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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7
votes
1answer
443 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 ...
13
votes
1answer
280 views

Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums?

Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be ...
3
votes
3answers
316 views

Jacobi's theorem on sums of two squares (reference request)

One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals $$4(d_1(n)-d_3(n)),$$ where the function $d_i$ counts the number ...
0
votes
0answers
90 views

Weyl sums with polynomial coefficients

Let $$ f(x,N) = \sum_{0 \leq n\leq N} e(x n^2).$$ Weyl's inequality gives an estimate for $f(x,N)$ when $x$ is near a rational with small denominator. My question is: What estimates are ...
1
vote
0answers
75 views

Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$

Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let $$ \mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} . $$ We ...
1
vote
1answer
117 views

Level dependence in the Ramanujan-Petersson Conjecture for GL(2) Maass forms

Suppose $f(z) = \sum_{n \geq 1} A(n)n^{\frac{k-1}{2}} e(nz)$ is a weight $k$ holomorphic cusp form on $\text{GL}(2)$. Then the Ramanujan-Petersson conjecture (proved in this case by Deligne) says ...
3
votes
2answers
360 views

Relation of these two Dirichlet $L$-functions

Let $\chi$ and $\psi$ be two quadratic Dirichlet characters and let $L(s,\chi)$ and $L(s,\psi)$ their associated Dirichlet $L$-functions. Is there a realtion between these two Dirichlet $L$-functions:...
2
votes
0answers
368 views

Analytically continuing the limit of this series?

Main Question I believe the following formula gives the right answer: $$ \lim_{k \to \infty} \lim_{n h \to k} \left( \sum_{r=1}^n c_r f(hr) h\right) = \int_0^\infty f(x) \, dx \times \sum_{r=1}^\...
2
votes
1answer
317 views

Some identities with the Riemann-Hurwitz zeta function

The only definition that I have ever seen of this Riemann-Hurtwitz zeta-function is this, For $0 < a \leq 1$ we have the identity $$ \zeta(z, a) = \frac{2 \Gamma(1 - z)}{(2 \pi)^{1-z}} \left[\sin ...
1
vote
1answer
240 views

Density of Sophie Germain $3\bmod 4$ primes

Is there any reason to expect the density of primes $p$ such that $2p+1$ is also a prime where $p=3\bmod4$ holds would be different from case of $p=1\bmod4$? What if $2p+1$ is replaced by $2p-1$ and ...
0
votes
0answers
56 views

Is there a generalized Mac-Lauren summation formula for this sum?

Let $\chi$ be a Dichlet character and $f$ be a derivable function $k+1$ times. Is there a generalized Mac-Lauren summation formula for this sum $\sum_{n \leq x}\chi(n) f(n)$ i.e is there an ...
7
votes
1answer
175 views

Main term in the number of sign changes of $\psi(x) - x$

Define $N_\Delta(T)$ to be the number of sign changes of $\psi(x) - x$ in the interval $[1, T]$. Landau's Theorem says $N_\Delta(T)$ is $\Omega(\log T)$ [1]. But perhaps that estimate is too crude. ...
2
votes
2answers
187 views

Lower bound for the number of representations of integers as sum of squares

Let $k\geq 4$. As usual, let $r_k(n)$ denote the number of ways to represent $n$ as the sum of $k$ squares. Is this true that for every $\varepsilon>0$, one has $r_k(n) \gg n^{\frac{k}{2}-1-\...
1
vote
1answer
467 views

What is the best currently proven bounds on prime gaps?

