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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524 views

Representations with Triangular Numbers

A well known theorem of Gauss says that any natural number $n$ may be written as the sum of three triangular numbers - $$ n={a_{1} \choose 2}+{a_{2} \choose 2}+{a_{3} \choose 2} $$ The following ...
4
votes
0answers
203 views

Translation of an article by Wolfgang Schmidt on normality for real numbers in different bases.

I would greatly appreciate a pointer to a translation from German into English of the article by Wolfgang Schmidt, Über die Normalität von Zahlen zu verschiedenen Basen, from Acta Arithmetica VII, ...
7
votes
2answers
558 views

Question about the Hardy-Littlewood method (quite basic)

Hi, I have a question about the Hardy-Littlewood method. Writing $R_s(n)$ for the number of ways to write $n$ as a sum of $s$ $k$-th powers and $f(\alpha )$ for the sum $\sum _{m=1}^Ne(\alpha m^k)$, ...
19
votes
4answers
2k views

The Riemann Hypothesis and the Langlands program

On page 263 of this book review appears the following: Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic ...
0
votes
1answer
699 views

Can infinite polynomials be expressed as a product of its linear factors?

Background: In the 1700s, Euler solved the Basel Problem, which was to solve $\sum_{n=1}^\infty\frac{1}{n^2}$ in closed-form. Euler showed that it was equal to $\frac{\pi^2}{6}$ by first expressing ...
6
votes
2answers
245 views

Minimal period of arithmetic progressions occurring in sets of positive density.

Let $A$ be a subset of ${\mathbb N}$ with positive upper-Banach density, and for each integer $k\geq3$, define $R_k=R_k(A)$ to be the smallest positive integer $r$ such that $A$ contains a length $k$ ...
2
votes
1answer
300 views

Asymptotic formula in Analytic Number Theory

Hi, Could anyone tell me in what sense the following is an "asymptotic formula": Theorem 1 from ...
2
votes
1answer
316 views

Siegel-Walfisz Theorem

Hi, Could anyone explain to me how A and B are the same/different/equivalent? A = The Siegel-Walfisz Theorem as stated in Wikipedia (this is the statement in Davenport) ...
10
votes
4answers
669 views

Using Quotient of Prime Numbers to Approximation Reals

We know a positive rational number can be uniquely written as $m/n$ where $m$ and $n$ are coprime positive integers. Particularly, we can pick out those numbers with $m$ and $n$ both prime. Question ...
6
votes
1answer
372 views

Lower bound for exponential sums.

Let $D$ be a subset of $\mathbb Z/n \mathbb Z$ containing $0$. For $m$ an integer, set $$\alpha(m,D)=\sum_{d \in D} e\left (\frac{m d }{n}\right ),$$ where as usual $e(x) = e^{2 i \pi x}$ This is an ...
3
votes
1answer
235 views

Ratio of consecutive divisors and average

Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard: (1) ...
4
votes
1answer
559 views

Bernoulli number formula involving roots of Taylor polynomial of $\exp-1$

Please prove, give more symbolic or numeric support (counterexample!?), simplify or drop me a reference (or some vague hunch). We have $$B_n = n!\sum_{\lambda} \frac{\lambda^{2n}}{p'_n(\lambda)}$$ ...
3
votes
0answers
132 views

Estimate needed on a sum involving the prime counting function

Let $F(x) = \sum_{1 < n \leq x} (-1)^{\pi (n)}$ where $\pi (n) $ is the prime counting function. I am trying to understand how $ F(x) $ behaves as $ x \to \infty$. In particular, what are the ...
4
votes
1answer
269 views

Distribution function for divisors of an Integer

For a fixed $n$, let $D_n(x) = \{ d|n : d \leq x \}$ . We assume here $p \leq x \leq n/p$, where $p$ is the smallest prime factor of $n$. For example if $n = p^i$ for some prime $p$ then $D_n(x) ...
11
votes
2answers
467 views

Average orders of multiplicative functions

For a multiplicative function $f$ and $x>0$ let $$S_f(x)= \sum_{n \leq x} f(n).$$ Studying sums of this type is a favourite pastime of analytic number theorists. I'm trying to understand what kind ...
14
votes
1answer
894 views

Is there a Montgomery's conjecture for Dirichlet characters and Artin representations ?

Edit: as GH noticed, the way I tried to state Montgomery's conjecture is wrong. There were some mistakes in the references I used, which compounded with some mistakes of mine, gave a very poor post. ...
3
votes
0answers
175 views

Is this extension of the Selberg class trivial?

