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

5
votes
2answers
297 views

Asymptotics of product of Euler's totient function (A001088)?

Conjecture: \begin{align} \lim_{n\to \infty } \, \frac{\left(\prod _{k=1}^n \phi (k)\right){}^{1/n}}{n}\sim 0.2059\text{...} \end{align} The numerical result from 100000 terms is: My questions ...
12
votes
1answer
394 views

Does the sum $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converge?

Prove, if possible in an elementary way, that $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converges/diverges, where $p_n$ denotes the $n^{\textrm{th}}$ prime.
1
vote
1answer
254 views

Asymptotics of “ugly” function elucidate Goldbach's conjecture?

Question We now define the following "ugly" function: $$ A_c(s,r,n,m) = \begin{cases} 1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise} \end{cases} $$ How does the ...
11
votes
1answer
1k views

What did Euler do with multiple zeta values?

When reading about multiple zeta values, I often find the claim that the case of length two $$ \zeta(s_1, s_2)=\sum_{n>m \geq 1} \frac{1}{n^{s_1}m^{s_2}}, \qquad s_1 \geq 2, \quad s_2 \geq 1 $$ ...
7
votes
1answer
256 views

Riemann zeta function: pair correlations vs. neighbor spacings

Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the ...
10
votes
1answer
416 views

Why do the Maynard-Tao weights work so well?

I am looking for an intuitive reason for why the Maynard-Tao weights work well to capture many primes of the form $n+h_1, \ldots , n+h_k$, where $(h_1, \ldots , h_k)$ is any admissible $k$-tuple. For ...
5
votes
1answer
174 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
1answer
169 views

Convergence of a double sum involving prime numbers

This has been moved from math.stackexchange; I am attempting to prove/disprove convergence of the following sum $$ \lim_{n \to \infty} \frac{1}{n} \sum_{p \leq n} \sum_{k=0}^\infty \ln p ...
0
votes
0answers
67 views

The growth rate of almost periods for almost periodic function

A subset $A \subset \mathbb{R}^2$ is relative dense if there exists $L>0$ such that for every $p\in \mathbb{R}^2$ there exists $p' \in A$ such that $|p-p'|<L.$ A continuous function $f : ...
3
votes
1answer
197 views

An exponential sum over squares

I have the following exponential sum: $\sum _{M<n\leq N}e\left (x/n^2\right )=\sum f(n),$ say, where $M$ and $N$ are something like $x^{1/4}$ and $x^{1/2}$. My question is basically, how do I ...
3
votes
1answer
480 views

Euler product for sum of multiplicative function times log

(Cross-posted from StackExchange). Let $g$ be a multiplicative function which satisfies $0 \le g(p) \ll 1/p$ and $$ \sum_{p\le x} g(p) = \log \log x + C + O((\log x)^{-10}). $$ Iwaniec and ...
2
votes
0answers
167 views

Short Kloosterman sum

Let $K(m,n,c)=\sum_{p\in A}\exp(2\pi i(pm+np^{-1}))$, where $m,n\in\mathbb{Z}$, $c\in\mathbb{N}$, and $A\subset (\mathbb{Z}/c\mathbb{Z})^\times$. Does anybody know any nontrivial bound for this kind ...
1
vote
0answers
210 views

Is this a proof of the Hardy-Littlewood inequality? [closed]

V.V. Miasoyedov posted a paper to the arXiv claiming a proof of the Hardy-Littlewood conjecture $\pi(x+y) \le \pi(x)+\pi(y)$. It seems a bit off, and not only because the conjecture is widely believed ...
4
votes
1answer
160 views

Improvement of a bound on divisor distributions from “Divisors” (Hall and Tenenbaum)?