I did some digging around on the internet but I found tons of different equations on both lower and upper bounds for the largest possible prime gap g(n). I was wondering what are currently the best ...
0
votes
0answers
42 views

On a count of certain number of primes in an interval

Fix a prime $p$, $\alpha\in(0,1)$ and $\beta\in(1,2)$ and let $\mathcal U$ be primes in $[p^\alpha,\beta p^\alpha]$ such that if $b\in\mathcal U$ and if $d$ is multiplicative inverse of $b$ in $\Bbb ...
3
votes
1answer
184 views

Divisibility of Dirichlet L-functions

Let $k$ be an even integer and $p$ a prime number such that $p-1|k$. Suppose that $p$ does not divide $L(1-k,\chi)$ and $L(1-k,\psi)$, where $\chi$ and $\psi$ are quadratic characters. Can we deduce ...
6
votes
1answer
143 views

Density of prime divisors of $a^n + b$

Suppose $a > 1, b \neq 0$ be two rational numbers. Is it known in general that the set of prime divisors of (the numerator of) $a^n + b$ has a positive relative density?
2
votes
2answers
310 views

How many integer solutions of $a^2+b^2=c^2+d^2+n$ are there?

Are there any references to study the integer solutions (existence and how many) of Diophantine equations like $a^2+b^2=c^2+d^2+2$, $a^2+b^2=c^2+d^2+3$, $a^2+b^2=c^2+d^2+5$...? Actually, I can prove ...
2
votes
1answer
116 views

Clarification request on sign changes of Hecke eigenvalues

In their paper 'Sign changes of Hecke eigenvalues', Matomaki and Radziwill established in Lemma 6.2 the following result: There exists absolute positive constants $c$ and $\eta$ such that uniformly in ...
3
votes
1answer
108 views

Moving from $\Re(s) = 1+\epsilon$ to $\Re(s) = \frac{1}{2}$ in the proof of the Weil-Guinand explicit formula

In all proofs of the Weil-Guinand explicit formula, there's this step (this is from Paul Garrett's notes): Now consider this: (1) $\frac{\zeta^\prime(s)}{\zeta(s)}$ has poles at $s=1$ and $s=\...
3
votes
0answers
169 views

Effective prime number theorem

The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...
3
votes
0answers
183 views

Number of solutions to $x_1x_2=x_3x_4\bmod n$

In https://www.math.ksu.edu/~cochrane/research/xyuvmodp.pdf it is shown $x_1x_2=x_3x_4\bmod p$ where $p$ is a prime has $\frac{|\mathcal B|}p+O(\sqrt{|\mathcal B|}\log^2p)$ solutions $(x_1,x_2,x_3,...
0
votes
0answers
47 views

Density of set of primes which avoid given finite set of residues modulo powers of all primes

Let $k\in\mathbb{N}$, $k\ge2$ and $S\subseteq\mathbb{Z}$ be a finite set of integers. For every prime $p$ let $c_p$ be a number of invertible residue classes mod $p^k$ that contain some element of $S$....
2
votes
0answers
261 views

An explicit formula for $\zeta(2m+1)$ with good convergence

The question: Is the following formula known? $$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!}\cdot \sum\limits_{v=k}^m \frac{(2^{2v-2k+...
3
votes
1answer
196 views

Non-vanishing of L-function of modular form

There is a theorem by Langlands and Shalika (link) that the L-function of a cuspidal automorphic representation does not vanish on the line $\mathrm{Re}( s)=1$ (in their normalization which might be ...
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)...
0
votes
0answers
49 views

Estimating exponential sum of the form $\sum e( \alpha_1 f + \alpha_2 L)$, where $f$ is quadratic and $L$ is linear, on the minor arcs

Let $f(x_1, x_2, x_3)$ be a degree two homogeneous polynomial with coefficients in $\mathbb{Z}$. Let $L(x_1, x_2, x_3)$ be a linear homogeneous polynomial with with coefficients in $\mathbb{Z}$. Let ...
3
votes
1answer
153 views

Are there L-functions of degree 1 that aren't Hecke L-functions?