I came across the following modification of the Selberg class in some of my work (see below), and while I've moved on in some sense -- I submitted the paper in question -- I can't get it off of my ...
7
votes
1answer
592 views

Least prime primitive root

For $p$ a prime number, let $G(p)$ be the least prime $q$ such that $q$ is a primitive root mod $p$, that is $q$ generates the multiplicative group $(\mathbb Z/p\mathbb Z$)* . Is it known that ...
2
votes
1answer
381 views

The tightest prime zipper

Define a prime zipper as an increasing function $f(n)$ mapping $\mathbb{N}$ into $\mathbb{N}$ with the property that, for every $n \ge 1$, there is at least one prime within the inclusive interval $[ ...
5
votes
2answers
320 views

Density of integers $n$ whose totient $\varphi(n)$ is larger than $\alpha n$

Fix $0 < \alpha < 1$ a real. Let $S_\alpha$ the set of integers $n \geq 1$ such that be $\phi(n)>\alpha n$. For $x>0$, let $S_\alpha(x)$ be the number of positive integers $n$ less han $x$ ...
20
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1answer
994 views

Provable zero-free region for any entire function that analytically is similar to zeta(s)

Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$: $f(z)$ is bounded when $\Re z>1+\delta$ $f(z)$ is unbounded when $\Re z=1$ $f(z)$ grows ...
9
votes
1answer
560 views

Prime Power Gaps

In 2000, Baker, Harman and Pintz proved that there is always a prime in the interval $(n-n^{0.525}, n)$. There are also conditional results implying smaller intervals. Nevertheless, I could not find ...
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vote
1answer
243 views

Does FE of Selberg Zeta function imply Trace formula?

Does the functional equation of the Selberg Zeta function imply the Selberg trace formula? BTW, the trace formula implies the functional equation.
13
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1answer
822 views

Small primes in arithmetic sequences

Fix an integer $a>1$. For $n \geq 1$ an integer, let $\pi_{n,1}(an)$ the number of primes $p \leq an$ such that $p \equiv 1 \pmod{n}$, and $\pi(an)$ the number of all primes $p \leq an$. Let ...
4
votes
1answer
401 views

An application of Mobius Inversion in a paper of Shintani

I've been reading about Shintani zeta functions and in particular with respect to finding the density of cubic discriminants as in the theorem of Davenport-Heilbronn. In Shintani's paper "On ...
3
votes
1answer
394 views

About Theorem 3.2 in 'introduction to spectral theory of automorphic forms' by Iwaniec

In Theorem 3.2 of 'Introduction to spectral theory of automorphic forms' by Iwaniec,the first bound is about the coefficients of automorphic forms $$\sum_{|n|\le ...
23
votes
2answers
2k views

Number of elements in the set $\{1,\cdots,n\}\times\{1,\cdots,n\}$

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?
14
votes
1answer
820 views

A question about Speiser's 1934 result on the Riemann hypothesis

A number of sources concerning Speiser's 1934 result state that the Riemann Hypothesis (RH) implies $\zeta'(s)\neq 0$ for all $0<\text{Re}(s)<1/2$. But I have seen some (possibly less reliable) ...
1
vote
2answers
378 views

Analytical predicate for integers over complex numbers

A complex number $z$ is an integer if and only if $\sin(\pi z)=0$. It follows that a complex number $z$ is an integer if and only $\sin^2(\pi z) = 0$. So for a real analytic function $f$ and any real ...
5
votes
1answer
227 views

Large gaps between P2s

Gaps between consecutive primes are $O(n^{\theta+\varepsilon})$ for $\theta=0.525$ and any $\varepsilon>0.$ I was wondering if a better result is known for gaps between numbers with at most two ...
4
votes
2answers
718 views

Kronecker's Jugendtraum for real quadratic fields?

Kronecker's Jugendtraum (or Hilbert's 12'th problem) is to find abelian extensions of arbitrary number fields by adjoining `special' values of transcendental functions. The Kronecker-Weber theorem was ...
28
votes
7answers
2k views

How should an analytic number theorist look at Bessel functions?

(And a related question: Where should an analytic number theorist learn about Bessel functions?) Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...
17
votes
1answer
942 views

Number of distinct values taken by $\alpha$ ^ $\alpha$ ^ $\dots$ ^ $\alpha$ with parentheses inserted in all possible ways, $\alpha\in\mathbf{Ord}$

Let $\alpha\in\mathbf{Ord}$ and $n\in\mathbb{N}^+$. Let $F_\alpha(n)$ be the number of distinct values taken by ordinal exponentiation $\underbrace{\alpha \hat{\phantom{\hat{}}} \alpha ...
5
votes
3answers
664 views

Better error bounds for partial sums of reciprocals of primes?

