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.

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3
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3answers
537 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
106 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
257 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 ...
20
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
134 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
509 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
106 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) = e^...
5
votes
0answers
130 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
157 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 ...
34
votes
1answer
2k 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
160 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 $≡ 1 \pmod p$. ...
9
votes
2answers
550 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
144 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
120 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
182 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
81 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 $\mathsf{prime}$...
0
votes
1answer
129 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 n)^c}\...
4
votes
1answer
205 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, ...
0
votes
1answer
165 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 ...
1
vote
0answers
98 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: http://math.stackexchange.com/questions/1519724/integrating-a-...
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vote
0answers
91 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 $Q(...
3
votes
2answers
284 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 $31$ ...
7
votes
1answer
204 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
286 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
201 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: $W_{n}(z)...
5
votes
1answer
217 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 $$ x^2+y^...
4
votes
1answer
286 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
47 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
243 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
88 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
155 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 $f(n)=\...
2
votes
2answers
130 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 $a\...
4
votes
1answer
218 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
268 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
1answer
456 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
326 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
332 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
319 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
396 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
110 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
326 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
154 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
497 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 z)^s=\sum_{(c,d)\...
1
vote
0answers
34 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
269 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
281 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
689 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
171 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
101 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 ...