I don't know of any examples and I don't know of any results which prohibit them
-1
votes
1answer
274 views

Consequences of the degree conjecture

the title is quite explicit: I would like to know the consequences of the degree conjecture for the Selberg class. Thank you in advance.
1
vote
2answers
189 views

Implicit constant in Tenenbaum's result

In his famous book 'Introduction to Analytic and Probabilistic Number Theory', Gérald Tenenbaum established the following result (Theorem III.3.5): Let $g$ be a positive multiplicative function and ...
3
votes
1answer
695 views

Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function

Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as $$ D(n) = \sum_{k=1}^{n}d(k) , $$ where $$ d(n) = \sum_{k|n}^{n}1. $$ One can observe the following pattern in the values of $...
1
vote
3answers
441 views

Chebychev Function in Arithmetic Progressions

In the paper "exponential sums with multiplicative coefficients", Maier claims that the Chevbychev function in arithmetic progression satisfies $\theta(x, r, a) = x/(r-1) +O(x^{1-1/r^\epsilon})$ for ...
3
votes
1answer
349 views

Dirichlet divisor problem with multiple factors

Let $D(x)$ denote the number of pairs of natural numbers $(j,k)$ such that $j k\leq x$. It is well known that $D(x) =x \log x + (2 \gamma-1) x + $lower order terms in the large $x$ limit (Dirichlet ...
1
vote
2answers
284 views

Is the singular integral that come up in circle method independnet of the representatin of the equations?

Let $F_1(\mathbf{x}), F_2(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous polynomial. For the system of equations $$F_1(\mathbf{x})= F_2(\mathbf{x}) =0,$$ we have the ...
2
votes
3answers
343 views

Primes in arithmetic progressions in number fields

My general question is how does one prove equi-distribution results for primes in arithmetic progressions in number fields? I am interested in the equi-distribution of prime elements of the ring of ...
7
votes
1answer
162 views

Do arbitrary $K$-twin primes exist?

Polignac's conjecture states that for any positive even integer $K$, there exist infinitely many pairs of primes such that their difference is $K$. I am interested the status in a much weaker form of ...
2
votes
1answer
112 views

On the construction of a certain sequence of integers

Let $\alpha \in \mathbb{R}$ be a fixed (positive) number. For each $k \in \mathbb{N}$ we choose $\varepsilon_k >0$ with the property that $\lim_k \varepsilon_k =0$. If $\alpha \in \mathbb{R} \...
10
votes
1answer
456 views

What is the motivation behind Ramanujan's conjecture?

One motivation I have seen given for Ramanujan's conjecture for the estimate $$ |a_p|< C p^{k - \frac{1}{2}} $$ for the Fourier coefficients of a cusp form of weight $2k$ is that it allows one to ...
2
votes
0answers
115 views

$f(x)$-th largest number of prime factors

Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words,...
42
votes
15answers
6k views

Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$? Background: The non-vanishing ...
14
votes
3answers
4k views

Are the nontrivial zeros of the Riemann zeta simple?

A few years ago, I found on arXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann zeta function were all ...
4
votes
1answer
140 views

What explains the asymptotic and the pattern in this sequence related to Riemann zeta zeros?

As the starting point for my experiment I assumed that the imaginary parts of the Riemann zeta zeros are of the form: $$\Im \{ \rho_n \} = \frac{2\pi}{\log x_n}$$ where $x_n$ is unknown. Therefore I ...
19
votes
3answers
512 views

Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...
18
votes
0answers
374 views

Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all $x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...
4
votes
2answers
275 views

The best possible density in Hilbert's Irreducibility Theorem

Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen (http://plms.oxfordjournals.org/content/...
2
votes
7answers
2k views

Bertrand postulate

I believe there was an old conjecture that there's always a prime number between N and 2N. What's the history and how is this ...
20
votes
1answer
1k views

How many primes can there be in a short interval?

Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$. What is $$ \limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \...
1
vote
1answer
67 views

Asymptotic for zeroes of $L(s,\chi)$ in a disk $|s|<R$

In 'Remarks on Weil's quadratic functional..' p.191 Bombieri claims any given $L$-function $L(s,\chi)$ has at least $$\big(\frac{1}{\pi}+o(1)\big)R\log R$$ zeroes in a disk $|s|<R$. Is there a ...
15
votes
2answers
2k views

How did Riemann calculate the first few non-trivial zeros of the zeta-function?

Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi (z)...
0
votes
0answers
155 views

Asymptotic value of sum over Möbius function

Consider the sum $$ S(x)=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}|\mu(d)|, $$ where $P(x)$ is the product of all primes less than or equal to $x$, and $\mu(d)$ is the Möbius function. Q:...