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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4
votes
2answers
154 views

Well-distribution of square of an interval $[1,p^{1-\varepsilon}]$ modulo a prime $p$

Let $p$ be a large prime. I would like to say that the multi-set $[1,p^{1-\varepsilon}]^2 = \{ab \mod p: a, b \in [1,p^{1-\varepsilon}]\}$ is close to uniformly distributed, i.e. that every nonzero ...
5
votes
0answers
42 views

Primes with prescribed digits in two bases

A result of Harman and Katai tells us that if $n$ is large, and $A$ is a subset of $\{1,\ldots, n\}$ is of size $|A|=r<c \sqrt{n}/\log n$, then the number of primes up to $N=2^n$ with the $j$-th ...
15
votes
6answers
1k views

primorial puzzlement

Let $x_n$ be the smallest positive integer which is not a quadratic residue modulo any of the first $n$ odd primes. The question is: is there any bound on how quickly $x_n$ grows as a function of $n?$ ...
7
votes
2answers
512 views

On bounds for idoneal integer

What is the best known lower bound and upper bound known for such a number if it exists and have there been any attempts (computational including) to eliminate the existence of such a number in known ...
5
votes
0answers
90 views

Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x

Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form $$ g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i ...
2
votes
2answers
187 views

Finiteness of number of consecutive primes with gap $4$

Assuming Riemann Hypothesis Hardy showed primes $3\bmod 4$ are more common than primes $1\bmod 4$ https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequences_of_the_generalized_Riemann_hypothesis. ...
5
votes
1answer
160 views

Dynamics of the distribution of prime factorization types in increasing intervals

I've tagged this as reference request as surely this question must be very well investigated, I just don't know how to look for it. Most likely the perfect answer will be in form of a keyword for ...
8
votes
0answers
150 views

Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?

Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
2
votes
0answers
57 views

Questions about holomorphy and zeros of the symmetric power $L$-function

Let $f$ be a primitive form of an even weight $k$ for the full modular group and let $L(Sym^rf,s)$ be the symmetric $r$th $(r\geq 2)$ power $L$-function associated to $f.$ I have three questions ...
7
votes
1answer
242 views

Sums of reciprocals involving divisor sums

This question was asked at MSE but never received an answer. Let $A\subset\mathbb{N}$ be a subset of the natural numbers, and let $\sigma(n)$ denote the sum of divisors of $n$. Recall that we have ...
1
vote
1answer
111 views

Need an explanation of a deduction

When I was reading the paper of Winfried Kohnen, Yuk-Kam Lau and Igore E. Shparlinski (ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS), I found this result (which is Theorem 2 of the ...
14
votes
2answers
481 views

Number of distinct factors

Denote $\omega(m)$ to be number of distinct factors of $m$ as defined in http://mathworld.wolfram.com/DistinctPrimeFactors.html. At every $c>0$, given $n\in\Bbb N$ define $$S(n,c)=\big\{m\in\Bbb ...
0
votes
1answer
98 views

Question about sign change of Hecke eigenvalues

I want to write a survey on the subject 'Sign changes for coefficients of symmetric power $L$-functions'. So, I browse the Web and I got some papers. I read it and I gave special interest to the paper ...
3
votes
2answers
310 views

Trivial zeroes of the Riemann Zeta function are simple

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that ...
7
votes
1answer
223 views

Numerically double-checking formula with L-values

I'm working with a special case of Ichino's triple product formula, which for classical holomorphic newforms $f$, $g$ ,$h$ of weights $k$, $m-k$, $m$ (and central characters satisfying $\chi_f \chi_g ...
7
votes
1answer
355 views

Bounded gaps between primes in arithmetic progressions

Has Zhang's work on bounded gaps between primes been extended to the following theorem? For any arithmetic progression $an+b,\gcd(a,b)=1$, there is a constant $H$ (depending only on $a$) such that ...
1
vote
1answer
137 views

Question about mean square estimate for sums of Dirichlet coefficients of Symmetric Power $L$-functions

I have a question related to Coefficients of Symmetric power $L$-functions and I would be grateful if you could answer it. Let $\lambda_{Sym^rf}(n)$ be the $n$th Dirichlet coefficient of ...
6
votes
1answer
223 views

A question about $(0,1]$-valued multiplicative functions

Suppose $f:\mathbb{N}\to [0,1]$ is a multiplicative function (i.e. $f(nm)=f(n)f(m)$ whenever $m$ and $n$ are coprime). Suppose $f$ has non-zero mean, which means $$ \lim_{N\to\infty}\frac{1}{N} ...
3
votes
1answer
324 views

