**5**

votes

**1**answer

165 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

**0**answers

159 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

**0**answers

59 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

**1**answer

252 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

**1**answer

121 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 ...

**13**

votes

**2**answers

516 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 N:...

**0**

votes

**1**answer

108 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

**2**answers

345 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 $\xi(-2k)=\xi(1+...

**7**

votes

**1**answer

239 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

**1**answer

413 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

**1**answer

150 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 $L(Sym^rf,s).$...

**6**

votes

**1**answer

226 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} \sum_{...

**3**

votes

**1**answer

337 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

**0**answers

96 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

**1**answer

107 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

**1**answer

81 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

**0**answers

95 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 ...

**9**

votes

**3**answers

492 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

**1**answer

520 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

**1**answer

224 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 ...

**19**

votes

**3**answers

598 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

**0**answers

152 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, ...

**2**

votes

**1**answer

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

**1**answer

647 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

**0**answers

80 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

**0**answers

51 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

**1**answer

370 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

**1**answer

251 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, $\sigma_{k}(n)=\sigma(\sigma(\sigma(\...

**7**

votes

**0**answers

579 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

**0**answers

105 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

**1**answer

184 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

**2**answers

658 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 \mathbb{N}...

**1**

vote

**0**answers

129 views

### Bounded discrepancy multiplicative functions

A rather specific question, concerning the second remark of Tao in
http://www.michaelnielsen.org/polymath1/index.php?title=...

**12**

votes

**2**answers

490 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

**1**answer

355 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

**0**answers

135 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

**2**answers

284 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

**0**answers

229 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
$\sum_{m\in\mathbb F_c} S(m,n,c)^k$, $\sum\sum_{m_1,...

**5**

votes

**0**answers

168 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

**1**answer

221 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

**0**answers

131 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

**2**answers

580 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 $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...

**4**

votes

**1**answer

107 views

### Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$, does $L_c(s, \chi)$ necessarily equal $1$?

Consider an analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, let $c \...

**4**

votes

**2**answers

685 views

### Number of prime numbers in a range

Denote by $A_n$ the number of prime numbers between $n$ and $n + \log n$.
Is it true that $A_n < const$?
UPD: Is it true that $A_n > \log \log n$ (or something another) for infinite number $n$?...

**5**

votes

**3**answers

428 views

### Logarithmic integral, $π(x)$ and $x/(\ln x)$

The function $\text{Li}$ (logarithmic integral) is defined for $x>0$
by
$$
\text{Li}(x)=\int_2^{x}\frac{dt}{\ln t}.
$$
The prime number theorem, proven by Hadamard and de la Vallée-Poussin in 1896 ...

**7**

votes

**0**answers

110 views

### Approximation to a certain Weyl-sum

Let $ S(\alpha)=\sum_{x\leq X}\sum_{y\leq Y}e \left(\alpha x y^3 \right)$, for some $X,Y \geq 1$ and write $\alpha = a/q + \beta$ for $(a,q)=1$, as usual.
For the 'classical' cubic Weyl-sum $f(\alpha)...

**4**

votes

**1**answer

331 views

### Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?

This question expands on this one from MSE.
In the literature about Dirichlet $L$-series, I found that their Euler products:
$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$
...

**4**

votes

**0**answers

440 views

### Best known bounds on certain exponential sums

What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...

**32**

votes

**2**answers

950 views

### Can we find two positive integers $n$ and $m$ ($n,m>1$) such that $n^\pi = m$? [duplicate]

I came across this apparent random question in some math questions website. At first, i thought it was easy to show that there are not integer solutions to this equation, but then i realized that the ...

**9**

votes

**1**answer

571 views

### Do relaxed Liouville functions violate Chowla's conjecture?

Let $\lambda$ be the Liouville function. One version of Chowla's conjecture says that for each set of distinct natural numbers $h_1 , \dots , h_k$,
$$\sum_{n\leq x} \lambda(n+h_1) \dots \lambda(n+h_k)...