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

-1
votes
0answers
59 views

Estimating an exponential sum of a particular type

I was trying to estimate the following exponential sum: For given irrationals $\alpha$ and $\beta$, and given integer $x$ , let $$S(x,\alpha,\beta)=\sum_{n\leq N}\sum_{m\le M}A(m)B(m,n)e(x(m\alpha-n\...
3
votes
0answers
159 views

Artin conjecture on L-functions

Artin conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group $\Gamma$ of a number field admits ...
0
votes
0answers
79 views

What is the sharpest bound of this sum?

Fix $y\geq 1$ and let $\delta$ be a small enough positive real number. Put $$\mathcal{D}^{+}=\left\{d=p_{1}...p_{l}: p_{l}<...<p_{1},\ p_{m} \leq y_{m} \ \textrm{for all odd} \ m \right\},$$ ...
-1
votes
1answer
155 views

When the Ratio of Two Factors is a power of $2$ (i.e. $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$)

We can write, $n!= 2^s \times a \times b$ where $gcd(a,b)=1$ and $2^{s+1} \nmid n!$. Problem: Is there infinite number of $n$ when $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$? Question: 1.How ...
0
votes
0answers
148 views
+50

Deduction formula for Goldbach counting function

Assume $N\geq 1$ is integer and $P\geq 1$ is square-free integer. Goldbach counting function, $S_P(N,x)$, is defined to be the number of $n$ between 1 and $x$ such that $(N-n)(N+n)$ is co-prime to $P$....
5
votes
0answers
120 views

Chowla's Construction of prime having least quadratic non-residue $\gg \log p$

This paper by NC Ankeny mentions that " S. Chowla has proved that there exist infinitely many primes $k$ where the first $c_1\log k$ residues $(\bmod k)$ are all quadratic residues". I recently ...
5
votes
1answer
191 views

Numbers divisible only by primes of the form 4k+1

Let $A(N)$ denote the number of positive integers $n\le N$ composed of prime numbers $p\equiv 1\pmod 4$ only. Is there an asymptotic formula for $A(N)$ (as $N$ tends to infinity)?
2
votes
0answers
118 views

Spectral decomposition on GL(n)

If $\Delta_1, \ldots, \Delta_{n-1}$ are a basis of the ring of commuting bi-$SL(n,R)$-invariant differential operators, $L_0^2=L_0^2(SL(n,Z)\backslash SL(n,R))$ is the space of cuspidal automorphic ...
2
votes
2answers
267 views

Asymptotic estimate of the probability of $(n, P(\sqrt{x})) \leq x$?

Let $P(x)$ be the product of all primes less or equal to $x$. The probability of $(n, P(\sqrt{x})) \leq x$ for an arbitrary $n$ is then given exactly by $$ \prod_{p\mid P(\sqrt{x})}{\left(1-\frac{1}{p}...
2
votes
0answers
75 views

Relation between the sign of the Stieltjes constants and some zero-free region of $\zeta$

One may recall that the Stieltjes constants $\gamma_{k}$ appear as the scaled coefficients in the regular part of the Laurent series expansion of the Riemann zeta function about $s = 1$: $$ \begin{...
0
votes
0answers
75 views

Prime Ideal Theorem for Real Quadratic Number Rings over Hyperbolic sectors?

As a follow-up to my previous question about variants of Prime Ideal Theorems about Imaginary quadratic number rings found here, I am now asking about the existence of such a companion result for Real ...
7
votes
0answers
198 views

A variant of the equidistribution of primes in an imaginary quadratic number ring

It is known that the arguments of prime elements of $\mathbb{Z}[i]$ are equidistributed in $(0,2π)$ (by Theorem 5.36 of Iwaniec and Kowalski, or one of Kubilius' papers cited below). This theorem ...
7
votes
2answers
373 views

Power sums of p-th roots of unity

The following question was asked by a colleague of mine. For any prime $p$ consider $$ M_p:=\min_{z_1,\dots,z_p}\max_{j,k}\left|z_1^k+\dots+z_j^k\right|,$$ where $z_1,\dots,z_p$ are the complex $p$-th ...
13
votes
1answer
396 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
332 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
107 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
85 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
126 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 ...
0
votes
0answers
57 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
181 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
194 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
474 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 ...
1
vote
1answer
244 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 ...
3
votes
1answer
187 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 ...
7
votes
1answer
152 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?
3
votes
2answers
410 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
2answers
313 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
118 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
0answers
177 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 ...
0
votes
0answers
48 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 $S$....
3
votes
0answers
184 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,...
3
votes
1answer
110 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=\...
0
votes
0answers
51 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 ...
2
votes
0answers
378 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}^\...
1
vote
2answers
191 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 ...
1
vote
2answers
299 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 ...
7
votes
1answer
164 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 ...
4
votes
1answer
202 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 ...
2
votes
3answers
349 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 ...
2
votes
1answer
114 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
462 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,...
4
votes
1answer
142 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 ...
18
votes
0answers
375 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
278 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
0answers
264 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+...
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 ...
0
votes
0answers
165 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:...
1
vote
0answers
80 views

Intersection of modular polynomial roots

Let $l,l'$ and $p$ be three distinct prime numbers and $\Phi_k(X,Y)$ is $k$-th modular polynomial defined over $GF(p)$. Suppose that we know $\Phi_l(X,j)$ and $\Phi_{l'}(X,j)$ have two roots. Is this ...