# Tagged Questions

**0**

votes

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41 views

### Estimates related to sum over a primes from a fixed, possibly sparse set

Let $E$ be a fixed infinite sequence of primes such that $\sum_{p \in E} \frac{1}{p} = \infty$. Assume that $\sigma > 1$ depends on some parameter $x \rightarrow \infty$ in such a way that $\sigma ...

**-5**

votes

**1**answer

231 views

### How to prove twin prime conjecture or Goldbach conjecture if we assume prime distribution is completely random? [closed]

If we assume that prime number distribution is COMPLETELY random (subject to 1/log(x) restriction), can we prove twin prime conjecture or Goldbach conjecture ?
My feeling is that, this will be ...

**4**

votes

**0**answers

87 views

### The probability distribution of LCM of uniformly distributed integers in $\{1,\ldots,n\}$

In the recent paper by Fernandez and Fernandez here on ArXiv, the following formula which was first proved by Diaconis and Erdos appears, on page 2.
For $0<t\leq 1$ the distribution of the lcm of ...

**3**

votes

**2**answers

149 views

### What is the asymptotic growth rate of the product of divisor function up to n [duplicate]

This was asked in mathstackexchange (see here) but was not satisfactorily answered beyond my basic observations.
Let $\tau(k)$ be the number of divisors of the positive integer $k$.
How does ...

**2**

votes

**2**answers

127 views

### How large can a subset of $\{1,\ldots,N\}$ be if all pairwise LCMs of its elements are lower bounded?

Consider the set $\{1,2,\ldots,N\}$. Let $LCM(a,a')$ denote the lowest common multiple of the integers $a,a'.$
We say that $A\subset \{1,2,\ldots,N\}$ is $M-$good if $LCM(a,a')\geq M,$ for all ...

**4**

votes

**1**answer

214 views

### Estimates of a sum involving both the Möbius function and Mertens function

I want to ask on the estimates of the sum
$$ \sum_{n=1}^{\infty} \mu(n)M\Big(\frac{x}{n}\Big)=\frac{1}{2\pi i }\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{x^s}{s\zeta(s)^2}ds.$$
It seems that the ...

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votes

**1**answer

263 views

### Primes in arithmetic progression with a moduli equal to a power of 2

I am currently looking for a result stronger than Siegel-Walfisz theorem, which gives an upper bound on the error term $|\pi(x,a,b)-\frac{\pi(x)}{\phi(a)}|$ for particular $a$.
The Siegel Walfisz is ...

**12**

votes

**1**answer

450 views

### Two Vinogradovs? Is one the son of the other? [closed]

Forgive me for my ignorance, but I'm very surprised to learn that there are two Vinogradovs, both famous in the field of analytic number theory. Guessing from their names and the Russian naming ...

**4**

votes

**1**answer

305 views

### Upper bound for the first Hardy-Littlewood conjecture

About the Hardy-Littlewood conjecture by Terence Tao:
Conjecture 2 (Prime tuples conjecture, quantitative form) Let ${k_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed ...

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votes

**0**answers

324 views

### How big can a set of integers be if all pairs have bounded gcd

In this recent MO question, it was shown that the maximal cardinality of a subset $A(M,N)$ of $[1,N]$ where the pairwise GCD's of all set elements are upper bounded by $M,$ with $M^2\leq N$ has size ...

**2**

votes

**1**answer

308 views

### infinite product of (1-1/(p+1)) over a density 0 set of primes

Let $S$ be a density zero set rational primes, in the concrete situation
$\#\{p<X,p\in S\}=\mathcal{O}(x/(\log x)^{3/2-\delta})$ for all $\delta>0$.
Then can $\prod_{p\in S}(1-\frac{1}{p+1})$ ...

**4**

votes

**1**answer

395 views

### About factorization in Zhang's proof of weak Twin Prime conjecture

Why does it need to firstly factorize the number n into two factors q and r( Lemma 4 in the paper,see the following)? What's the motivation. What if it doesn't do this factorization?

**0**

votes

**1**answer

107 views

### Sum of digits of a power [closed]

Are there any explicit formula for a sum of digits for a power in the given base? A problem to be specific: find a sum of digits for a number $2^{100}$ in the system with a base 5. In the system with ...

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votes

**2**answers

320 views

### Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa?

I know that better and better bounds of the Chebyshev Theta and Psi functions are implied by knowing that the first (insert large number here) zeta zeroes lie on the Critical Line. These bounds, ...

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votes

**0**answers

142 views

### On sets of coprime numbers

We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$
Denote by ...

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votes

**1**answer

490 views

### How much can an Eisenstein series be truncated?

For ease of exposition, I will stick to the simplest case: consider the Eisenstein series for $SL_2(\bf R)$
$$E(z,s)=\sum_{\gamma\in P_{\bf Z}\backslash SL_2(\bf Z)}\text{Im}(\gamma ...