One of Mertens' theorems gives that $$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k)$$ where $B$ is a constant near $0.26$ in value and $E(k)$ is an error term whose size is ...
2
votes
2answers
240 views

Are there formulas for the derivatives $\zeta_{F}^{(n)}(0)$ of Dedekind zeta functions?

Let $F/\mathbb{Q}$ be a number field. I'm interested in knowing if there are formulas for the values of the derivatives $\zeta_{F}^{(n)}(0)$ of the Dedekind zeta function of $F$ at zero. Maybe if in ...
7
votes
3answers
916 views

Sato-Tate measure for GL(3) Automorphic forms

As we have known, the Sato-Tate measure for GL(2) turned out to be the half circle measure $\frac{1}{2\pi} \sqrt{4-x^2}dx$ on [-2,2], which appears in various versions of equi-distribution problems ...
6
votes
1answer
268 views

equidistribution on the unit circle of particular sequences of finite subsets

Given a strictly convex function $g : [0, 1] \to \mathbb{R}$, I'm curious about the asymptotic distribution of the points $\exp{(2 \pi i N g(n / N))}$ for $n = 1, 2, \dots, N$, counted with ...
5
votes
1answer
764 views

Request for the proof of a result from Ramanujan's letter to Hardy.

Srinivasa Ramanujan in his first letter to G.H. Hardy stated many results for which he didn't give proofs. Among them the result taken from this link seems interesting : If ...
2
votes
2answers
522 views

Liouville's Theorem in Diophantine Approximation

Liouville's Theorem states that for any algebraic $\alpha \in \mathbb{R}$ of degree $n$, there exists a positive constant $c:=c(\alpha)$ such that $$|\alpha-\frac{p}{q}|>\frac{c}{q^n}$$ for any $p ...
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vote
2answers
709 views

Trying to debunk a claim

Claim: Take any function $f(t) > 0$ for $t > 0$, such that $f(t) \to \infty$ as $t \to \infty$, then for $\sigma > 0$ $$|\zeta(\sigma + it)| = o(f(t))$$ Is there any already existing ...
6
votes
3answers
1k views

Sum of the sum-of-divisors function

I was looking at the abstract of a paper [1] which claims that [2] and [3] prove $$ \sum_{n\le x}\sigma(n)-\frac{\pi^2}{12}x^2=\Omega(x\log\log x). $$ But I cannot find the above—or indeed, ...
6
votes
2answers
521 views

On a sum involving prime numbers

I find myself needing the asymtotics of the following summation for my work. Let $a$ be a positive real number and $p_n$ be the $n$-th prime. $$ \sum_{k=1}^{n} [k^a - (k-1)^a]p_k $$ At $a=1$, this ...
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votes
0answers
194 views

Limit of an inverse Mellin transform

In Edwards' very nice book ``Riemann's zeta function'' the following integral comes up in section 1.14. Suppose $\beta = \sigma + i\tau$ with $\sigma > 0$. Suppose $x > 1$. Fix some real number ...
7
votes
1answer
574 views

Asymptotics of the n-th prime using the gamma function

In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that. $$ p_n = n ...
4
votes
1answer
466 views

Progress on Bouniakowsky's Conjecture

Has there been any progress on the Bouniakowsky conjecture? In particular, has anyone been able to prove something for a particular polynomial - or for a class of them? (I can't seem to find ...
6
votes
2answers
588 views

Number of integers coprime to l

A long time ago I've seen a paper considering, given $\ell$ fixed, estimates for $$ \sum_{n \leq x, (n, \ell) = 1} 1 $$ Of course, this is easy to estimate with a trivial error term of ...
8
votes
1answer
1k views

What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers?

Could somebody explain to me, from a mathematical stand-point, what is a quasi-crystal, and how it relates to the set of Pisot numbers, and the Riemann Hypothesis? I've heard Freeman Dyson say that ...
2
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1answer
574 views

Generalization of Mertens' theorem

One classical Mertens' theorem tells us that $$\prod_{p \leq n} (1-\frac{1}{p})^{-1} = e^\gamma \log n + \mathcal{O}(1).$$ It is now very natural to ask, whether we have some good estimate to ...
13
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1answer
887 views

Certain functional equations for the Riemann Zeta function?

Referring to this question I asked on math.SE. I am posting a more generalized question here, for answers and further inquiry. For the Riemann zeta function, we know of the standard functional ...
3
votes
1answer
205 views

Minor Arc Estimates for an Exponential Sum for a Quadratic Polynomial Over the Primes

Let $f$ be a quadratic polynomial with leading coefficient $\alpha$, and suppose $\alpha$ is in a "minor arc" in the sense that $\alpha$ is not within $\frac{(\log N)^A}{q N^2}$ of any rational number ...