Questions on the use of the methods of real/complex analysis in the study of number theory.

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54 views

Twisting modular forms by Dirichlet characters

Let $\chi,\chi_1$ be Dirichlet characters modulo $M$ and $N$. In Koblitz's book "Introduction to Elliptic Curves and Modular Forms", Proposition III.3.17, it is proved that if ...
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0answers
24 views

Upper bounds for Euler's totient function on an increasing sequence of numbers

Let $\omega(k)$ be the number of distinct prime divisors of an integer $k$. Let $\delta > 0$ and suppose we have an increasing sequence of positive numbers $S = \{a_n \}_{n=1}^{\infty}$ with the ...
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1answer
41 views

Upper bounds for Euler's totient function on a set of numbers with unbounded number of prime divisors

If we take an infinite set $S$ of positive numbers with the property that the number of prime divisors of the elements is unbounded above, then can we make $\phi(n)/n$ arbitrarily small for infinitely ...
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0answers
50 views

Number of prime factors of Mersenne numbers

Let $p$ be a prime and let $M_p = 2^p-1$. Is it known whether the number of prime factors of $M_p$ is unbounded above as $p \to \infty$? Also do the probabilities estimating the chance that $M_p$ is ...
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1answer
63 views

Is any elementary proof important (beside Selberg's work) ?

Is any elementary proof important (beside Selberg's work) ? Plus, why is the elementary proof of prime number theory by Selberg so important ? Selberg was awarded the Field medal is mainly because ...
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30 views

Upper bounds for Euler's totient function on a set of composite numbers with bounded number of prime divisors

For an integer $c$, denote with $\omega(c)$ the number of distinct prime divisors of $c$. Now fix an integer $k \geq 2$ and let $W_k$ be some set, with $\#W_k = \infty$, of positive composite integers ...
3
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1answer
89 views

Divisor function asymptotics

Define $\tau_{r}(n) = \sum_{d_1...d_r = n}1$. One exercise in a book on sieve theory asked for an elementary proof by induction of the fact that $$\sum_{n\le x}\tau_r(n) = \frac{1}{(r - 1)!}x(\ln ...
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1answer
95 views

Computing infinite product over primes

How can I compute $$ \prod_p \left(1+\frac{k}{p}\right)\exp(-k/p) $$ where $0<k<e$ and the product is over all primes $p$? Background L. G. Sathe proved [1] that there are $$ ...
2
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2answers
34 views

Inequality $k!\pi+\frac{\pi}{6}\le{m!}\le{k!}\pi+\frac{5\pi}{6}$

there Is the following statement is true? $\forall k \in \mathbf{N},\exists{m}\in\mathbf{N}, k!\pi+\frac{\pi}{6}\le{m!}\le{k!\pi}+\frac{5\pi}{6}$ I tried by descendant proof but was not satisfied ...
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3answers
62 views

Smallest prime factor of a Mersenne number

The Mersenne numbers $M_n$ are integers of the form $2^n-1$, where $n$ is a positive integer. In the case when $n$ is a prime, are there any results known on the smallest prime factor, $p_n$, of ...
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1answer
35 views

Can we expect to find $r,$ large enough, so, $\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}\leq C (1+y^{2})^{s} $ for all $y\in \mathbb R$?

Fix $y\in \mathbb R$ and $s>1.$ Consider the series: $$I(y)=\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}.$$ My Question is: Can we expect to find $r$ large enough, so that ...
2
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1answer
43 views

A slightly various form of Dirichlet's theorem on arithmetic progressions

Are there infinitely many primes of the form $2n(n+1)+1$?
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26 views

how that if $P(x_1,…,x_n) \in C[x_1,…,x_n]$ takes only prime values at all non-negative integer values $x_i$, then $P$ is constant.

Show that if $P(x_1,...,x_n) \in C[x_1,...,x_n]$ takes only prime values at all non-negative integer values $x_i$, then $P$ is constant. To start, how would you express $P(x_1,...,x_n)$? I really ...
2
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1answer
41 views

Prove that for $P(X) \in \mathbb{Z}[X]$ the set $S = \left\{p : \text{prime and }p \mid P(n) \text{ for } n \in \mathbb{Z}^+\right\}$ is infinite

Prove that for $P(X) \in \mathbb{Z}[X]$, $P(x)$ non-constant, the set $S = \left\{p : \text{prime and }p \mid P(n) \text{ for some } n \in \mathbb{Z}^+\right\}$ is infinite. could someone please give ...
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1answer
82 views

Size of N in primes in arithemtic progression algorithm

I've been implementing the search for Primes in Arithmetic Progression (PAP) as explained by Weintraub (1976), and in his paper he refers to a number N which he sets to what seems to be an arbitrary ...
2
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1answer
84 views

Dirichlet series and Riemann zeta function

Im trying to show, for $\Re(s)>1$, that $\displaystyle\sum_{n=0}^{\infty} \frac{d(n^2)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)}$, where $d(n)= |\{k \mid k|n \}|$, number of positive integers that ...
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1answer
39 views

Integral inequality with gamma function

I have some trouble with paper I'm reading. The goal is this: let $s=\frac{1}{2}+\frac{1}{\log n}+it$. $M$ is a function such that $M(s)=O(\log^{3}(N(|t|+2)))$. Define $$U(s)=\frac{1}{2\pi ...
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0answers
28 views

Distribution of $\lfloor n^{\log{n}} \rfloor$ modulo $q$.

