**7**

votes

**1**answer

366 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

174 views

### Density of Sophie Germain $3\bmod 4$ primes

Is there any reason to expect the density of primes $p$ such that $2p+1$ is also a prime where $p=3\bmod4$ holds would be different from case of $p=1\bmod4$?
What if $2p+1$ is replaced by $2p-1$ and ...

**0**

votes

**0**answers

32 views

### On a count of certain number of primes in an interval

Fix a prime $p$, $\alpha\in(0,1)$ and $\beta\in(1,2)$ and let $\mathcal U$ be primes in $[p^\alpha,\beta p^\alpha]$ such that if $b\in\mathcal U$ and if $d$ is multiplicative inverse of $b$ in $\Bbb ...

**3**

votes

**1**answer

169 views

### Divisibility of Dirichlet L-functions

Let $k$ be an even integer and $p$ a prime number such that $p-1|k$.
Suppose that $p$ does not divide $L(1-k,\chi)$ and $L(1-k,\psi)$, where $\chi$ and $\psi$ are quadratic characters.
Can we deduce ...

**6**

votes

**1**answer

123 views

### Density of prime divisors of $a^n + b$

Suppose $a > 1, b \neq 0$ be two rational numbers. Is it known in general that the set of prime divisors of (the numerator of) $a^n + b$ has a positive relative density?

**3**

votes

**2**answers

208 views

### Relation of these two Dirichlet $L$-functions

Let $\chi$ and $\psi$ be two quadratic Dirichlet characters and let $L(s,\chi)$ and $L(s,\psi)$ their associated Dirichlet $L$-functions.
Is there a realtion between these two Dirichlet ...

**2**

votes

**2**answers

275 views

### How many integer solutions of $a^2+b^2=c^2+d^2+n$ are there?

Are there any references to study the integer solutions (existence and how many) of Diophantine equations like $a^2+b^2=c^2+d^2+2$, $a^2+b^2=c^2+d^2+3$, $a^2+b^2=c^2+d^2+5$...? Actually, I can prove ...

**2**

votes

**1**answer

112 views

### Clarification request on sign changes of Hecke eigenvalues

In their paper 'Sign changes of Hecke eigenvalues', Matomaki and Radziwill established in Lemma 6.2 the following result: There exists absolute positive constants $c$ and $\eta$ such that uniformly in ...

**3**

votes

**1**answer

104 views

### Moving from $\Re(s) = 1+\epsilon$ to $\Re(s) = \frac{1}{2}$ in the proof of the Weil-Guinand explicit formula

In all proofs of the Weil-Guinand explicit formula, there's this step (this is from Paul Garrett's notes):
Now consider this:
(1) $\frac{\zeta^\prime(s)}{\zeta(s)}$ has poles at $s=1$ and ...

**3**

votes

**0**answers

163 views

### Effective prime number theorem

The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...

**3**

votes

**0**answers

179 views

### Number of solutions to $x_1x_2=x_3x_4\bmod n$

In https://www.math.ksu.edu/~cochrane/research/xyuvmodp.pdf it is shown $x_1x_2=x_3x_4\bmod p$ where $p$ is a prime has $\frac{|\mathcal B|}p+O(\sqrt{|\mathcal B|}\log^2p)$ solutions ...

**0**

votes

**0**answers

44 views

### Density of set of primes which avoid given finite set of residues modulo powers of all primes

Let $k\in\mathbb{N}$, $k\ge2$ and $S\subseteq\mathbb{Z}$ be a finite set of integers. For every prime $p$ let $c_p$ be a number of invertible residue classes mod $p^k$ that contain some element of ...

**2**

votes

**0**answers

257 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)!}\cdot \sum\limits_{v=k}^m ...

**2**

votes

**0**answers

307 views

### Analytically continuing the limit of this series?

Main Question
I believe to following formula gives the right answer:
$$ \lim_{k \to \infty} \lim_{n h \to k} \left( \sum_{r=1}^n c_r f(hr) h\right) = \int_0^\infty f(x) \, dx \times ...

