**2**

votes

**0**answers

207 views

### On a sequence of L-functions having same zeros in critical strip and GRH

I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ?
Let's ...

**0**

votes

**0**answers

103 views

### Abel summation formula versus Perron's formula to bound a partial sum

Taking $\chi$ a primitive character, with Abel summation it is easy to show that for $\epsilon >0$, there is a constant $M$ such that for all $x$ we have :
$$|\sum_{n<x} \frac{\chi(n)}{n^{\...

**15**

votes

**2**answers

2k views

### Could this unexpected bias in the distribution of consecutive primes have any impact on the security of encryption algorithms?

In a recent paper a quite unexpected result about a new pattern in prime numbers emerged:
Unexpected biases in the distribution of consecutive primesby Oliver, R. J. L.; Soundararajan, K. (Submitted ...

**7**

votes

**1**answer

110 views

### Hilbert symbol averages

Let me call a pair of integers $a, b$ acceptable if the equation $ax^2 + by^2 = z^2$ has a non-trivial rational solution. Theorem 4.5.4 of Cojocaru-Murty's book on Sieves says that the number of ...

**22**

votes

**1**answer

468 views

### Why is there a Parity Problem in Sieve Theory and not a Mod p problem for any other p?

The "parity problem" in sieve theory, so far as I understand it, is the fact that sieves can't distinguish between primes and $2$-almost primes, numbers with exactly two prime factors, and will always ...

**4**

votes

**0**answers

210 views

### Asymptotic estimate for a random model of primes

Question
Let
$$
\pi_{rm_c}(x) = \sum_{ \substack{ {n\leq x}\\{(n+a,P(\sqrt{n}))=1}}} 1-1,
$$
where $P(x)$ is the product of all primes less or equal to $x$ and $a$ is a random integer constrained to ...

**4**

votes

**0**answers

83 views

### Large Gaps Between Almost Primes

What is the best lower bound for the longest interval contained in $[1,x]$ free of primes and products of two primes? In other words I am asking for the best lower bounds in a variant of the ...

**1**

vote

**0**answers

63 views

### Behavior of partial Euler product in the critical strip (with Dirichlet Character)

Consider a primitive Dirichlet Character $\chi$ (non principal) and the partial Euler product attached to the L-function $L(\chi,s)$ ($p_i$ are the prime numbers) :
$$P(\chi,N)=\prod_{i=1}^{N} \frac{...

**5**

votes

**1**answer

335 views

### About the logarithmic derivative of the Riemann zeta function

Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that $$\frac{\zeta'}{\zeta}\left(s\right)=\sum_{\left|t-\gamma\...

**7**

votes

**1**answer

319 views

### The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?

I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a ...

**14**

votes

**1**answer

477 views

### Growth of $\zeta_{\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]}(2)$

Let $K_n$ be the field $\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]$ (the real subfield of the cyclotomic field $\mathbf Q[e^{\frac{i\pi}{2^{n+1}}}]$).
Is there anything known about the growth of the ...

**15**

votes

**1**answer

370 views

### Is the number of representations as the sum of two elements of a polynomial sequence always small?

Let $f(x) \in \mathbb{Z}[x]$ be a degree $d>1$ polynomial with integer coefficients. Define
$$r(n) := | \{x,y \in \mathbb{Z} : f(x)+f(y) = n \}|. $$
My question is:
Is it true that $r(...

**1**

vote

**0**answers

62 views

### summability and analytic continuation

Let $d_n=LCM(1,\cdots,n)$. It is well-known that $d_n=e^{\Psi(n)}$ where $\Psi$ est the second Chebyshev function. One knows that $\Psi(x)=\sum_{k\le x}\Lambda(k)$ where $\Lambda$ is the Von Mangold ...

**19**

votes

**3**answers

512 views

### Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...

**1**

vote

**1**answer

183 views

### Is it proved that for every integer $p>0$ there exists an integer $k>0$ such that every integer $n>0$ can be expressed as $j_1^p+\dots+j_k^p$?

It has been shown, by elementary methods, that every positive integer can be expressed as the sum of $4$ squares. This type of result has been proven for many different powers $p$, for example, when $...

**1**

vote

**0**answers

106 views

### Asymptotic of a sequence related to $LCM(1,\cdots,n)$ [closed]

Let $d_n=LCM(1,2,\cdots, n)$ and $u_n$ be a sequence such that $u_n=o(d_n)$. Some testing in maple suggests the following asymptotic:
$$\lim_{n\to+\infty}\frac{\log(\sum_{k=0}^n(d_k+u_k)\binom{n}k)}{...

**4**

votes

**2**answers

154 views

### Evolution of partial sum of a sequence of induced Dirichlet characters

Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$.
Lets consider the sequence of induced characters $\chi^{P_N} $ obtained ...