Let $q$ be an arbitrary integer. I want to investigate the distribution of the set $\mathcal{S} = \{\lfloor n^{\log{n}} \rfloor : n \in \mathbb{N}\}$. After a few explicit computations with SAGE, it ...
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0answers
45 views

Montgomery&Vaughan's Multiplicative number theory theorem 13.3

I can't understand well the proof of theorem 13.3 There exist a constant $C>0$ s.t. if RH is true, then for every $x\ge 2$ the interval $(x,x+Cx^{1/2}\log x)$ contains at least $x^{1/2}$ prime ...
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1answer
69 views

the average order of divisor function

In Analytic number theory by Apostol there's a theorem: $$\sum_{n\le x} \sigma(n)= \frac{1}{2} \zeta(2)x^2 + O(x\log x)$$ and then it claims that because we know that $\zeta (2)= \frac{\pi^2}{6} $ ...
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1answer
56 views

an exercise from apostol analytic number theory

this is the second exercise of chapter 3: if $x\ge 2$ prove that $$\sum_{n\le x} \frac{d(n)}{n}= \frac{1}{2} {\log^2 x} + 2C\log x +O(1)$$ where $C$ is Euler's constant. here's what i've done to ...
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1answer
268 views

Notation in Terry Tao's exposition on the PNT

The exposition I'm talking about can be found here (page 6): http://www.math.ucla.edu/~tao/preprints/Expository/prime.dvi Essentialy, Tao proves the prime number theorem in the elementary way, ...
2
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1answer
60 views

Identities for L-series and Euler product

It is a mabe a stupid question for many experts here. There is something wrong in the following reasoning, and now I could not find it. Could someone help me out? Any advice will be highly ...
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0answers
22 views

Gamma function whose argument is a reciprocal power with integer base and exponent

Consider the analytic continuation of the factorial function $n!$ given by $\Gamma(z)$ (note $n!=\Gamma(n+1)$), and suppose $z=a^{-n}$, where $a,n\in\mathbb{N}$ are positive integers. Are there any ...
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0answers
107 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
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0answers
41 views

Riemann functional equation question?

I was looking through the derivation of the Riemann functional equation, and I understand how to obtain the result $$ \pi^{-\frac s2} \Gamma (\frac s2) \zeta(s) = \pi^{-\frac{1-s}{2}} ...
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1answer
16 views

Given $d$, for how many $m$'s is $d$ a quadratic residue mod $m$?

Let $d$ be a fixed, square-free integer, and let $M$ be some very large number. I would like to count the numbers $m \leq M$ such that $m \perp d$ and $d$ is a quadratic residue modulo $m$. Call this ...
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1answer
48 views

Rearrangements of Dirichlet Eta Function

I was wondering if explicit closed forms for rearrangements of $\eta(s)$, for $s$ such that the series is not absolutely convergent, are useful in studying the Dirichlet $\eta$ function. I am asking ...
4
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1answer
85 views

Analytic Continuation of Zeta Function using Bernoulli Numbers

In my complex analysis textbook by Stein and Shakarchi, as an exercise, I am supposed to extend $\zeta(s)$ to the entire complex plane using Bernoulli numbers, but I am stuck. I can prove that $$ ...
2
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1answer
65 views

Congruences of weights of modular forms modulo primes

I'm trying to prove that for two modular forms $f$ and $g$ of weight $k$ and $k'$ respectively, that are congruent modulo a prime $\ell\ge 5$, their weights are congruent modulo $\ell-1$. This is what ...
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1answer
48 views

Transformation property for classical Siegel modular forms of weight 2

Let $\mathbb{H}_g = \{ \tau \in GL_g(\mathbb{C}) | \; {^t\tau} = \tau, Im(\tau) >0\}$ be the Siegel upper half space. There are Eisenstein series $$ E_{2k}(\tau) := \sum_{\gamma\in (P_0\cap ...
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1answer
55 views

Riemann Zeta Function at Real Values of the variable s

My question is: Is the Riemann Zeta function for real values of $s$ $( s = \sigma + 0\,i)$ a monotone function of $\sigma \,$?
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48 views

