**1**

vote

**1**answer

173 views

### Bombieri-Vinogradov in short intervals

In 1985 Perelli, Pintz & Salerno proved a short-interval form of the Bombieri-Vinogradov theorem with $\theta \in (7/12, 1]$. Have there been any improvements on this, in particular with the ...

**6**

votes

**1**answer

164 views

### When is $\vartheta(x)>x$? [Skewes number analog]

Let $\vartheta(x)=\sum_{p\le x}\log p$. What is known about the first time $\vartheta(x)>x?$
Bays & Hudson give good upper bounds (slightly improved by Chao & Plymen) on the first crossing ...

**5**

votes

**0**answers

698 views

### Zeta function double product

Is it possible to write the following double product in terms of the zeta function?
\begin{align}
&\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}}
\end{align}
Extending the ...

**4**

votes

**0**answers

134 views

### The divisors of $p-1$ and high-degree residues modulo $p$

Here is a somewhat more explicit version of a question that I asked a while ago.
Suppose that $p$ is a prime of the form $p=2n(n+1)+1$, with a positive integer $n$. Can every odd prime divisor of ...

**1**

vote

**1**answer

221 views

### Residues and values of Riemann Zeta function at some points

I need the following computational results for proving something.
Let $1/2 + i\gamma_0$, be the first nontrivial zero of Riemann zeta function, $\zeta(s)$,
i.e. $\gamma_0\sim 14.134...$.
1) what is ...

**2**

votes

**1**answer

134 views

### Bounds on imaginary parts of partial Kloosterman sums?

For a prime $p$ and integers $a,m$, $0<a,m<p$ define the (partial Kloosterman) sum
$$ S_p(a,m) = \sum_{0<k<m} \exp\left(\frac{2\pi\mathrm{i}}{p}(a x + x^{-1})\right), $$
where $x^{-1}$ is ...

**4**

votes

**1**answer

193 views

### References to proofs of upper and lower bounds on the number of coprimes in an interval?

On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...

**2**

votes

**0**answers

122 views

### Siegel Walfisz Theorem for algebraic number fields

Is there a generalization of the Siegel Walfisz to algebraic number fields? This has been done for the prime number theorem in the prime ideal theorem.

**0**

votes

**0**answers

109 views

### A question about the duality principle

Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by ...

**1**

vote

**0**answers

83 views

### number of divisors

Let $d(n,Q)$ denote the number of divisors of $n$ that is smaller than $Q$. Let $D(n,Q)$ denote the number of positive integers smaller than $Q$ that has the same prime factors with $n$.
My question ...

**3**

votes

**0**answers

90 views

### Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$

Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding
$$
\int_0^T L\left(\tfrac{1}{2} + it, f ...

**4**

votes

**1**answer

183 views

### Voronoi formula and twists by additive characters

I was wondering if there are any references for the error term in the problem
$$\sum_{n\leq x} r(n) \exp(2\pi i\frac{a}{q}n)$$
where $r(n)$ is the number of representations of $n$ as a sum of two ...

**2**

votes

**1**answer

168 views

### Bound for sums of bounded multiplicative functions that are zero at primes

Let $h:\mathbb{N}\rightarrow\mathbb{C}$ be a bounded multiplicative function with $h(p)=0$. The motivation for this question is just a general enquiry and, since I suppose it has already been ...

**7**

votes

**0**answers

279 views

### Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?

(General) Lindelof Hypothesis which says for any $L$-function we have $$L(1/2+it)\ll Q(t)^{\epsilon}$$ for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $t$.
For a Maass form $\phi$ ...

**6**

votes

**1**answer

158 views

### Computing certain integrals over high-dimensional polyhedra

Let $\delta>0$ be a small real number and consider the $k$-dimensional region consisting of points for which
$$\delta\leq x_1\leq x_2\leq\ldots \leq x_k$$
and
$$x_1+\ldots+x_k\leq 1.$$
I am ...

**3**

votes

**0**answers

124 views

### Computing local volumes : the case of Hecke p-adic subgroups

I am quite interested in knowing how to compute some volumes of groups defined on local fields $K$, mainly in order to evaluate the identity term in trace formulas. It is something well done in the ...

**4**

votes

**1**answer

197 views

### Asymptotic behaviour of $K$-Bessel function in transition range

It is known that the famous mistake of Iwaniec-Sarnak in their paper of $L^\infty$ norm of eigenfucntion of non-cocompact arithmetic surfaces in lemma (A1) is because of they did not consider the bump ...

