**3**

votes

**3**answers

537 views

### How do estimates on $N_\chi(\alpha,T)$ lead to the Dirichlet prime number theorem for arithmetic sequences?

Let $ N_\chi(\alpha,T)$ be the number of zeros of $L(s=\sigma+it,\chi) = \sum \frac{\chi(n)}{n^s}$ where $c > 0$ and $(\sigma,t) $ are in the rectangle $ [\alpha,1] \times [-T,T]$.
In various ...

**1**

vote

**1**answer

106 views

### Exponential sum estimates similar to the one for $\sum_p (\log p) e(p \alpha)$, but for different sequences

Obtaining a non-trivial estimate for $\sum_p (\log p) e(p \alpha)$ over the minor arcs is one of the estimates required for obtaining the ternary Goldbach for $n$ sufficiently large via the circle ...

**1**

vote

**2**answers

257 views

### estimate sum of $\log \log p/p$

It is known that $$\sum_{p\leq x} \frac{\log p}{p}=\log x+c.$$
Are any tight bounds on
$$\sum_{p\leq x} \frac{\log \log p}{p}$$ known?
I haven't managed to find anything in the literature. Trying to ...

**20**

votes

**1**answer

1k views

### What's special about the circle problem?

Let $K$ be a number field, and let
$$\zeta_{K}(s):= \sum_{0
\neq I \text{ ideal of }O_K} \frac{1}{N_{K/\mathbb{Q}}(I)^s} = \sum_{n \ge 1} \frac{a_n}{n^s}$$
be the Dedekind zeta function of $K$. The ...

**9**

votes

**0**answers

134 views

### Newly defined $L$-function in terms of $L$-function, does it have any obvious zeros or poles?

Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ideal, let $[\mathfrak{a}]$ denote its ...

**13**

votes

**1**answer

509 views

### What is known about $\sum_{n \leq x} \mu(n) \varphi(n)$?

Let $\mu(n)$ denote the Möbius function and $\varphi(n)$ the Euler-phi function. What is known about $f(x) = \sum_{n \leq x} \mu(n) \varphi(n)$? For example:
Is it known that $f(x)$ grows without ...

**5**

votes

**0**answers

106 views

### Sums of twisted products of Kloosterman Sums

For $m,n,c \in \mathbb{N}$, let $S(m,n;c)$ denote the Kloosterman sum
$$
S(m,n;c) := \sum_{\substack{1 \leq a < c \\ \gcd(a,c) = 1}} e \left( \frac{ma + n\overline{a}}{c} \right)
$$
where $e(n) = e^...

**5**

votes

**0**answers

130 views

### Particular case of the class number formula, Dirichlet characters

Let $\chi$ be a Dirichlet character modulo $4$ such that $\chi(-1) = -1$, and let $\chi'$ be a Dirichlet character modulo $5$ such that $\chi'(-1) = 1$, $\chi'(2) = \chi'(3) = -1$. How do I see the ...

**0**

votes

**0**answers

157 views

### How to compute this sum over numbers?

When I was doing some task of analytic number theory I was stuck on computing this sum
$$S:=\frac{1}{L} \sum_{q \in \mathcal{Q}} \phi(q) \overline{a}^{\frac{1}{2}},$$
where $\overline{a}$ is the ...

**34**

votes

**1**answer

2k views

### The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?

Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only ...

**4**

votes

**1**answer

160 views

### Reference to a variant of Abel's summation formula

Edit. A stronger version of the formula is true (details follow).
Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, $(\lambda_n)_{n \ge 1}$ a nondecreasing sequence of positive reals such that $...

**21**

votes

**2**answers

1k views

### A conjecture based on Wilson's theorem

Definitions:
Lagrange's theorem implies that for each prime $p$, the factors of $(p − 1)!$ can be arranged in unequal pairs, with the exception of $±1$, where the product of each pair $≡ 1 \pmod p$. ...

**9**

votes

**2**answers

550 views

### Fourier transform of the critical line of zeta?

This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here.
Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...

**3**

votes

**0**answers

144 views

### Almost primes in short intervals

Define an integer $n$ to be a $k$-almost prime if it has at most $k$ distinct primes factors. A detecting function for the set of such numbers is the generalized von Mangoldt function given by $\...

**1**

vote

**1**answer

120 views

### Upper and lower bounds for $|L(1+it,\chi)|$ for complex primitive character $\chi$?

I would guess this is some standard fact related to the zero-free region. But cannot find it in the textbooks I read.

**1**

vote

**1**answer

182 views

### Least simultaneous quadratic non-residue

Suppose $p,q$ are distinct primes with least quadratic non-residues $n_p$ and $n_q$ respectively. Can one bound the least $n$ for which $\left(\frac{n}{p}\right)=\left(\frac{n}{q}\right)=-1$ in terms ...

**0**

votes

**0**answers

81 views

### Probability distribution associated with total divisors of an integer

Is there a generalization to https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem which gives distribution function for $$\omega(n)=\big|\{d\in\mathsf{prime}:d|n\}\big|$$ where $\mathsf{prime}$...

