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.

learn more… | top users | synonyms

0
votes
0answers
39 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
1answer
202 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
0answers
83 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
2answers
139 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
2answers
126 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
1answer
199 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 ...
8
votes
1answer
252 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 ...
11
votes
1answer
432 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
1answer
277 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 ...
2
votes
0answers
299 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
1answer
292 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
1answer
390 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
1answer
98 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 ...
0
votes
2answers
305 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, ...
2
votes
0answers
131 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 ...
10
votes
1answer
469 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 ...
1
vote
0answers
30 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
2answers
262 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
1answer
258 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
1answer
650 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 $a\mapsto P_+(a)$ ...
5
votes
0answers
157 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, ...
5
votes
0answers
93 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 ...
7
votes
2answers
189 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 ...
6
votes
1answer
187 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 ...
1
vote
1answer
158 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 ...
0
votes
0answers
108 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
1answer
212 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
1answer
183 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
0answers
71 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
1answer
278 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 ...
7
votes
3answers
520 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
1answer
150 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
0answers
114 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
0answers
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
1answer
473 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 ...
2
votes
0answers
49 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 ...
2
votes
0answers
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
0answers
79 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
1answer
193 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
1answer
218 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
1answer
317 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
1answer
309 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
1answer
202 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 ${\mathbb R}^2$). ...
1
vote
0answers
87 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
1answer
459 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 ...
35
votes
3answers
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
1answer
178 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
1answer
347 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
1answer
333 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
2answers
892 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 ...