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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2
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0answers
96 views

Primitive triples in a region [duplicate]

Are there at least $cn$ Pythagorean triples and at least $dn$ Primitive Pythagorean triples $(A,B,C)$ with $$2^{\frac n2}<A<2^{\frac n2+1}<2^n<B<C<2^{n+1}$$ with some fixed ...
4
votes
0answers
89 views

Shifted convolution problem for Coefficients of automorphic forms

The shifted convolution problem for coefficients of modular forms is well studied and many estimates were established for the shifted convolution sums of Hecke eigenvalues. So, one may ask about the ...
7
votes
1answer
204 views

Behaviour of $\zeta(1-it)/\zeta(1+it)$?

I am trying to understand the behaviour of $$\int^\infty_{-\infty}\frac{\xi(1-it)}{\xi(1+it)}h(t)\frac{dt}{t}$$ where $h$ is a Schwartz function on $\mathbb R$, and $\xi(s)$ the completed Riemann zeta ...
6
votes
1answer
221 views

Logarithmic weights on number theoretic sums

Suppose we are interested in the sum $\sum _{n\leq x}a_n.$ The study of the sum $\sum _{n\leq x}a_n\log (x/n)$ may be easier. What can one say about the first sum from knowing the behaviour of ...
3
votes
1answer
377 views

Greatest number of coprime numbers between two numbers

We know that from prime number theorem that the number of primes below $n$ is approximately $$\frac{n}{\log_en}.$$ $\star$ Given $n,m$, what is the largest list of pairwise coprime numbers that one ...
14
votes
1answer
324 views

Connection between Bernoulli numbers and Riemann-Siegel theta function?

I have come across a strange approximation for the Riemann-Siegel theta function involving the Bernoulli numbers - namely that $$\frac{1}{2} \log \left| B_{2 n}\right|\approx \vartheta (2n)\ ,\quad n ...
6
votes
1answer
243 views

Lattice points on the boundary of an ellipse

How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). ...
1
vote
0answers
95 views

More general better error bounds for partial sums of reciprocals of primes

In this question Gerhard Paseman asks for explicit estimates on $E(k)$ in $$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k).$$ There are some nice answers, but my question is on ...
6
votes
1answer
487 views

An elementary lower bound on the number of primes

Recall the second Chebyshev function: $$\psi(x) = \sum_{p \leq x} \lfloor \log_p x \rfloor \log p$$ where $x$ is a positive integer, and $p$ runs over all primes $\leq x$. In a hunt for an ...
37
votes
3answers
1k views

Is this integral representation of $\zeta(2n+1)$ known?

Background: I'm an undergraduate at an institution with no researchers in analytic number theory, and no ties to the analytic number theory community. I believe I have found what is, as far as I can ...
6
votes
1answer
184 views

What is the mean value of a pair of Ramanujan Sums when summed over squares?

Does anyone know of the mean value of two Ramanujan Sums when summed over the square of integers? In my research on the Landau problem regarding nearly square primes, I have run into the mean value ...
17
votes
1answer
355 views

Smoothed exponential sums: bounds and sources?

Let $f:\mathbb{R}\to\mathbb{C}$ be differentiable $k$ times, with $f, f',\dotsc,f^{(k)}\in L^1$. Let $\alpha\in \mathbb{R}/\mathbb{Z}$, $\alpha\ne 0$. In "Every odd number..." (Math. Comp. 83, 2014), ...
1
vote
1answer
340 views

Proof that $p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log {p_n}$ [closed]

I cannot find a proof of this theorem. May anyone assist? $p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log{p_n}$
10
votes
2answers
911 views

Update for 2015: least prime of form nq+1, with q prime?

I have received a complaint about my 2011 answer least prime in a arithmetic progression which, indeed, gives conflicting reports about this: given a prime $q,$ what can we say about an upper ...
7
votes
2answers
454 views

What is wrong with this deterministic algorithm efficiently generating large primes?

According to PolyMath (Strong) conjecture. There exists deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in ...
11
votes
2answers
618 views

Many representations as a sum of three squares

Let $r_3(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n \}\right|$. I am looking for the maximum asymptotic size of $r_3(n)$. That is, the maximum number of representations that a number can ...
1
vote
1answer
148 views

Primes in simultaneous arithmetic progressions

Suppose we're given four positive integers $a$, $b$, $c$, $d$ such that $a$ and $b$ are coprime, and $c$ and $d$ are coprime. Is there a non-negative integer $k$ such that both $ak+b$ and $ck+d$ are ...
4
votes
2answers
161 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
518 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
93 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
198 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
164 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
155 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
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
1answer
251 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
120 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
2answers
504 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
107 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
335 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
235 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
394 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
149 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
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} ...
3
votes
1answer
335 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
94 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
106 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
80 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
93 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
3answers
490 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
513 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
221 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
559 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
151 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
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
641 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
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
0answers
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
1answer
360 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 ...