

**0**

votes

**0**answers

131 views

### shortest relation between poly-Bernoulli numbers and Euler numbers

Poly-Bernoulli numbers which introduced by M.Kaneko
are $B_k^{(n)}$ which satisfies in generating function ${Li_{k}(1-e^{-x}) \over 1-e^{-x}}=\sum_{n=0}^{\infty}B_{n}^{(k)}{x^{n}\over n!}$
where Li ...

**2**

votes

**1**answer

218 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

34 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

750 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

199 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 ...

**6**

votes

**2**answers

568 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 ...

**0**

votes

**0**answers

79 views

### Zeroes of a homogeneous function

I am interested in the zero-set of a homogeneous function $f(x_1, \cdots, x_n)$, where $f$ is not necessarily a polynomial. In particular, I would like to know if there are any general results ...

**6**

votes

**6**answers

749 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

164 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?

**0**

votes

**0**answers

95 views

### Bounding number of solutions to an equation:

I have an equation that I think should not have too many solutions, but I don't see a way to argue this.
Given $a, b, c, N \in \mathbb{N}$, how many positive integer solutions $x, y \leq N$ can the ...

**1**

vote

**0**answers

76 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

380 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

106 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) = ...

**3**

votes

**1**answer

195 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

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 ...

**3**

votes

**1**answer

259 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

192 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

264 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. ...