**10**

votes

**1**answer

434 views

### Parity of the Prime Counting Function

I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers.
Let:
$\ \ E_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 ...

**0**

votes

**1**answer

119 views

### A conjectural convergence condition for a weakened Elliott-Halberstam conjecture

For $a$ and $q$ positive integers such that $a\lt q$ and $(a,q)=1$, let $\pi(x;q,a)$ be the number of primes $p\equiv a\pmod q$ below $x$. One can show that $\pi(x;q,a)\sim \dfrac{\pi(x)}{\varphi(q)}$ ...

**0**

votes

**1**answer

300 views

### Is there a formula that can predict the primes in the sequence of ratios of consecutive superior highly composite numbers? : $2, 3, 2, 5, 2, 3, 7,…$

This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers:
$2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, ...$
The $n^{th}$ ...

**3**

votes

**0**answers

172 views

### Effective version of the Bombieri-Vinogradov theorem

Is there an effective version of the Bombieri-Vinogradov Theorem, in that have bounds on the implied constant been found?

**7**

votes

**1**answer

967 views

### What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?

The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...

**4**

votes

**1**answer

200 views

### A lower bound on the $L^2$ norm of a Dirichlet polynomial

The Question. Suppose $0 < \alpha < \beta$ are fixed, and $a_n$ is an arbitrary sequence of real numbers. Is it known how to bound from below
\begin{equation*}
\int_0^{T} \Big| \sum_{\alpha T ...

**1**

vote

**1**answer

143 views

### trigonometric sum and inequalities

let $x\in\mathbb{R}-\mathbb{Z}$ and $e(x)=e^{2\pi ix}$. If we have this sum $$\left|\overset{q}{\underset{h=1}{\sum}^{*}}e\left(h\, x\right)\underset{\underset{p\equiv h\,\textrm{mod}\, q}{p\leq ...

**5**

votes

**1**answer

177 views

### The sixth power integral moment of automorphic L-function attached to Maass Forms

It is known that the sixth power integral moment of automorphic L-function attached to Cusp Forms has been proved by M. Jutila, that is $\int_{0}^{T}|L(1/2+it,f)|^{6}dt \ll T^{2+\varepsilon}$.
And ...

**7**

votes

**1**answer

372 views

### What does the sum of the reciprocals of all the highly composite numbers converge to?

I've calculated the sum of the reciprocals of all the $156$ first highly composite numbers up to $10^{18}$:
$\sum \dfrac{1}{HCC(n)} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{12} + ...

**2**

votes

**0**answers

107 views

### The behavior of series involving special subsets of the prime numbers

It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ ...

**3**

votes

**1**answer

135 views

### Lower bound of first moment of $L$-function on $\mathrm{GL}(3)$

Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$.
Do we know any case that
\[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\]
holds unconditionally?
I know the ...

**3**

votes

**2**answers

250 views

### Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $

Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold?
$$
\int_{k + 1/2}^{k + 3/2}
\frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}}
...

**1**

vote

**0**answers

100 views

### Some computational results on Hurwitz zeta function and questions

Let
\begin{equation}
\zeta(\sigma - it, a) := \sum_{m \geq 1} \frac{1}{(m + a)^{\sigma - i t}},
\quad \text{Re}(s) > 1.
\end{equation}
When arg a > 0 and $t \to + \infty$, $|\zeta(\sigma - it, ...

**3**

votes

**1**answer

569 views

### what would be the consequences on the distribution of primes of $\Lambda=\infty$?

It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the ...

**10**

votes

**0**answers

156 views

### Weyl law for Maass forms with nontrivial character

The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...

**1**

vote

**0**answers

65 views

### density of zeroes of Epstein zeta functions on vertical strips

There are many results (e.g. Davenport and Heilbronn) asserting that Epstein zeta functions in general have zeroes outside the line $\mathrm{Re\ } s = n/4$. Is there any result about the density of ...

**-3**

votes

**1**answer

195 views

### Andrica's and Legendre's Conjectures [closed]

My question is, which of these two conjectures is stronger, Andrica's or Legendre's? Could proving one prove the other? If the upper bound for the prime gap above any given natural number $n$ were to ...