**13**

votes

**1**answer

362 views

### Counting lattice points inside a three-dimensional ellipsoid

I want to answer the following simple question:
Given a three-dimensional ellipsoid defined by $Q(x, y, z) \leq Z$ for a positive definite quadratic form $Q$, how many lattice points in $\mathbb{Z}...

**16**

votes

**2**answers

853 views

### Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields?

Suppose I am given a machine that gives me the coefficients $a_1$, $a_2$, $a_3$, ... of a Dirichlet series
$$\sum_1^{\infty} \frac{a_n}{n^s} $$
and assume that I know that this Dirichlet series is the ...

**1**

vote

**0**answers

34 views

### Best values in the estimate of Vinogradov-Korobov

Let $C(N)=\sum_{1<n\le N}{n^{-it}}$.
Vinogradov- Korobov estimate is
$$|C(N)| \le KN\exp\left(-\gamma \frac{\ln^3 N}{\ln^2 t}\right).$$
What are the best values of $K$ and $\gamma$ ? I have ...

**3**

votes

**1**answer

186 views

### On partial sum of non-primitive Dirichlet characters

Consider a Dirichlet character, $\chi(n)$, and the partial sum :
$$S(\chi,x)=\bigg |\sum_{n=1}^{x} \chi(n)\bigg|$$
There are many works to bound this sum when $\chi$ is a primitive character, but ...

**9**

votes

**0**answers

272 views

### Independence between the number of prime factors of $n$ and $n+2$

I am interested in having an upper bound for the cardinality of
$\#\left\{n\leq x\,:\quad\omega(n)=k, \omega(n+2)=\ell\right\}$
for $k,\ell\geq 1$,
where $\omega(n)=\sum_{p\vert n}1$ counts the ...

**2**

votes

**2**answers

174 views

### A lower bound involving the divisor function and primorial numbers

It is known that
$\lim$ $\sup \dfrac{\sigma(N_k)}{e^{\gamma}N_k \log\log N_k}=\frac{6}{\pi^2}$,
where $\gamma$ is the Euler-Mascheroni constant and $N_k$ is the $k-th$ primorial number.
But is it ...

**16**

votes

**1**answer

716 views

### A converse of the abc conjecture?

Let ${\rm rad}(n)$ denote the radical of a positive
integer $n$, i.e. the product of its distinct prime divisors.
Given positive integers $a$ and $b$, the triple $(a,b,a+b)$ is
called an abc triple if ...

**0**

votes

**0**answers

68 views

### the shifted convolution sums and the sub convexity problem for l functions

in the paper of gergely harcos, an additive problem in the fourier coefficients of cusp forms, a bound for the shifted convolution sums for hecke eigenvalues was explicited and i thought that his ...

**1**

vote

**0**answers

95 views

### On exponential sum weighted with von-Mangoldt function

Suppose we have $\alpha \in \mathbb{R}$ such that $|\alpha - a/q| < 1/q^2$,
where $(a,q)=1$. Then we know that the exponential sum
$$
S(\alpha) = \sum_{1 \leq n \leq X} \Lambda(n) e(n \alpha)
$$
...

**22**

votes

**2**answers

794 views

### Elementary congruences and L-functions

In a recent article, Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element
$$ S = \prod_{k=1}^{\frac{N-1}2} k^k $$
...

**6**

votes

**3**answers

328 views

### Asymptotics for the number of abelian groups of order at most $x.$

The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see http://oeis.org/A000688), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for $a(...

**3**

votes

**1**answer

195 views

### A decreasing sequence involving the divisor function?

Define $N_k \geq 6$ to be the $k-th$ primorial number and let $\sigma(n)$ be the divisor function.
It seems that $u_k = \dfrac{\sigma(N_k)}{N_k \log\log N_k}$ is a decreasing function ?
By ...

**4**

votes

**1**answer

112 views

### Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$

Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...

**5**

votes

**1**answer

241 views

### For a sufficiently large $a$, are there distinct (mod $a$) integers such that all powers up to the $n$-th are “close” modulo $a$?

Given $n\in\Bbb N$ is there an $a_n\in\Bbb N$ such that for every $a>a_n$ there are two distinct integers $0<b<c<a$ such that $b^i\bmod a,c^i\bmod a\in(\sqrt a,\sqrt a\log a)$ for every $i\...

**6**

votes

**3**answers

555 views

### If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal

On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark:
One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and ...

**7**

votes

**2**answers

399 views

### Asymptotics of product of Euler's totient function (A001088)?

Conjecture:
\begin{align}
\lim_{n\to \infty } \, \frac{\left(\prod _{k=1}^n \phi (k)\right){}^{1/n}}{n}\sim 0.2059\text{...}
\end{align}
The numerical result from 100000 terms is:
My questions ...

**12**

votes

**1**answer

527 views

### Does the sum $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converge?

Prove, if possible in an elementary way, that $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converges/diverges, where $p_n$ denotes the $n^{\textrm{th}}$ prime.