On Heath-Brown's “Prime twins and Siegel zeros”

With a graduate student, I'm going through the paper (Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224.) We are quite baffled by what follows Lemma 3 on p. 198. Here's the background and ...
1
vote
0answers
90 views

Question about expression of a sum of two Hecke eigenvalues

I did some computations but I am stuck in finding the exression of the sum $$\lambda_f(n^2)+\lambda_f(n)^2 $$ in terms of $\lambda_f(n),$ where $f$ is a modular form for the full modular group. Any ...
0
votes
1answer
103 views

Discussion for the sign of a specific sum

Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$ I want to ...
0
votes
1answer
74 views

Expression of a sum of Hecke eigenvalues in terms of one Hecke eigenvalue

Let $f$ be a modular form of an even weight $k$ over the modular group $SL_2(Z).$ Denote $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ I am doing some calculations and I am stack in ...
0
votes
0answers
90 views

Explicit formula for the product of three Hecke eigenvalues

I am interested in Hecke eigenvalues $\lambda_f(n)$ which are normalized Fourier coefficients of modular forms of an even weight $k$ for the full modular group. I often know that these coefficients ...
8
votes
3answers
475 views

Polynomials vanishing modulo some integer $n$

It is well-known that a polynomial $q \in \mathbb Z[t]$ vanishes modulo $p$ only if it lies in the ideal $J_p$ generated by $p$ and $t^p-t$. This means that either the degree is large (at least $p$) ...
7
votes
1answer
492 views

Are there effective small intervals in which primes are dense?

As mentioned in Terry Tao's comment to this question, it is constructively known that there are primes between sufficiently large cubes. $\:$ According to wikipedia, "there exists a constant $\: ...
1
vote
1answer
213 views

reference on Dirichlet theorem on primes in arithmetic progression

I appreciate if you could help me to find a reference (and a proof). Combining Dirichlet theorem on primes in arithmetic progression with Chebotarev densitiy theorem, we know that given two positive ...
18
votes
3answers
522 views

Lower bounding the probability that $\gcd(t,N)≤B$, for a random $t$ and fixed (large) $N$

$\newcommand{\Prb}[1]{\mathcal{P}_{#1}}$ I have the following number theory problem, related to Odlyzko's improvement on Shor’s factoring algorithm (see this cstheory.sx question for details). Let ...
0
votes
0answers
150 views

arithmetic progressions with few primes

Is this true ? Let $\beta_0$ be a positive number. One may find $\beta>\beta_0$, $0<\lambda<1$, and infinitely many $q>1$ so that there exists an arithmetic progression of step $q$, $a_1, ...
1
vote
0answers
156 views

Probability of correlated residues

Given $N,c\in\Bbb N$, where $c\ll(\log N)^{1/b}$ for any $b>1$ is fixed, what is the probability that given $A_1,A_2,A_3\in\Bbb N$ with ...
2
votes
1answer
77 views

Elaboration of a certain section of a paper by Thanigasalam

In section 11 of this paper by Thanigasalam, it says "... we get $G(10)\le 105$, and this implies that $H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that $G(10)\le ...
13
votes
1answer
623 views

Elementary Proof of Infinitely many primes $\mathfrak{p} \in \mathbb{Z}[i]$ in the sector $\theta < \arg \mathfrak{p} <\phi $

A quick look at the primes in $\mathbb{Z}[i]$ suggests they might be evenly distributed by angle if we zoom out on a coarse enough scale. I would like ask about the much weaker statement forgetting ...
0
votes
0answers
79 views

Find motivation for calculating $\int_{2}^{X} A^2(t) A(\alpha t)dt$

I read a thesis of Kong Kar Lun (student of Tsang K.M) about the some mean value theorems for certain errors terms in analytic number theory and in which he gave the asymptotic formulas of the ...
1
vote
0answers
48 views

Interest to know explicit values of certain coefficients

Sorry if my question is stupid but it comes to my mind whenever I read about the theory of symmetric power $L$ functions. Out of curiosity, I did a web search and found only the explicit expression of ...
1
vote
1answer
345 views

Analytic Number Theory without Pigeonhole Principle [closed]

I don't know if this is an appropriate question for this website, but I will try my luck. I am an undergraduate student, and recently I became interested in analytic number theory. When I started ...
6
votes
1answer
246 views

Is $\liminf \frac{\sigma_{k}(n)}{n}$ finite for every $k$?

