4
votes
1answer
221 views

Lower bound on the irrationality measure of $\pi$

There seems to be a lot of work on the upper bound for the irrationality measure of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. Looks like it is the best ...
3
votes
0answers
213 views

Is it possible to find explicit formula for the product $\prod_{d|n,\ d>1} (1-\mu(d)/\varphi(d))^{\varphi(d)}$?

I am trying to calculate the following product $$ \prod_{d|n}_{d>1} \left( 1-\frac{\mu(d)}{\varphi(d)} \right)^{\varphi(d)} $$ where the functions $\varphi$ and $\mu$ are Euler's totient and ...
3
votes
1answer
294 views

Lower bounds on the error term of the prime number theorem

Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t. $$f(x)\ll |\psi(x) - x|$$ where $\psi$ is the Chebyshev function.
3
votes
1answer
334 views

Smallest prime in an arithmetic progression

Let $\{a_n\}_{n\in\mathbb{N}}$ be defined as $a_n = a + bn$ for some $a, b >0,(a, b) = 1$. Are there good bounds on the minimal $k$ s.t. $a_k$ is prime. It is well known that there are infinitely ...
6
votes
1answer
201 views

lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$?

Are there known any lower and upper bounds for $$ \sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k, $$ where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$? Or at least is it known ...
6
votes
2answers
205 views

Divisor sums over values of binary forms of primes

Let $\tau$ be the divisor function, that is $$ \tau(n)=\sharp\{d \in \mathbb{N}, d|n\}. $$ I was wondering if anyone has ever proved an asymptotic estimate for the sum $$S(x):=\sum_{p,q\leq ...
11
votes
1answer
619 views

Szemeredi's theorem in the Gaussian integers

Suppose that $S\subseteq\mathbb{Z}[i]$ has the following properties: For convenience, let $A_n = \{z : z\in\mathbb{Z}[i], \text{Nm}(z)\le n\}$ $$\limsup_{n\rightarrow\infty} \frac{|S\cap ...
4
votes
3answers
165 views

Expression for the derivative of Eisenstein series $G_2$

I am new to number theory, so I am guessing this is a standard formula. I would be grateful for a reference: We know that the Eisenstein series $G_2$ is quasimodular of level $SL_2(\mathbb Z)$, so ...
9
votes
1answer
298 views

Squarefree numbers $n$ such that $432n+1$ is also squarefree

This is a second attempt (see Primes $p$ such that $432 p +1$ is prime) Is the set of squarefree numbers $n$ such that $n(432 n+1)$ is also squarefree known to be infinite? Fact: the number of such ...
-1
votes
1answer
290 views

Primes $p$ such that $432 p +1$ is prime [closed]

Is the set of prime numbers $p$ such that $432 p + 1$ is also prime infinite? It doesn't follow from Dirichlet's theorem as far as I can tell.
5
votes
2answers
563 views

The shortest interval for which the prime number theorem holds [closed]

It is well known that the prime number theorem on the form \begin{align*} \pi(x+y) - \pi(x) \sim \frac{y}{\log (x+y)} \end{align*} breaks down for short enough intervals, e.g. taking $y=(\log ...
6
votes
1answer
638 views

Probability that a positive integer is the euler phi function of another positive integer

Define $f(n) = |\{m : m\le n, \exists k \text{ s.t. }\phi(k) = m\}|$. Clearly, $f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1$ since $\phi(n)$ is even for all $n > 2$. Is ...
1
vote
2answers
110 views

Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions

For any real number $x$, let's define $Om_{k}(x)$ as the number of positive integers $m$ below $x$ such that $\Omega(m)-\omega(m)=k$, where $\omega(n)$ is the number of distinct primes dividing $n$, ...
8
votes
2answers
390 views

Incomplete Kloosterman sum

I am interested in an upper bound on the following incomplete Kloosterman sum $$ \sum_{\substack{x=1 \\ x+_{_{\bf Z}}x^{-1}>p}}^{p-1}e\left(\frac{x+x^{-1}}{p}\right).$$ Using the Weil's bound it ...
2
votes
0answers
150 views

Kloosterman-like sum with inverse to different moduli

In some recent work, the following strange-looking exponential sum arose: $$ \sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg). $$ Here $e(x) = e^{2\pi i x}$ as ...
2
votes
0answers
140 views

Arguments for the second Hardy–Littlewood conjecture being false?

Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that $$\pi(x + y) - \pi(y) \leq \pi(x).$$ We can easily justify this heuristically, since $$ ...
17
votes
1answer
647 views

The conjecture of Montgomery and Soundararajan on primes in short intervals: Empirical inconsistencies?

Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...
0
votes
0answers
264 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
2
votes
1answer
226 views

Proof of the Friedlander–Iwaniec theorem

Does anybody know where I could find the proof of the Friedlander–Iwaniec theorem. The link that I find when I search for it is http://www.pnas.org/content/94/4/1054.full.pdf+html, but this seems more ...
5
votes
0answers
238 views

Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...
9
votes
2answers
466 views

Bound on gcd of two integers

Well this is a problem I was fiddling with. I came up with it but it probably is not original. Suppose $a\in \mathbb{N}$ is not a perfect square. Then show that : ...
16
votes
1answer
378 views

Covering a set with geometric progressions

Consider the set $S_n=\{1,2,\cdots ,n\}$. What is the minimum number of distinct geometric progressions that cover $S_n$? Let us call this number $a_n$. I was wondering about this number after doing a ...
8
votes
0answers
153 views

Symmetric Fifth Power Lift of GL(2) Automorphic Form

Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that $$L(s, \pi, ...
1
vote
0answers
111 views

Order of individual Fourier coefficient of a Maass form

Let $D$ be a definite quaternion division algebra over $\mathbb{Q}$ and $\mathcal{O}$ be an Eichler order of $D$. Let $F$ be a Maass form in $L^2(PGL_2(\mathcal{O})\backslash ...
4
votes
1answer
174 views

looking for reference on dihedral, tetrahedral, or octahedral forms

I am looking for a reference on dihedral, tetrahedral, or octahedral forms. As far as I read, they are some cuspidal automorphic forms on $GL(2)$ induced from $GL(1)$. Dihedral is from $GL(1)/K$ to ...
4
votes
1answer
150 views

Log weight removal in general (weaker) prime number theorem

Let $a_n$ be a sequence of non-negative numbers. Assume that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p\log p}{X}\leq 1.$$ Can we prove that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} ...
3
votes
1answer
125 views

A question on the big-O value of the complex integral especially in the number theory

My question is quite simple and elementary. Let $A(x)=\sum_{1}^{x}a(n)$ and $\alpha(s)=\sum_{1}^{\infty}a(n)n^{-s}$. Then, as we know, $$ A(x)= ...
7
votes
1answer
240 views

Is the adjoint L-function on GL(m) holomorphic?

Let $\pi$ be an automorphic representation on GL($m$)/$\mathbb{Q}$. Define $$L(s,\pi,Ad):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an L-function with euler products of degree $m^2-1$. ...
5
votes
1answer
206 views

absolute convergence of Rankin-Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representation on $GL(n)$ and $GL(n')$ over $\mathbb{Q}$. I heard that the rankin-selberg convolution L-function $L(s,\pi\times\pi')$ is absolutely ...
6
votes
0answers
119 views

Eisenstein series over a definite division algebra

Let $D$ be the definite quaternion division algebra over $\mathbb{Q}$. $\mathcal{O}$ is a maximal order inside $D$, let's fix $\mathcal{O}$ to be the Hurwitz quaternion. Let ...
7
votes
1answer
171 views

standard zero free region of automorphic L-function on GL(N)

Let $L(s,\pi)$ be the standard(Godement-Jacquet) $L$-function of $\pi$, where $\pi$ is a cuspidal automorphic represetation of $GL(m,A_Q)$. What's the standard zero-free region for $L(s,\pi)$? any ...
4
votes
1answer
191 views

Uniformity of the distribution of the prime numbers on the prime residue classes (mod $m$)

Given positive integers $m$, $r$ and $n$, let $\pi(m,r,n)$ denote the number of prime numbers $p \leq n$ in the residue class $r$ (mod $m$). Further let $1 = r_1 < r_2 < \dots < ...
2
votes
0answers
211 views

Relation between Maier's theorem and a conjecture of Montgomery and Soundararajan

Let us consider the number of primes in the interval $[N,N+h]$, with $h\leq N$. According to the answer given by Lucia to a previous question on the distribution of primes, it is natural to consider ...
1
vote
0answers
100 views

Prime counting function with a form of finite product using perron's formula

There's a form of complex integral what Riemann obtained to finding $\pi (x)$, $$ \pi^{*}(x)=\int_{L}\frac{\log \zeta (s)}{s}x^{s}ds, (1)$$ we already know that it lead us to the Prime Number ...
4
votes
1answer
566 views

What is the critical idea behind Hardy-Littlewood circle method?