**1**

vote

**1**answer

297 views

### Asymptotics of “ugly” function elucidate Goldbach's conjecture?

Question
We now define the following "ugly" function:
$$ A_c(s,r,n,m) =
\begin{cases}
1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise}
\end{cases}
$$
How does the "ugly"...

**11**

votes

**1**answer

1k views

### What did Euler do with multiple zeta values?

When reading about multiple zeta values, I often find the claim that the case of length two
$$
\zeta(s_1, s_2)=\sum_{n>m \geq 1} \frac{1}{n^{s_1}m^{s_2}}, \qquad s_1 \geq 2, \quad s_2 \geq 1
$$
...

**8**

votes

**1**answer

304 views

### Riemann zeta function: pair correlations vs. neighbor spacings

Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the ...

**10**

votes

**1**answer

477 views

### Why do the Maynard-Tao weights work so well?

I am looking for an intuitive reason for why the Maynard-Tao weights work well to capture many primes of the form $n+h_1, \ldots , n+h_k$, where $(h_1, \ldots , h_k)$ is any admissible $k$-tuple.
For ...

**7**

votes

**1**answer

443 views

### Lattice points near a curve

Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve in the plane. Does it generalize to a bound for the count of lattice points near (say within a distance ...

**1**

vote

**1**answer

189 views

### Convergence of a double sum involving prime numbers

This has been moved from math.stackexchange;
I am attempting to prove/disprove convergence of the following sum
$$ \lim_{n \to \infty} \frac{1}{n} \sum_{p \leq n} \sum_{k=0}^\infty \ln p \left\{\frac{...

**0**

votes

**0**answers

78 views

### The growth rate of almost periods for almost periodic function

A subset $A \subset \mathbb{R}^2$ is relative dense if there exists $L>0$ such that for every $p\in \mathbb{R}^2$ there exists $p' \in A$ such that $|p-p'|<L.$
A continuous function $f : \...

**3**

votes

**1**answer

238 views

### An exponential sum over squares

I have the following exponential sum:
$\sum _{M<n\leq N}e\left (x/n^2\right )=\sum f(n),$
say, where $M$ and $N$ are something like $x^{1/4}$ and $x^{1/2}$.
My question is basically, how do I ...

**3**

votes

**1**answer

533 views

### Euler product for sum of multiplicative function times log

(Cross-posted from StackExchange). Let $g$ be a multiplicative function which satisfies $0 \le g(p) \ll 1/p$ and
$$ \sum_{p\le x} g(p) = \log \log x + C + O((\log x)^{-10}). $$
Iwaniec and ...

**2**

votes

**0**answers

181 views

### Short Kloosterman sum

Let $K(m,n,c)=\sum_{p\in A}\exp(2\pi i(pm+np^{-1}))$, where $m,n\in\mathbb{Z}$, $c\in\mathbb{N}$, and $A\subset (\mathbb{Z}/c\mathbb{Z})^\times$.
Does anybody know any nontrivial bound for this kind ...

**1**

vote

**0**answers

218 views

### Is this a proof of the Hardy-Littlewood inequality? [closed]

V.V. Miasoyedov posted a paper to the arXiv claiming a proof of the Hardy-Littlewood conjecture $\pi(x+y) \le \pi(x)+\pi(y)$. It seems a bit off, and not only because the conjecture is widely believed ...

**4**

votes

**1**answer

177 views

### Improvement of a bound on divisor distributions from “Divisors” (Hall and Tenenbaum)?

In the classic text referred to in the title of this question, the bound
$$
H(x,y,2y) \ll \frac{x}{(\log y)^{\delta}\sqrt{\log \log y}},\quad (3\leq y\leq \sqrt{x})
$$
is given, where $\delta=1-\frac{...

**3**

votes

**0**answers

117 views

### exponential sum of primes

Fix $\alpha \notin \mathbb{Q}$. I would like to know a reference that shows $$\mathbb{E}_{n\leq N} \Lambda(n) e^{2\pi i \alpha n} \to 0,$$ as $N$ tends to infinity.
I am familiar with Vinagradov's ...

**3**

votes

**0**answers

255 views

### Colmez conjecture and endomorphism rings

It is given by the Colmez conjecture that if $A$ has a CM-type $(k, \phi)$ and $\text{End}(A) = \mathcal{O_k}$ $$\displaystyle h_{\text{fal}}(A) = \sum_{\text{irr} \hspace{1 mm}\rho: G(k^{\text{nor}}/\...

**4**

votes

**1**answer

180 views

### Well-spacing of the roots of a quadratic congruence

On pages 956-957 of this paper, it is established that for any two $v_1, v_2$ satisfying $v_1^2 + 1 \equiv 0\operatorname{(mod} d_1), v_2^2 + 1\equiv 0\operatorname{(mod} d_2)$, $$\left\lVert \frac{...

**8**

votes

**1**answer

398 views

### On (a generalization of) the Gauss Circle Problem

Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...