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vote

**0**answers

33 views

### On collision-free sets of residues of integers

Denote $\pi_m$ to be collection of primes in $[2^{m},2^{m+1}]$.
Denote $\psi_{n,m}$ to be collection of integers of form $ab$ where $a\in\pi_n$, $b\in\pi_m$.
Given $n,t\in\Bbb N$, what is the ...

**3**

votes

**2**answers

267 views

### multiplicative functions of powers

Suppose I have a multiplicative function $f(n),$ and I want to understand the behavior of
$$
\sum_{n<x} f(n^k),
$$ for some integer $k.$ This seems like it should be easy (since the Dirichlet ...

**4**

votes

**1**answer

273 views

### Fourier expansion of automorphic forms

we know that for $r \in \{1,2,3,4\},$ $\lambda_{Sym^rf}$ is an automorphic form (here $f$ is a modular form for the full modular group) and this fact is conjectured for $r\geq 5$ by Langlands and ...

**22**

votes

**1**answer

683 views

### How big can a set of integers be if all pairs have small gcd?

Suppose $A\subset[1,N]$ is a set of integers. If for any distinct $a,b\in A$ we have $(a,b)\leq M$ then how big can $|A|$ be?
If $M=1$ then $|A|$ is at most $\pi(N)$ since the map $a\mapsto P_+(a)$ ...

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165 views

### Effective bound of $L(1,\chi)$

Let $d$ be a fundamental discriminant and let $\chi$ be the associated primitive real character of modulus $\vert d \vert$. Assuming GRH, Littlewood proved that as $\vert d \vert$ grows large,
$$L(1, ...

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votes

**0**answers

98 views

### Methods of variational calculus in analytic number theory

What methods of calculus of variations have been used in analytic number theory?
I mean do Hamilton-Jacobi theory of PDE found usage in analytic number theory, which raises yet another question has ...

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votes

**2**answers

192 views

### How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root

(This is an extension and specification of a question which I initially asked in MSE having now one comment (which I could not yet digest completely) and which I also detailed further (after working ...

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**1**answer

188 views

### $N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$

Assume that $ab \neq 0$. What is $$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$$I need this result, but unfortunately I am not a number theorist. Could ...

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vote

**1**answer

160 views

### Values of the completed Riemann $\xi(1+it)$ for small t?

I'm editing this question heavily for clarity:
I am looking for methods to compute $\zeta(1+it)$, or the (partially) completed Riemann zeta function
$$\pi^{-s/2}\Gamma(s/2)\zeta(s)$$
along the line ...

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votes

**0**answers

109 views

### Bounding $\dfrac{r(x)}{\pi(x+r(x))-\pi(x-r(x))}$ with $1\ll r(x)\ll \log^{4}(x)$

I would like to know whether it is possible to obtain the bounds $\sqrt{r(x)}\ll k(x)\ll r(x)$ where $k(x)={\pi(x+r(x))-\pi(x-r(x))}$ and $1\ll r(x)\ll \log^{4}(x)$ and thus ...

**5**

votes

**1**answer

228 views

### Another question on Heath-Brown's “Prime twins and Siegel zeros”

With a graduate student, I'm going through the paper (Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224.)
Here's the background and notation.
We have a quadratic character $\chi$ modulo $q$, ...

**4**

votes

**1**answer

186 views

### Examples when one can use the the symmetric power $L$-functions to study topics related to the number theory

"The symmetric power $L$-functions are a powerful tool for studying algebraic or geometric objects through analytic methods." I read this sentence in the introduction of a Master thesis. I want to ...

**2**

votes

**0**answers

77 views

### Determining coefficients of a Dirichlet series based on values on a vertical line

Let us suppose we have a Dirichlet series
$$ D(s) = \sum_{n \geq 1} \frac{a(n)}{n^s},$$
and that we know the values of $D(\tfrac{1}{2} + im)$ for $m \in \mathbb{Z}$. Can we recover the coefficients ...

**3**

votes

**1**answer

281 views

### Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?

http://mathworld.wolfram.com/ChoquetTheory.html
Is the claim in the link true? Here's the reference given there:
https://www.renyi.hu/~p_erdos/1934-01.pdf
Erdős proved that there exist at least ...

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votes

**3**answers

537 views

### Quantitative and elementary proofs of the Prime Number Theorem

I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there ...

**5**

votes

**1**answer

175 views

### Average of Short Character Sum over All Dirichlet Characters Mod n

Cross-posted from M.SE.
Given $a,n$ coprime positive integers, let $L = \{(x,y)\in \mathbb{Z}^2, ax=y(n)\}$ be the lattice of all points satisfying $ax=y\pmod{n}$.
I want to find an ...

**5**

votes

**0**answers

115 views

### Moments of completed L-functions?

This is a follow up question to this one.
It seems that results on moments of L-functions, that is, estimates for integrals of the form
$$\int^{T}_1|\zeta(\sigma+it)|^{2k}dt$$
are typically for the ...