**0**

votes

**1**answer

129 views

### Linear forms that avoid numbers with lot of factors

Is following true?
For every given $c>0$ there is an $n_c>0$ such that for every $n>n_c$ there are integers $n<a,b<2n$ such that there are two positive integers $\frac{n}{2(\log n)^c}\...

**4**

votes

**1**answer

205 views

### Goldbach for certain classes of $n$

Asked on MSE without response here.
$\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$.
The Wiki article on the Goldbach conjecture states that
In 1975, ...

**0**

votes

**1**answer

165 views

### On the number of divisors in a given range

Given $\alpha\in\Bbb N$, can there be more than $(\log N)^4$ divisors (composites allowed) of $N$ in $\big[\frac\alpha2,\alpha\big]$ when $\sqrt N\in\big[\frac\alpha2,\alpha\big]$?
What is the ...

**1**

vote

**0**answers

98 views

### Integrating a series expansion of $\mbox{frac}(x)\lfloor x\rfloor$ coming from Fourier series of sawtooth function

Let me preface this question by saying that I am not exactly sure it counts as research level. It is crossposted on mathstackexchange: http://math.stackexchange.com/questions/1519724/integrating-a-...

**1**

vote

**0**answers

91 views

### Bounds on the number of zeros of a quadratic form

Let $Q(x_1, \dots, x_n)$ be a non-degenerate indefinite quadratic form with integer coefficients. Let $N(Q,T)$ be the set of vectors $x=(x_1, \dots, x_n) \in {\mathbb Z}^n$ such that $|x|<T$ and $Q(...

**3**

votes

**2**answers

284 views

### Primes $P_{2n-1}$ that are $2$ mod $3$

Are infinitely many primes $P_{2n-1}$ expressible as $3k-1$?
The primes $P_{2n-1}$ are every other prime beginning with $2$: $2,5,11,17,23,31,\cdots$. The first few are of the form $3k-1$, but $31$ ...

**7**

votes

**1**answer

204 views

### Hecke characters and Conductors

Motivation: Let $\ell$ be an odd prime. There is a conductor-preserving correspondence between primitive Dirichlet characters of order $\ell$
and cyclic, degree $\ell$ number fields $K/\mathbb{Q}$.
...

**12**

votes

**0**answers

286 views

### No Siegel-Landau zeros for $\mathrm{GL}(n)$

The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact:
There ...

**1**

vote

**0**answers

201 views

### Is the difference of these two real-rooted functions real-rooted?

During our on-going search of approximations to the Riemann $\Xi(z)$ function, we discovered a family of functions $W_n(z)$ as shown in (1).
Our final goal is to prove that:
Proposition 1: $W_{n}(z)...

**5**

votes

**1**answer

217 views

### The number of integral solutions to $x^2+y^2-az^2=0$

I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation
$$ x^2+y^...

**4**

votes

**1**answer

286 views

### Show the upper bound of cardinality of $A$ is $C\sqrt{n\log{n}}$

$\forall l,m,n\in \Bbb{Z_+}$, let $A:=\{k: m+1\leq k\leq m+n\text{ and }l-k^2\text{ is a square number}\}$.
Please prove that the number of elements in $A$ is not more than $C\sqrt{n\log n}$, where $...

**0**

votes

**0**answers

47 views

### Estimates related to sum over a primes from a fixed, possibly sparse set

Let $E$ be a fixed infinite sequence of primes such that $\sum_{p \in E} \frac{1}{p} = \infty$. Assume that $\sigma > 1$ depends on some parameter $x \rightarrow \infty$ in such a way that $\sigma ...

**-5**

votes

**1**answer

243 views

### How to prove twin prime conjecture or Goldbach conjecture if we assume prime distribution is completely random? [closed]

If we assume that prime number distribution is COMPLETELY random (subject to 1/log(x) restriction), can we prove twin prime conjecture or Goldbach conjecture ?
My feeling is that, this will be ...

**4**

votes

**0**answers

88 views

### The probability distribution of LCM of uniformly distributed integers in $\{1,\ldots,n\}$

In the recent paper by Fernandez and Fernandez here on ArXiv, the following formula which was first proved by Diaconis and Erdos appears, on page 2.
For $0<t\leq 1$ the distribution of the lcm of ...

**3**

votes

**2**answers

155 views

### What is the asymptotic growth rate of the product of divisor function up to n [duplicate]

This was asked in mathstackexchange (see here) but was not satisfactorily answered beyond my basic observations.
Let $\tau(k)$ be the number of divisors of the positive integer $k$.
How does $f(n)=\...

**2**

votes

**2**answers

130 views

### How large can a subset of $\{1,\ldots,N\}$ be if all pairwise LCMs of its elements are lower bounded?

Consider the set $\{1,2,\ldots,N\}$. Let $LCM(a,a')$ denote the lowest common multiple of the integers $a,a'.$
We say that $A\subset \{1,2,\ldots,N\}$ is $M-$good if $LCM(a,a')\geq M,$ for all $a\...