Can someone show me how to prove that $$\liminf_{n \to \infty} \frac{\sigma_{k}(n)}{n} < \infty$$ for every natural number $k$? Or is this problem open? Here, ...
7
votes
0answers
558 views

“Forthcoming paper” of Goldston-Graham-Pintz-Yıldırım

The above-named authors of [1] and its (significantly different) published version [2] write: In a forthcoming paper, we will show how the methods here can be extended to prove corresponding ...
2
votes
0answers
101 views

Distribution of residue classes of totients of (univariate) polynomials

Let $\phi$ denote Euler's totient function and $f$ a non-constant univariate polynomial with integer coefficients such that $f(u) \in \mathbf N^+$ for all $u \in \mathbf N^+$ (assume $f$ is ...
4
votes
1answer
178 views

Equidistribution of representations by a binary cubic form

Let $f(x,y)\in\mathbb{Z}[x,y]$ be a binary cubic form with nonzero discriminant, and for a positive integer $m$ consider the integral representations $f(x,y)=m$. Assume that the number of ...
7
votes
2answers
651 views

Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial

The starting point for this question is the following (false) statement $\forall n\in \mathbb{N} (n^2 + n + 41 \text{ is prime}).$ Given a polynomial function $p:\mathbb{N} \to ...
1
vote
0answers
120 views

Bounded discrepancy multiplicative functions

A rather specific question, concerning the second remark of Tao in ...
0
votes
0answers
64 views

Finite Ramanujan expansions over a finite field

I'm wondering if we could have an analogy in finite fields. After all, the Discrete Fourier Transform (DFT) has been generalized to finite fields as well (with essentially identical properties as in ...
12
votes
2answers
473 views

Infinitely many irreducible polynomials of the form f(X^2) + X mod 3?

Are there infinitely many polynomials $f \in \mathbb{F}_3[X]$ for which $f(X^2) + X$ is irreducible?
2
votes
1answer
313 views

Best known zero-free region for Dirichlet $L$-functions in the $q$-aspect

It is classical that there is a $c > 0$ such that for all Dirichlet characters $\chi$ except for at most one exception, one has that $L(s,\chi)$ has no zeroes for $\sigma > 1 - \frac{c}{\log{q} ...
0
votes
0answers
119 views

A (weak?) lower bound on primes in arithmetic progressions in short intervals

I was wondering if the following could be established by the methods that go into e.g. Linnik: $\textbf{Claim. } \text{Let $\chi$ be a nonprincipal quadratic character of conductor $q$, and (e.g.) $c ...
3
votes
2answers
257 views

Approximations to the Mertens function

The Mertens function $M(x)$ is the summatory Möbius function i.e. $$M(x) = \sum_{k=1}^{x} \mu (k)$$ The conjecture that $M(x) = \mathcal{O}\left(x^{\frac{1}{2} + \epsilon}\right)$ was shown to be ...
3
votes
0answers
199 views

Ramanujan conjecture and covariance of Kloosterman sums

There has been interest in moments and covariances/correlations of Kloosterman sums $S(m,n,c)=\sum_{ad=1\ (\text{mod}\ c)} e(\frac{ma+nd}{c})$ like $\sum_{m\in\mathbb F_c} S(m,n,c)^k$, ...
5
votes
0answers
156 views

Explicit bounds for exceptional zeros and/or $L(1,\chi)$ for real $\chi$

I would like to have an explicit upper bound (that is, one with explicit constants) for a possible real zero $\beta$ for $L(1,\chi)$ for real Dirichlet characters $\chi$. I need such a bound for real ...
3
votes
1answer
211 views

Least prime for which a square-free integer is a non-residue

Suppose $a$ is a square-free integer and $\left(\frac{a}{p}\right)=1$ for the primes $p\leq k$. I'll call $a$ a quasi-square of order $k$. What I am interested in is the maximum value of $k$ in terms ...
3
votes
0answers
130 views

Goldbach's problem in algebraic number fields [duplicate]

Are there any results on the representation of numbers in algebraic number fields as the sum of primes in the ring of integers in that field? There are some results for Waring's problem in other ...
11
votes
2answers
532 views

Easiest way to see that $\zeta_{\mathbb{Z}[i]}(s) = \zeta(s) L(s, \chi)$?

As the question suggests, what is the easiest way to see that$$\zeta_{\mathbb{Z}[i]}(s) = \zeta(s)L(s, \chi)?$$Here, $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...