**2**

votes

**0**answers

19 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

169 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

93 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

**2**answers

378 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 ...

**42**

votes

**4**answers

3k views

### If the Riemann Hypothesis fails, must it fail infinitely often?

That is must there either be no non-trivial zeros off the critical line or
infinitely many?
I'm sure that no one believes otherwise, but I've never seen a theorem in the
literature addressing this. ...

**0**

votes

**1**answer

78 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 ...

**-2**

votes

**0**answers

125 views

### $\mathsf{GCD}$ in arithmetic progression

Given $\mathsf{M\in\Bbb N}$, pick $\mathsf{r,s,A,B\in\Bbb N}$ randomly with $\mathsf{0<r<s<A<B<M}$ satisfying $\mathsf{gcd(A,B)=1}$.
Given $\mathsf{c\geq1}$, what is the probability ...

**2**

votes

**2**answers

266 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 ...

**3**

votes

**0**answers

151 views

### On covering by smooth numbers

Denote $P(n)=\mathsf{greatest}\mbox{ }\mathsf{prime}\mbox{ }\mathsf{factor}\mbox{ }\mathsf{of}\mbox{ }n$.
Denote $S(x,y)=\{n<x: P(n)<y\}$.
Denote $S_t(x,y)=\sum_{i=1}^tS(x,y)$ as $t$-fold ...

**7**

votes

**1**answer

190 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

300 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

110 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 ...

**19**

votes

**4**answers

1k views

### Given a prime $p$ how many primes $\ell<p$ of a given quadratic character mod $p$?

There was this question for which my response was unusally popular, so I dare to ask the following:
(1) Given a prime $p>2$, how many primes $\ell < p$ there exist which are quadratic residues ...

**6**

votes

**1**answer

202 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} ...

**22**

votes

**6**answers

3k views

### Explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros.
Because there are various explicit formulae ...

**3**

votes

**1**answer

224 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 the proof of Lemma 3 on p. 198.
Here's the background and ...

**3**

votes

**1**answer

593 views

### Number of solutions in a sum of squares Diophantine equation

Let $n$ be an integer. I'm interested in upper bounding the number of integer solutions of
\begin{equation*}
x_1^2+x_2^2+\ldots+x_p^2=y_1^2+y_2^2+\ldots+y_p^2,
\end{equation*}
where for all ...

**1**

vote

**0**answers

83 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

96 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 ...

**35**

votes

**3**answers

3k views

### Number of elements in the set $\{1,\cdots,n\}\cdot\{1,\cdots,n\}$

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?

**0**

votes

**1**answer

66 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 ...

**1**

vote

**0**answers

149 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 ...

**0**

votes

**0**answers

81 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 ...

**7**

votes

**3**answers

383 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 $p$) ...

**44**

votes

**3**answers

7k views

### Is a “non-analytic” proof of Dirichlet's theorem on primes known or possible?

It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} ...

**5**

votes

**1**answer

440 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 $\: ...

**13**

votes

**2**answers

491 views

### What are Mean Values of Ideal Densities in Galois Extensions?

In an unfinished (and as of now unpublished) article intended for the encyclopedia of mathematics, Arnold Scholz wrote:
"Classifying extensions according to the Galois group
of their normal closure ...

**1**

vote

**1**answer

189 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

**3**answers

405 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 ...

**26**

votes

**2**answers

2k views

### Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mathematica knows that:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$
The von Mangoldt function should then be:
...

**0**

votes

**0**answers

147 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, ...

**9**

votes

**2**answers

629 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 ...

**2**

votes

**1**answer

66 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 ...

**17**

votes

**1**answer

1k views

### Quantitative lower bounds related to Zhang's theorem on bounded gaps

Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ ...

**13**

votes

**1**answer

578 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

75 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

45 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 ...

**5**

votes

**3**answers

363 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 ...

**11**

votes

**2**answers

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

**1**

vote

**1**answer

315 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 ...

**5**

votes

**1**answer

165 views

### Bounding a Sum of Adjoint L-Function Values

Fix integers $k\geq2$ and $N>1$, and let $S(k,N)$ denote the normalized new Hecke eigenforms in $S_k(\Gamma_1(N))$. [If it makes my question easier to answer, feel free to replace this with ...

**5**

votes

**1**answer

211 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

**0**answers

505 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 ...

**85**

votes

**4**answers

5k views

### If $2^x $and $3^x$ are integers, must $x$ be as well?

I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number.
If $n^x$ is an integer for ...

**2**

votes

**0**answers

94 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

172 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 ...

**9**

votes

**2**answers

871 views

### Asymptotics of the n-th prime using the gamma function

In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that.
$$
p_n = n ...

**19**

votes

**2**answers

6k views

### Why Is $e^{\pi\sqrt{232}}$ an Almost Integer?

We have already discussed why $e^{(\pi\sqrt{163})}$ is an almost integer.
Why are powers of $\exp(\pi\sqrt{163})$ almost integers?
Basically $j(\frac{1+\sqrt{-163}}{2} ) \simeq 744 - ...

**12**

votes

**1**answer

804 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 ...

**1**

vote

**0**answers

76 views

### Bounded discrepancy multiplicative functions

A rather specific question, concerning the second remark of Tao in
...