Tau Summatory Function

It is well known that the divisor summatory function can be calculated in $O(x^{1/2})$ via $$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^{\lfloor \sqrt{x}\rfloor} \lfloor\frac{x}{k}\rfloor - \lfloor ...
3
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0answers
91 views

Is this the chord G Major I am hearing as base tones from interference of zeta zeros times eigenvalues of the von Mangoldt function matrix?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: ...
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0answers
54 views

Wolstenholme Number

Does Wolstenholme Numbers have perfect squares other than 1 and 49? The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749 seems to be a complicated problem
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1answer
41 views

Twin primes : prove the convergence of $ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} (\log(p)+\frac{1}{p})²$

let $T$ be the twin primes set : $p \in T $ if and only if $ p$ and $p+2$ are primes. Can you help me establish the convergence of : $$ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} ...
5
votes
2answers
211 views

least common multiple $\lim\sqrt[n]{[1,2,\dotsc,n]}=e$

The least common multiple of $1,2,\dotsc,n$ is $[1,2,\dotsc,n]$, then $$\lim_{n\to\infty}\sqrt[n]{[1,2,\dotsc,n]}=e$$ we can show this by prime number theorem, but I don't know how to start I ...
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0answers
35 views

Growth rate of arithmetical function

I'm interested in how one would estimate the growth rate of $$f(n)=\sum_{k\le n}\mu^2(k)\log(k)$$ I.e. sum of logarithms of square free integers. I can think of some trivial methods in my head ...
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0answers
28 views

A question on the big-O value of the complex integral especially in the number theory

My question is quite simple and elementary. Let $A(x)=\sum_{1}^{x}a(n)$ and $\alpha(s)=\sum_{1}^{\infty}a(n)n^{-s}$. Then, as we know, $$ A(x)= ...
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1answer
72 views

Sum of reciprocals of primes for known primes.

I was reading through some old analytic number theory notes earlier and found the interesting fact that even though $\sum\frac{1}{p}$ diverges: $\sum_{\text{known primes}}\frac{1}{p} < 4$. ...
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1answer
61 views

Problem from Apostol's analytic number theory book

Im trying to solve the exercise 13.2 in Apostol's analytic number theory book: Let $A(x)=\sum_{n\leq x}a(n)$, where $a(n)$ is zero unless $n=p^k$ for some prime $p$, in that case $a(n)=1/k$. Prove ...
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0answers
25 views

Riemann's hypotesis and some equivalences [duplicate]

Use that $\psi(x)=x+O(\sqrt{x}\:log^{2}x)$ to show that $\pi(x)=li(x)+O(\sqrt{x}\:logx)$ where $li(x)=\int_{2}^{x}\frac{dt}{logt}$ I tried but I get confused. Many pdf's say it's very easy to show. I ...
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1answer
58 views

On Newman/Zagier's proof of PNT

I have just got this paper: http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf and I have a serious doubt: When proving that soft Tauberian theorem he explicitly uses ...
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0answers
26 views

Zeta Riemann Function

Use that $\zeta(s)=1+\frac{1}{s-1}-s\int_{1}^{\infty}\frac{\left\{u\right\}}{u^{s+1}}du$ if $Re(s)>0$ to show that. 1) $\zeta(s)=s\int_{1}^{\infty}\frac{\left[ u\right]}{u^{s+1}}du$ for ...
2
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1answer
43 views

Based on prime number theorem

I have a problem... If $A(x)=\sum_{n\leq x}a(n)$ where $a(n)=\frac{1}{k}$ if $n=p^{k}$ and $a(n)=0$ in other case, show that $A(x)=\pi(x)+O(\sqrt{x}\:log\:logx)$ I think I should use the theorem of ...
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1answer
35 views

Summation formula in dimension 2

One of the most common tools in analytic number theory is the summation by parts, my question is what is the similar formula when we are, for example, in dimension two and we have the sum $$ ...
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1answer
22 views

bounding gaps between points in an interval

I've been reading Davenport's Multiplicative Number Theory and came across something that I didn't understand. On p. 108, there is an argument for finding a lower bound on the imaginary parts $\gamma$ ...
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2answers
110 views

A set with zero density

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that $$n \mid a^{f(n)}-1.$$ Prove that $S$ has density $0$; that ...
0
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0answers
56 views

Finding an asymptotic formula for $f(m,n)=\sum_{\substack{d\mid m \\ d\leq n}}1$?

$$f(m,n)=\sum_{\substack{d\mid m \\ d\leq n}}1$$ Here $n<m$ and $m$, $n$ are positive integers.
3
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0answers
49 views

Prove that the set has zero density

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that $$n \mid a^{f(n)}-1.$$ Prove that $S$ has density $0$; that ...