**1**

vote

**0**answers

71 views

### Sieving question

How many integers $n\leq X$ are there with the property that $\prod_{p\in S} p \geq n^{1/2-\epsilon}$? Here (to keep notation readable) I've written $p\in S$ if and only if $p||n$ (that is, $p|n$ and ...

**1**

vote

**0**answers

72 views

### Uniform bounds of number of integral points on affine varieties

In Duke-Rudnick-Sarnak 93, Density of integer points on affine homogeneous varieties, one of the consequences is the following,
Consider the variety $V_{n,k} = \{A \in Mat_n(\mathbb{Z}): det(A) = ...

**2**

votes

**1**answer

204 views

### Lower bound of Hecke eigenvalues of Maass form

If $f$ is a Maass form and $p$-Hecke eigenvalue (i.e. Hecke eigenvalue of usual Hecke operator $T_p$) of $f$ is $\lambda_f(p)$, do we know anything about lower bound of the sum$$S(x) = \sum_{x\le p\le ...

**4**

votes

**1**answer

214 views

### polynomials in many variables and Hasse principle

I was wondering whether there exists any result of the form
"if $f \in \mathbb{Z}[x_1, ..., x_k]$ is a polynomial (not form! I don't require homogeneity) of total degree $n$, with $k \geq \delta ...

**12**

votes

**2**answers

602 views

### Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line ...

**1**

vote

**1**answer

209 views

### is $x_{n}\ll \overline{x}_{n}^{2}$?

This question is a cross-post from MSE, cause I didn't get any answer there. I hope it is well suited for MO:
Let $(x_{n})_{n\ge 1}$ be an increasing sequence of positive integers and ...

**0**

votes

**0**answers

51 views

### Definition of Degenerate hyper-Kloosterman Sum

If $(a,q)=1$, we all know the usual hyper-Kloosterman sum
$$Kl_n(a,q)= \sum_{x_1...x_n\equiv a \mod q} e(\frac{x_1+\dots+x_n}{q}).$$
I wonder what would be the correct definition when $(a,q)\neq 1$.
...

**1**

vote

**1**answer

107 views

### Expression and growth bound for $r_{p^m,k}(n)$

Let's define , $$R_{p^m,k}(n)=\#\{(a_1,\dots,a_k)\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2\le n \ \text{and} \ p^m|\sum_{i=1}^ka_i^2\}$$
what will be growth bound of $R_{p^m,k}(n)$? This can be thought as a ...

**4**

votes

**1**answer

142 views

### Inequality due to Siegel (assumptions) and upper bounds on number field discriminants

In Siegel's 1969 paper, Abschätzung von Einheiten, on page 73, he states the inequality
$$\log\sqrt d\le n-1+{n\over 2}\log\pi+r_2\log 2\qquad (*)$$
and compares with the bound due to Minkowski that
...

**0**

votes

**0**answers

70 views

### Bounding the absolute value of a polynomial involving a Diophantine equation

Let $\mathbf{z}\in\mathbb{C}^n$ with entries $z_1,z_2,\ldots,z_n$. I would like to bound the following quantity
\begin{equation*}
...

**4**

votes

**1**answer

457 views

### Number of solutions in a sum of squares Diophantine equation

Let $n$ be an integer. I'm interested in upper bounding the number of integer solutions of
\begin{equation*}
x_1^2+x_2^2+\ldots+x_p^2=y_1^2+y_2^2+\ldots+y_p^2,
\end{equation*}
where for all ...

**0**

votes

**0**answers

79 views

### Bounding Random Quadratic Gauss sums

I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$ and $|\epsilon_k|=1$ for all $k=1,2,\ldots,n$. We have
\begin{align*}
...

**1**

vote

**0**answers

173 views

### Davenport's proof that almost all integers are the sum of 4 cubes

Where can I find a pdf that describes Davenport's proof that almost all integers are the sum of $4$ cubes?

**2**

votes

**2**answers

424 views

### Is there a von Koch-type theorem for the generalized Riemann hypothesis?

Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound
$$
\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).
$$
Q1: ...

**5**

votes

**0**answers

258 views

### $x^2+1$ attaining almost prime values

Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...

**2**

votes

**1**answer

161 views

### Growth of $r_k(n)$

What is the best known growth bound of $r_k(n)$, where $$r_k(n)=\#\{(a_1,\dots,a_k\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2=n\}?$$ Please provide some reference if known. Thanks.