# Tagged Questions

**4**

votes

**1**answer

112 views

### What explains the asymptotic and the pattern in this sequence related to Riemann zeta zeros?

As the starting point for my experiment I assumed that the imaginary parts of the Riemann zeta zeros are of the form:
$$\Im \{ \rho_n \} = \frac{2\pi}{\log x_n}$$
where $x_n$ is unknown. Therefore I ...

**18**

votes

**3**answers

387 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, ...

**14**

votes

**3**answers

4k views

### Are the nontrivial zeros of the Riemann zeta simple?

A few years ago, I found on arXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann zeta function were all ...

**7**

votes

**1**answer

344 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 ...

**17**

votes

**0**answers

332 views

+50

### Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all
$x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...

**4**

votes

**2**answers

260 views

### The best possible density in Hilbert's Irreducibility Theorem

Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen ...

**3**

votes

**0**answers

185 views

### An explicit formula for $\zeta(2m+1)$ with good convergence

The question: Is the following formula known?
$$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!} \sum\limits_{v=k}^m ...

**2**

votes

**7**answers

2k views

### Bertrand postulate

I believe there was an old conjecture that there's always a prime number between N and 2N.
What's the history and how is this ...

**20**

votes

**1**answer

1k views

### How many primes can there be in a short interval?

Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$.
What is
$$
\limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} ...

**1**

vote

**1**answer

63 views

### Asymptotic for zeroes of $L(s,\chi)$ in a disk $|s|<R$

In 'Remarks on Weil's quadratic functional..' p.191 Bombieri claims any given $L$-function $L(s,\chi)$ has at least
$$\big(\frac{1}{\pi}+o(1)\big)R\log R$$
zeroes in a disk $|s|<R$. Is there a ...

**15**

votes

**2**answers

2k views

### How did Riemann calculate the first few non-trivial zeros of the zeta-function?

Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi ...

**0**

votes

**0**answers

149 views

### Asymptotic value of sum over Möbius function

Consider the sum
$$
S(x)=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}|\mu(d)|,
$$
where $P(x)$ is the product of all primes less than or equal to $x$, and $\mu(d)$ is the Möbius function.
...

**1**

vote

**0**answers

75 views

### Intersection of modular polynomial roots

Let $l,l'$ and $p$ be three distinct prime numbers and $\Phi_k(X,Y)$ is $k$-th modular polynomial defined over $GF(p)$. Suppose that we know $\Phi_l(X,j)$ and $\Phi_{l'}(X,j)$ have two roots. Is this ...

**2**

votes

**1**answer

206 views

### Weil Conjectures Analog for Multivariate Zeta Functions

We know that the Riemann zeta function can be generalized to multivariate zeta functions.
Is there a multivariate analog of the Weil conjectures?

**3**

votes

**0**answers

111 views

### How to show that $h(-D)\geq \displaystyle\sum_{a\in A_1\\, 1\leq a\leq{\frac{\sqrt D}{2}}} 1$?

Here $A_1=\{u;p|u\Longrightarrow \chi(p)=1\}$ with $\chi$ a real quadratic character and $h(-D)$ the class number of the imaginary quadratic field of the fundamental discriminant. This problem occurs ...

**1**

vote

**0**answers

253 views

### Controlling residue class sizes

$\forall k>10^2$ is there $m_k$ such that at infinite primes $q>m_k$ $\exists$ pairwise coprime $a,b,c$ such that
$$(1)\quad q^{\frac14+\frac1k}<a,b,|a-b|<q^{\frac14+\frac2k} < ...

**2**

votes

**1**answer

106 views

### Maass form properties and their fourier coefficients

Some Maass form can be written ($K_{iR}$ is the K-Bessel function):
$$f(x+iy)=\sum_{n \ne 0}^{\infty} a_n \sqrt{y} \;K_{iR}(2\pi |n| y) \; e^{2 i\pi nx}$$
with the $a_n$ multiplicative, but inversly ...

**6**

votes

**0**answers

132 views

### Mahler's theorem in the primes

Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. Denote by $A_F$ the area of the region
$$\displaystyle \{(x,y) \in \mathbb{R}^2 : |F(x,y)| \leq 1\}.$$
It ...

**12**

votes

**3**answers

891 views

### At what point would an elementary generalization of Bertrand's Postulate be interesting?

I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$.
At what point would an improvement on Nagura's result be interesting? ...

**18**

votes

**1**answer

298 views

### On random divisor sums modulo $2^k$

Let $k,n,\ell$ be positive integers with $k,n\ge 2$ and $0\le \ell \le k-1$. For each integer $2\le j \le n$, choose a divisor $d_j$ of $j$, uniformly at random from the divisors of $j$. We denote by ...

