# Questions tagged [analytic-number-theory]

A beautiful blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.

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23 views

### What is meant by singular hyperplane of $c(w, \cdot)$? (global intertwining operator related to Eisenstein series)

Let $P_0$ be a minimal $\mathbb{Q}$-parabolic subgroup of $G$, a semisimple linear algebraic group over $\mathbb{Q}$. Then $P_0 = M_0 N_0$ where $M_0$ is a Levi subgroup of $P_0$. Let $E^G_{P_0}$ be ...

**1**

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75 views

### Primes with given Hamming weight

If I understand correctly, in the following MO-thread
Are There Primes of Every Hamming Weight?
two users of the site claim that it has been already proven that, for every sufficiently large $n \in \...

**4**

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**2**answers

460 views

### Estimate related to the Möbius function

I need to know, or at least have a good bound for, the asymptotic behaviour on $x$ of amount of integers less or equal than $x$ that are square free and with exactly $k$ primes on its decomposition. ...

**1**

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187 views

### An implication of the Zagier et al result on the hyperbolicity of Jensen polynomials for the Riemann zeta function?

In their paper recently published in the PNAS, Zagier et al demonstrated that
The Jensen polynomials $J_{\alpha}^{d,n}(X)$ of the Riemann zeta function of degree $d$ and shift $n$ are hyperbolic for ...

**4**

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**3**answers

304 views

### Positive proportion of logarithmic gaps between consecutive primes

For $x, \lambda > 0$, define
$$S_\lambda(x) := \#\{p_{n+1} \leq x : p_{n+1} - p_n \geq \lambda \log x\} ,$$
where $p_n$ is the $n$th prime number. It is known [1] that an uniform version of the ...

**7**

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**0**answers

164 views

### Primes $p\in(n,2n)$ with $(\frac{-n}p)=-1$

Bertrand's postulate proved by Chebyshev states that for any $x>1$ there is a prime $p$ in the interval $(x,2x)$. In 2012 I considered some refinements of this by imposing additional requirement ...

**4**

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**0**answers

256 views

### Question on an application of Langlands' result on the constant term of Eisenstein series (Is this a typo?)

I would like to understand an argument in https://link.springer.com/content/pdf/10.1007/BF01393904.pdf, which uses Langlands' result on the constant term of Eisenstein series, but I'm not getting it ...

**13**

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**1**answer

519 views

### Why do Maynard-Tao weights succeed?

I'm attempting to understand why the Maynard-Tao weights are successful in proving bounded gaps between primes, but the GPY weights are not.
These two posts do an excellent job in giving an overview ...

**2**

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**1**answer

99 views

### The minimal volume of the intersection of two $\mathscr{l}_1$-ball in high dimension

We define
$$B_1(r,c) = \{x\in\mathbb{R}^d : \Vert{}x-c\Vert_1 \le r\}$$
Now for arbitrary constant $r \ge s > 0$, given constant $\epsilon \in (r-s,r+s)$, considering $v \in \mathbb{R}^d$ such ...

**6**

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**1**answer

263 views

### Rate of convergence of the prime zeta function P(2)

For an application in statistical group theory, we need explicit upper and lower bounds that an expert in number theory (I am not one) may know how to prove.
Question 1: What are "good" bounds $f_1(x)...

**2**

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**1**answer

276 views

### Conditional bound on RH for $\Re\left(\sum_{p\leq\sqrt{x}}\frac{(1/2)}{p^{1+2it}}\right)$

I would like to prove that
Assume RH. Let $T$ large, $2\leq x \leq T^2$ and $T\leq t \leq 2T$, then
$$
\log|\zeta(1/2+it))|\leq \Re\left(\sum_{p\leq x}\frac{1}{p^{1/2+1/\log x+it}}\frac{\log(x/p)}...

**0**

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60 views

### General Term Formula for Sequences

Let $k_1,k_2,\cdots,k_n,\cdots$ be a sequence of known positive numbers. Define
$$
a_1:=k_1,\\
a_2:=C_2^2k_2+C_2^1k_1a_1,\\
a_3:=C_3^3k_3+C_3^2k_2a_1+C_3^1k_1a_2,\\
a_4:=C_4^4k_4+C_4^3k_3a_1+C_4^...

**1**

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**1**answer

117 views

### Complex integral of logarithmic derivative of $\zeta$

I want to prove that for any $x\geq 2$ we have
$$
\begin{split}
-\frac{\zeta^{\prime}}{\zeta}(s)&=\sum_{n\leq x}\frac{\Lambda(n)}{n^s}\frac{\log(x/n)}{\log x}+\frac{1}{\log x}\left(\frac{\zeta^{\...

**2**

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**0**answers

158 views

### Trying to understand why Eisenstein series is well defined

I am struggling to see why Eisenstein series is well defined, and I would greatly appreciate clarification.
Let
$$
E(x, \lambda) = \sum_{\delta \in P(\mathbb{Q}) \backslash G(\mathbb{Q}) }
e^{\...

**1**

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**1**answer

112 views

### Upper bound of signed exponential sums

I am wondering whether I can get the upper bound in closed form of
$$\sum_{n=1}^N(\alpha \exp(j2\pi n/N)) \text{ where } \alpha = +1\text{ or }-1 \text{ and } j^2=-1.$$
If alpha is just positive one,...