**3**

votes

**1**answer

184 views

### Non-vanishing of L-function of modular form

There is a theorem by Langlands and Shalika (link) that the L-function of a cuspidal automorphic representation does not vanish on the line $\mathrm{Re}( s)=1$ (in their normalization which might be ...

**1**

vote

**0**answers

197 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: ...

**0**

votes

**0**answers

48 views

### Estimating exponential sum of the form $\sum e( \alpha_1 f + \alpha_2 L)$, where $f$ is quadratic and $L$ is linear, on the minor arcs

Let $f(x_1, x_2, x_3)$ be a degree two homogeneous polynomial with coefficients in $\mathbb{Z}$. Let $L(x_1, x_2, x_3)$ be a linear homogeneous polynomial with with coefficients in $\mathbb{Z}$. Let ...

**3**

votes

**1**answer

153 views

### Are there L-functions of degree 1 that aren't Hecke L-functions?

I don't know of any examples and I don't know of any results which prohibit them

**-1**

votes

**1**answer

274 views

### Consequences of the degree conjecture

the title is quite explicit: I would like to know the consequences of the degree conjecture for the Selberg class.
Thank you in advance.

**1**

vote

**2**answers

183 views

### Implicit constant in Tenenbaum's result

In his famous book 'Introduction to Analytic and Probabilistic Number Theory', Gérald Tenenbaum established the following result (Theorem III.3.5):
Let $g$ be a positive multiplicative function and ...

**3**

votes

**1**answer

688 views

### Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function

Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of ...

**1**

vote

**3**answers

441 views

### Chebychev Function in Arithmetic Progressions

In the paper "exponential sums with multiplicative coefficients", Maier claims that the Chevbychev function in arithmetic progression satisfies
$\theta(x, r, a) = x/(r-1) +O(x^{1-1/r^\epsilon})$ for ...

**3**

votes

**1**answer

348 views

### Dirichlet divisor problem with multiple factors

Let $D(x)$ denote the number of pairs of natural numbers $(j,k)$ such that $j k\leq x$. It is well known that $D(x) =x \log x + (2 \gamma-1) x + $lower order terms in the large $x$ limit (Dirichlet ...

**1**

vote

**2**answers

251 views

### Is the singular integral that come up in circle method independnet of the representatin of the equations?

Let $F_1(\mathbf{x}), F_2(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous polynomial.
For the system of equations $$F_1(\mathbf{x})= F_2(\mathbf{x}) =0,$$
we have the ...

**2**

votes

**3**answers

328 views

### Primes in arithmetic progressions in number fields

My general question is how does one prove equi-distribution results for primes in arithmetic progressions in number fields? I am interested in the equi-distribution of prime elements of the ring of ...

**7**

votes

**1**answer

160 views

### Do arbitrary $K$-twin primes exist?

Polignac's conjecture states that for any positive even integer $K$, there exist infinitely many pairs of primes such that their difference is $K$.
I am interested the status in a much weaker form of ...

**2**

votes

**1**answer

108 views

### On the construction of a certain sequence of integers

Let $\alpha \in \mathbb{R}$ be a fixed (positive) number. For each $k \in \mathbb{N}$ we choose $\varepsilon_k >0$ with the property that $\lim_k \varepsilon_k =0$.
If $\alpha \in \mathbb{R} ...

**10**

votes

**1**answer

447 views

### What is the motivation behind Ramanujan's conjecture?

One motivation I have seen given for Ramanujan's conjecture for the estimate
$$ |a_p|< C p^{k - \frac{1}{2}} $$
for the Fourier coefficients of a cusp form of weight $2k$ is that it allows one to ...

**2**

votes

**0**answers

115 views

### $f(x)$-th largest number of prime factors

Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other ...

**2**

votes

**1**answer

296 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 ...

**42**

votes

**15**answers

6k views

### Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing ...

**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 ...

**4**

votes

**1**answer

137 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

465 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, ...

**18**

votes

**0**answers

372 views

### 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

268 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 ...

**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

67 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

155 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

77 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

208 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

116 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 ...

**2**

votes

**1**answer

114 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

135 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

898 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

302 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 ...

**3**

votes

**0**answers

103 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

128 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 ...