**10**

votes

**1**answer

398 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

342 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

459 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

409 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 ...

**6**

votes

**0**answers

227 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 ...

**10**

votes

**1**answer

391 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

932 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 ...

**17**

votes

**1**answer

767 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

84 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
...

**8**

votes

**1**answer

603 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

361 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

166 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

502 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

159 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

370 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.
...

**63**

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 ...

**2**

votes

**1**answer

223 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

763 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

233 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 ...

**9**

votes

**2**answers

687 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

827 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

171 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

84 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

397 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

120 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

219 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

159 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

275 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

270 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

191 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

250 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

303 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

161 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

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 ...

**8**

votes

**2**answers

572 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

282 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

402 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 ...

**4**

votes

**2**answers

235 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

640 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

715 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

661 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

561 views

**5**

votes

**0**answers

908 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: ...