**3**

votes

**0**answers

85 views

### Enumerating factors in intervals

Given $1<a<N-N^{1/\alpha}$ where $\alpha\geq2$, denote the number of distinct factors of $N$ in $[a,a+N^{1/\alpha}]$ as $\sigma_{0,a}(N,\alpha)$ denote ...

**4**

votes

**1**answer

487 views

### Green-Tao theorem for 1-central numbers

This question came to my mind this afternoon while trying to figure out a possible way to tackle de Polignac's conjecture, which states that every even positive integer can be written as the ...

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votes

**0**answers

50 views

### Primes which have common difference

Given $m,n\in\Bbb N$, assume $2<p_1<\dots<p_n<m$ with each $p_i$ a prime. If $p_i$ are picked randomly, what is average and worst case $d\in2\Bbb N$ such that each of $$p_i+d$$ is a prime ...

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**0**answers

96 views

### Primitive triples in a region [duplicate]

Are there at least $cn$ Pythagorean triples and at least $dn$ Primitive Pythagorean triples $(A,B,C)$ with $$2^{\frac n2}<A<2^{\frac n2+1}<2^n<B<C<2^{n+1}$$
with some fixed ...

**4**

votes

**0**answers

88 views

### Shifted convolution problem for Coefficients of automorphic forms

The shifted convolution problem for coefficients of modular forms is well studied and many estimates were established for the shifted convolution sums of Hecke eigenvalues. So, one may ask about the ...

**7**

votes

**1**answer

203 views

### Behaviour of $\zeta(1-it)/\zeta(1+it)$?

I am trying to understand the behaviour of
$$\int^\infty_{-\infty}\frac{\xi(1-it)}{\xi(1+it)}h(t)\frac{dt}{t}$$
where $h$ is a Schwartz function on $\mathbb R$, and $\xi(s)$ the completed Riemann zeta ...

**6**

votes

**1**answer

221 views

### Logarithmic weights on number theoretic sums

Suppose we are interested in the sum
$\sum _{n\leq x}a_n.$
The study of the sum
$\sum _{n\leq x}a_n\log (x/n)$
may be easier.
What can one say about the first sum from knowing the behaviour of ...

**3**

votes

**1**answer

363 views

### Greatest number of coprime numbers between two numbers

We know that from prime number theorem that the number of primes below $n$ is approximately $$\frac{n}{\log_en}.$$
$\star$ Given $n,m$, what is the largest list of pairwise coprime numbers that one ...

**13**

votes

**1**answer

318 views

### Connection between Bernoulli numbers and Riemann-Siegel theta function?

I have come across a strange approximation for the Riemann-Siegel theta function involving the Bernoulli numbers - namely that
$$\frac{1}{2} \log \left| B_{2 n}\right|\approx \vartheta (2n)\ ,\quad n ...

**6**

votes

**1**answer

227 views

### Lattice points on the boundary of an ellipse

How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). ...

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vote

**0**answers

95 views

### More general better error bounds for partial sums of reciprocals of primes

In this question Gerhard Paseman asks for explicit estimates on $E(k)$ in
$$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k).$$
There are some nice answers, but my question is on ...

**6**

votes

**1**answer

483 views

### An elementary lower bound on the number of primes

Recall the second Chebyshev function: $$\psi(x) = \sum_{p \leq x} \lfloor \log_p x \rfloor \log p$$ where $x$ is a positive integer, and $p$ runs over all primes $\leq x$.
In a hunt for an ...

**37**

votes

**3**answers

1k views

### Is this integral representation of $\zeta(2n+1)$ known?

Background: I'm an undergraduate at an institution with no researchers in analytic number theory, and no ties to the analytic number theory community. I believe I have found what is, as far as I can ...

**6**

votes

**1**answer

182 views

### What is the mean value of a pair of Ramanujan Sums when summed over squares?

Does anyone know of the mean value of two Ramanujan Sums when summed over the square of integers?
In my research on the Landau problem regarding nearly square primes, I have run into the mean value ...

**17**

votes

**1**answer

355 views

### Smoothed exponential sums: bounds and sources?

Let $f:\mathbb{R}\to\mathbb{C}$ be differentiable $k$ times, with $f, f',\dotsc,f^{(k)}\in L^1$. Let $\alpha\in \mathbb{R}/\mathbb{Z}$, $\alpha\ne 0$. In "Every odd number..." (Math. Comp. 83, 2014), ...

**1**

vote

**1**answer

338 views

### Proof that $p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log {p_n}$ [closed]

I cannot find a proof of this theorem. May anyone assist?
$p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log{p_n}$

**10**

votes

**2**answers

910 views

### Update for 2015: least prime of form nq+1, with q prime?

I have received a complaint about my 2011 answer
least prime in a arithmetic progression
which, indeed, gives conflicting reports about this:
given a prime $q,$ what can we say about an upper ...