In the classic text referred to in the title of this question, the bound $$ H(x,y,2y) \ll \frac{x}{(\log y)^{\delta}\sqrt{\log \log y}},\quad (3\leq y\leq \sqrt{x}) $$ is given, where ...
3
votes
0answers
102 views

exponential sum of primes

Fix $\alpha \notin \mathbb{Q}$. I would like to know a reference that shows $$\mathbb{E}_{n\leq N} \Lambda(n) e^{2\pi i \alpha n} \to 0,$$ as $N$ tends to infinity. I am familiar with Vinagradov's ...
3
votes
0answers
184 views

Colmez conjecture and endomorphism rings

It is given by the Colmez conjecture that if $A$ has a CM-type $(k, \phi)$ and $\text{End}(A) = \mathcal{O_k}$ $$\displaystyle h_{\text{fal}}(A) = \sum_{\text{irr} \hspace{1 mm}\rho: ...
4
votes
1answer
176 views

Well-spacing of the roots of a quadratic congruence

On pages 956-957 of this paper, it is established that for any two $v_1, v_2$ satisfying $v_1^2 + 1 \equiv 0\operatorname{(mod} d_1), v_2^2 + 1\equiv 0\operatorname{(mod} d_2)$, $$\left\lVert ...
7
votes
1answer
330 views

On (a generalization of) the Gauss Circle Problem

Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...
4
votes
3answers
491 views

How do estimates on $N_\chi(\alpha,T)$ lead to the Dirichlet prime number theorem for arithmetic sequences?

Let $ N_\chi(\alpha,T)$ be the number of zeros of $L(s=\sigma+it,\chi) = \sum \frac{\chi(n)}{n^s}$ where $c > 0$ and $(\sigma,t) $ are in the rectangle $ [\alpha,1] \times [-T,T]$. In various ...
1
vote
1answer
89 views

Exponential sum estimates similar to the one for $\sum_p (\log p) e(p \alpha)$, but for different sequences

Obtaining a non-trivial estimate for $\sum_p (\log p) e(p \alpha)$ over the minor arcs is one of the estimates required for obtaining the ternary Goldbach for $n$ sufficiently large via the circle ...
1
vote
2answers
240 views

estimate sum of $\log \log p/p$

It is known that $$\sum_{p\leq x} \frac{\log p}{p}=\log x+c.$$ Are any tight bounds on $$\sum_{p\leq x} \frac{\log \log p}{p}$$ known? I haven't managed to find anything in the literature. Trying to ...
19
votes
1answer
1k views

What's special about the circle problem?

Let $K$ be a number field, and let $$\zeta_{K}(s):= \sum_{0 \neq I \text{ ideal of }O_K} \frac{1}{N_{K/\mathbb{Q}}(I)^s} = \sum_{n \ge 1} \frac{a_n}{n^s}$$ be the Dedekind zeta function of $K$. The ...
9
votes
0answers
120 views

Newly defined $L$-function in terms of $L$-function, does it have any obvious zeros or poles?

Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ideal, let $[\mathfrak{a}]$ denote its ...
13
votes
1answer
475 views

What is known about $\sum_{n \leq x} \mu(n) \varphi(n)$?

Let $\mu(n)$ denote the Möbius function and $\varphi(n)$ the Euler-phi function. What is known about $f(x) = \sum_{n \leq x} \mu(n) \varphi(n)$? For example: Is it known that $f(x)$ grows without ...
5
votes
0answers
95 views

Sums of twisted products of Kloosterman Sums

For $m,n,c \in \mathbb{N}$, let $S(m,n;c)$ denote the Kloosterman sum $$ S(m,n;c) := \sum_{\substack{1 \leq a < c \\ \gcd(a,c) = 1}} e \left( \frac{ma + n\overline{a}}{c} \right) $$ where $e(n) = ...
5
votes
0answers
123 views

Particular case of the class number formula, Dirichlet characters

Let $\chi$ be a Dirichlet character modulo $4$ such that $\chi(-1) = -1$, and let $\chi'$ be a Dirichlet character modulo $5$ such that $\chi'(-1) = 1$, $\chi'(2) = \chi'(3) = -1$. How do I see the ...
0
votes
0answers
154 views

How to compute this sum over numbers?

When I was doing some task of analytic number theory I was stuck on computing this sum $$S:=\frac{1}{L} \sum_{q \in \mathcal{Q}} \phi(q) \overline{a}^{\frac{1}{2}},$$ where $\overline{a}$ is the ...
33
votes
1answer
1k views

The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?

Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only ...
4
votes
1answer
141 views

Reference to a variant of Abel's summation formula

Edit. A stronger version of the formula is true (details follow). Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, $(\lambda_n)_{n \ge 1}$ a nondecreasing sequence of positive reals such that ...
21
votes
2answers
1k views

A conjecture based on Wilson's theorem

Definitions: Lagrange's theorem implies that for each prime $p$, the factors of $(p − 1)!$ can be arranged in unequal pairs, with the exception of $±1$, where the product of each pair $≡ 1 \pmod p$. ...
8
votes
2answers
434 views

Fourier transform of the critical line of zeta?

This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here. Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...
3
votes
0answers
131 views

Almost primes in short intervals

Define an integer $n$ to be a $k$-almost prime if it has at most $k$ distinct primes factors. A detecting function for the set of such numbers is the generalized von Mangoldt function given by ...
1
vote
1answer
110 views

Upper and lower bounds for $|L(1+it,\chi)|$ for complex primitive character $\chi$?

I would guess this is some standard fact related to the zero-free region. But cannot find it in the textbooks I read.
1
vote
1answer
169 views

Least simultaneous quadratic non-residue

Suppose $p,q$ are distinct primes with least quadratic non-residues $n_p$ and $n_q$ respectively. Can one bound the least $n$ for which $\left(\frac{n}{p}\right)=\left(\frac{n}{q}\right)=-1$ in terms ...
0
votes
0answers
77 views

Probability distribution associated with total divisors of an integer

Is there a generalization to https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem which gives distribution function for $$\omega(n)=\big|\{d\in\mathsf{prime}:d|n\}\big|$$ where ...
1
vote
1answer
124 views

Linear forms that avoid numbers with lot of factors

Is following true? For every given $c>0$ there is an $n_c>0$ such that for every $n>n_c$ there are integers $n<a,b<2n$ such that there are two positive integers $\frac{n}{2(\log ...
4
votes
1answer
197 views

Goldbach for certain classes of $n$

Asked on MSE without response here. $\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$. The Wiki article on the Goldbach conjecture states that In 1975, ...
1
vote
1answer
146 views

On the number of divisors in a given range

Given $\alpha\in\Bbb N$, can there be more than $(\log N)^4$ divisors (composites allowed) of $N$ in $\big[\frac\alpha2,\alpha\big]$ when $\sqrt N\in\big[\frac\alpha2,\alpha\big]$? What is the ...
0
votes
0answers
66 views

Integrating a series expansion of $\mbox{frac}(x)\lfloor x\rfloor$ coming from Fourier series of sawtooth function

Let me preface this question by saying that I am not exactly sure it counts as research level. It is crossposted on mathstackexchange: ...
1
vote
0answers
86 views

Bounds on the number of zeros of a quadratic form

Let $Q(x_1, \dots, x_n)$ be a non-degenerate indefinite quadratic form with integer coefficients. Let $N(Q,T)$ be the set of vectors $x=(x_1, \dots, x_n) \in {\mathbb Z}^n$ such that $|x|<T$ and ...
3
votes
2answers
276 views

Primes $P_{2n-1}$ that are $2$ mod $3$

Are infinitely many primes $P_{2n-1}$ expressible as $3k-1$? The primes $P_{2n-1}$ are every other prime beginning with $2$: $2,5,11,17,23,31,\cdots$. The first few are of the form $3k-1$, but $31$ ...
5
votes
1answer
171 views

Hecke characters and Conductors

Motivation: Let $\ell$ be an odd prime. There is a conductor-preserving correspondence between primitive Dirichlet characters of order $\ell$ and cyclic, degree $\ell$ number fields $K/\mathbb{Q}$. ...
12
votes
0answers
272 views

No Siegel-Landau zeros for $\mathrm{GL}(n)$

The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact: There ...
1
vote
0answers
161 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: ...
5
votes
1answer
208 views

The number of integral solutions to $x^2+y^2-az^2=0$

I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation $$ ...
4
votes
1answer
281 views

Show the upper bound of cardinality of $A$ is $C\sqrt{n\log{n}}$

$\forall l,m,n\in \Bbb{Z_+}$, let $A:=\{k: m+1\leq k\leq m+n\text{ and }l-k^2\text{ is a square number}\}$. Please prove that the number of elements in $A$ is not more than $C\sqrt{n\log n}$, where ...
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
82 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 ...