**2**

votes

**1**answer

267 views

### Some identities with the Riemann-Hurwitz zeta function

The only definition that I have ever seen of this Riemann-Hurtwitz zeta-function is this,
For $0 < a \leq 1$ we have the identity
$$ \zeta(z, a) = \frac{2 \Gamma(1 - z)}{(2 \pi)^{1-z}} \left[\sin ...

**3**

votes

**0**answers

98 views

### Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...

**-1**

votes

**1**answer

124 views

### Find the values of $n$ that satisfy this inequality involving a product over prime numbers [closed]

Inequality
What values of $n$ satisfy the following inequality?
$$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$
$p$ are prime numbers and the notation $p_i$ indicates the $i$th ...

**4**

votes

**1**answer

235 views

### The values of $n$ which satisfy an inequality about prime numbers

For which values of $n$ does the following inequality hold for?
$$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right)$$
$p$ are prime numbers and the notation $p_i$ indicates the ...

**4**

votes

**0**answers

200 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 ...

**7**

votes

**0**answers

129 views

### k-Almost Primes in short intervals

According to this question every interval $[x, x + x^{0.45}]$ contains a product of two primes, and this has been improved further slightly. Are there better results available for $k$-almost primes? ...

**22**

votes

**1**answer

459 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 ...

**2**

votes

**0**answers

189 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 ...

**35**

votes

**3**answers

3k views

### Why such an interest in studying prime gaps?

Prime gaps studies seems to be one of the most fertile topics in analytic number theory, for long and in lots of directions :
lower bounds (recent works by Maynard, Tao et al. [1])
upper bounds ...

**3**

votes

**1**answer

299 views

### Relation between the binary Goldbach problem and binary version of Mobius sum

What I want to ask is about the structure of the Goldbach function that defined by
$$ R(x)=\#\{ p \mid x-p \in \mathbb{P} , \ p\leq x/2\}$$
for $x\in 2\mathbb{N}$, where $\mathbb{P}$ is the set of ...

**0**

votes

**0**answers

95 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} ...

**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

214 views

### Eisenstein series over a definite division algebra

Let $D$ be the definite quaternion division algebra over $\mathbb{Q}$. $\mathcal{O}$ is a maximal order inside $D$, let's fix $\mathcal{O}$ to be the Hurwitz quaternion. Let ...

**8**

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 ...

**12**

votes

**1**answer

347 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 ...

**10**

votes

**2**answers

2k views

### Why is $\frac{\sqrt{6}}{32}(29 + \sqrt{145}) \approx \pi$ ?

Apologies in advance if this is a stupid question; also, disclaimer: this is purely for fun; but:
Why is $\frac{\sqrt{6}}{32}(29 + \sqrt{145})$ such a good approximation to $\pi$?
(Correct to 8 ...

**4**

votes

**0**answers

81 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 ...

**15**

votes

**1**answer

359 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 ...

**1**

vote

**0**answers

60 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} ...

**20**

votes

**1**answer

1k views

### Wrong asymptotics of OEIS A000607 (number of partitions of an integer in prime parts)?

Sequence A000607 in the Online Encyclopedia of Integer Sequences is the number of partitions of $n$ into prime parts. For example, there are $5$ partitions of $10$ into prime parts: $10 = 2 + 2 + 2 + ...

**15**

votes

**1**answer

821 views

### Tight prime bounds

This is a cross-post of this question on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the ...

**21**

votes

**4**answers

5k views

### How does Yitang Zhang use Cauchy's inequality and Theorem 2 to obtain the error term coming from the $S_2$ sum

I have been reading Yitang Zhang's paper now for one and a half weeks and also volunteered to give a popular talk on the paper next week at Stockholm University.
Today I found a detail in the proof ...

**22**

votes

**5**answers

1k views

### Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line?

The proof that $\Gamma(z)\pm \Gamma(1-z)$ only has zeros for $z \in \mathbb{R}$ or $z= \frac12 +i \mathbb{R}$ has been given here:
Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real ...

**12**

votes

**5**answers

4k views

### Upper bounds for the sum of primes up to $n$

Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ ...

**4**

votes

**2**answers

151 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 ...

**7**

votes

**1**answer

293 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 ...

**5**

votes

**1**answer

284 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 ...

**14**

votes

**1**answer

468 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 ...

**1**

vote

**0**answers

61 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 ...

**1**

vote

**1**answer

182 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 ...