I want to know what the critical idea behind Hardy-Littlewood circle method is. It seems that they divide the circle into major arcs and minor arcs to ignore the singularities of generating function ...
0
votes
0answers
116 views

Non existence (or existence) of a set that is equidistributed modulo $q$ for every $q$

I have been thinking about some set that is equidistributed modulo $q$, uniformly in $q$ in some sense. I was starting to think this particular condition, which I describe below, is too strong and ...
7
votes
1answer
402 views

Major arcs in the proof that every odd number is the sum of at most 5 primes

In his proof that all odd numbers greater than 1 are the sum of at most 5 primes, Terence Tao uses one large major arc around 0 rather than small ones around the rationals, which I am more accustomed ...
3
votes
1answer
149 views

Wiener-Ikehara tauberian theorem and order of pole at s=1

In the introduction to Akshay Venkatesh's thesis "Limiting Forms of the Trace Formula" we have the following statement : "For, in summing over primes, the limit ...
25
votes
1answer
701 views

Is it possible to show that $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?

Let $\mu(n)$ denote the Mobius function with the well-known Dirichlet series representation $$ \frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}. $$ Basic theorems about Dirichlet series ...
3
votes
1answer
247 views

An introduction to sieve method and their application, Cojocaru & Murty

On page 188. Lemma 10.2.3 is $\sum_{\substack{\delta \leq x \\ 2 \nmid \delta}}\frac{\mu^{2}(\delta)}{\phi_{1}(\delta)} = A_{1}log(x) +A_{2} + O(\frac{1}{x^{1/4}})$ for positive $A_{1}$, $A_{2}$. ...
1
vote
1answer
144 views

Two-dimensional Perron formula

There is a well-known Perron formula, which connects a mean value of certain arithmetic function with its Dirichlet series: $$ \sum_{n\le x} f(n) = {1\over 2\pi i} \int_{c-i\infty}^{c+i\infty} F(s) ...
3
votes
2answers
227 views

Real character modular forms: Fourier coefficient real?

Let $f$ be a modular form of level $N$ and real character $\chi$ of mod $N$ and weight $k$. Does the Fourier coefficient or hecke-eigenvalue of $f$ have to be real? What I knew is that if $N=1$ and ...
7
votes
1answer
337 views

Prime races à la Mertens

I have just read the nice survey by Granville and Martin about prime races. I wonder what happens if one changes the rules for the prime races as follows. Fix $q$ a modulus (an integer $>1$). For ...
3
votes
1answer
133 views

Can the generalized divisor summatory function $D_z$ be expressed explicitly in terms of Zeta Zeros?

Mertens function has, by residues, an explicit formula of $M(n)=\displaystyle\sum_{\rho}\frac{x^\rho}{\rho\zeta'(\rho)}-2+\sum_{n=1}^\infty\frac{(-1)^{2 n}(2\pi)^{2n}}{(2n)! n \zeta(2n+1)x^{2n}}$ ...
0
votes
1answer
234 views

A convergence issue [Edited]

Let $\{x_n\}_{n=1}^\infty$ be a sequence of vectors in a Hilbert space $$l^2_{k^{-2}}:=\{z=\{z(k)\}_{k=1}^\infty:\sum\limits_{k=1}^\infty z(k)^2k^{-2}<\infty\}.$$ It is known that for some $x\in ...
4
votes
1answer
274 views

Existence of a certain subset of natural numbers equidistributed modulo $m$ for every $m$

I was talking to a friend and the following set $S$ came up. Let $f$ be some real valued function tending to infinity. Let $S$ be a subset of natural numbers such that $|S \cap [1,N]| = N^{\delta}+ ...
4
votes
1answer
257 views

Circle method on things other than the integers

The circle method is often used to estimate the number of solutions to the equation $$x_1 + x_2 + ... x_k = N$$ if for all $i$ $x_i\in A\subseteq\mathbb{N}_0$ and some subset of the nonnegative ...
8
votes
1answer
397 views

The sum over zeros in the explicit formula for $\zeta(s)$

The explicit formula for $\zeta(s)$ is: $$ \psi(x)=x-\sum_{|\operatorname{Im}\rho|<T}\frac{x^\rho}{\rho}-\log(2\pi)-\log\left(1-\frac{1}{x^2}\right)+O\left(\frac{x\log^2T}{T}\right), $$ where ...
11
votes
1answer
387 views

Roots of unity near 1 in Z / p Z

Let $r \ge 3$ be a fixed integer. I'm interested in primes p such that no integer in the interval $(-\sqrt{p}, \sqrt{p})$, except $1$ (and $-1$ if $r$ is even), is an r-th root of unity modulo p. The ...
2
votes
1answer
257 views

Square-free integers not divisible by any “small” primes

I have two very related questions: If $f(N)$ is the number of square-free integers in the interval $[1, N]$, it is well known that $$f(N) \sim \frac{6}{\pi^{2}} N.$$ My first question is, if we ...