**4**

votes

**1**answer

218 views

### Estimates of a sum involving both the Möbius function and Mertens function

I want to ask on the estimates of the sum
$$ \sum_{n=1}^{\infty} \mu(n)M\Big(\frac{x}{n}\Big)=\frac{1}{2\pi i }\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{x^s}{s\zeta(s)^2}ds.$$
It seems that the ...

**8**

votes

**1**answer

268 views

### Primes in arithmetic progression with a moduli equal to a power of 2

I am currently looking for a result stronger than Siegel-Walfisz theorem, which gives an upper bound on the error term $|\pi(x,a,b)-\frac{\pi(x)}{\phi(a)}|$ for particular $a$.
The Siegel Walfisz is ...

**12**

votes

**1**answer

456 views

### Two Vinogradovs? Is one the son of the other? [closed]

Forgive me for my ignorance, but I'm very surprised to learn that there are two Vinogradovs, both famous in the field of analytic number theory. Guessing from their names and the Russian naming ...

**4**

votes

**1**answer

326 views

### Upper bound for the first Hardy-Littlewood conjecture

About the Hardy-Littlewood conjecture by Terence Tao:
Conjecture 2 (Prime tuples conjecture, quantitative form) Let ${k_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed ...

**2**

votes

**0**answers

332 views

### How big can a set of integers be if all pairs have bounded gcd

In this recent MO question, it was shown that the maximal cardinality of a subset $A(M,N)$ of $[1,N]$ where the pairwise GCD's of all set elements are upper bounded by $M,$ with $M^2\leq N$ has size ...

**2**

votes

**1**answer

319 views

### infinite product of (1-1/(p+1)) over a density 0 set of primes

Let $S$ be a density zero set rational primes, in the concrete situation
$\#\{p<X,p\in S\}=\mathcal{O}(x/(\log x)^{3/2-\delta})$ for all $\delta>0$.
Then can $\prod_{p\in S}(1-\frac{1}{p+1})$ ...

**4**

votes

**1**answer

396 views

### About factorization in Zhang's proof of weak Twin Prime conjecture

Why does it need to firstly factorize the number n into two factors q and r( Lemma 4 in the paper,see the following)? What's the motivation. What if it doesn't do this factorization?

**0**

votes

**1**answer

110 views

### Sum of digits of a power [closed]

Are there any explicit formula for a sum of digits for a power in the given base? A problem to be specific: find a sum of digits for a number $2^{100}$ in the system with a base 5. In the system with ...

**0**

votes

**2**answers

326 views

### Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa?

I know that better and better bounds of the Chebyshev Theta and Psi functions are implied by knowing that the first (insert large number here) zeta zeroes lie on the Critical Line. These bounds, ...

**2**

votes

**0**answers

154 views

### On sets of coprime numbers

We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$
Denote by $...

**10**

votes

**1**answer

497 views

### How much can an Eisenstein series be truncated?

For ease of exposition, I will stick to the simplest case: consider the Eisenstein series for $SL_2(\bf R)$
$$E(z,s)=\sum_{\gamma\in P_{\bf Z}\backslash SL_2(\bf Z)}\text{Im}(\gamma z)^s=\sum_{(c,d)\...

**1**

vote

**0**answers

34 views

### On collision-free sets of residues of integers

Denote $\pi_m$ to be collection of primes in $[2^{m},2^{m+1}]$.
Denote $\psi_{n,m}$ to be collection of integers of form $ab$ where $a\in\pi_n$, $b\in\pi_m$.
Given $n,t\in\Bbb N$, what is the ...

**3**

votes

**2**answers

269 views

### multiplicative functions of powers

Suppose I have a multiplicative function $f(n),$ and I want to understand the behavior of
$$
\sum_{n<x} f(n^k),
$$ for some integer $k.$ This seems like it should be easy (since the Dirichlet ...

**4**

votes

**1**answer

281 views

### Fourier expansion of automorphic forms

we know that for $r \in \{1,2,3,4\},$ $\lambda_{Sym^rf}$ is an automorphic form (here $f$ is a modular form for the full modular group) and this fact is conjectured for $r\geq 5$ by Langlands and ...

**22**

votes

**1**answer

689 views

### How big can a set of integers be if all pairs have small gcd?

Suppose $A\subset[1,N]$ is a set of integers. If for any distinct $a,b\in A$ we have $(a,b)\leq M$ then how big can $|A|$ be?
If $M=1$ then $|A|$ is at most $\pi(N)$ since the map $a\mapsto P_+(a)$ (...

**5**

votes

**0**answers

171 views

### Effective bound of $L(1,\chi)$

Let $d$ be a fundamental discriminant and let $\chi$ be the associated primitive real character of modulus $\vert d \vert$. Assuming GRH, Littlewood proved that as $\vert d \vert$ grows large,
$$L(1, ...

**5**

votes

**0**answers

101 views

### Methods of variational calculus in analytic number theory

What methods of calculus of variations have been used in analytic number theory?
I mean do Hamilton-Jacobi theory of PDE found usage in analytic number theory, which raises yet another question has ...