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

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0
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1answer
31 views

Vanishing property of logarithmic derivative of zeta function

I was trying to derive the explicit formula for the integrated Chebyshev $\psi$ function, $\psi_1$ defined as \begin{equation}\psi_1(x)=\int_1^x\psi(y)dy\end{equation} But I have stumbled upon one ...
1
vote
2answers
120 views

Number of distinct prime divisors of an integer $n$ is $O(\log n/\log\log n)$

I strongly believe that the claim is true; but I'm neither a mathematician nor speaking French and hope that somebody can confirm it, since related questions (here, here and here) either don't have an ...
1
vote
1answer
38 views

Is there numerical evidence supporting the predicted density of the primes of the form $x^2+1$?

A famous conjecture (due I think to Hardy and Littlewood) states that if $P(x)$ denotes the number of primes of the form $n^2+1$ less than or equal to $x$, then $$P(x)\sim \frac{C\sqrt x}{\log x}$$ ...
0
votes
0answers
23 views

The estimation of $\sum^{K_0+K}_{k=K_0+1} \min\left\{ U, \frac{1}{\left< \alpha k + \beta \right>} \right\}$

I have some difficulty with understanding the proof of the following theorem: Suppose $\alpha$ is a real number which has the form $\alpha = \frac{h}{q} + \frac{\theta}{q^2}$, $(q,h)=1$, $q \geq ...
1
vote
1answer
93 views

Question concerning the Dirichlet density of a subset of the set of primes

I have the following question: I am reading Serre's book "A Course in Arithmetic" (see http://www.math.purdue.edu/~lipman/MA598/Serre-Course%20in%20Arithmetic.pdf). On page 75, it is stated that the ...
9
votes
2answers
136 views

Evaluating $\sum_{\gcd\left(m,n\right)=1}\frac{1}{m^2n^2}$

I was wondering how one would evaluate the sum $$\sum_{\gcd\left(m,n\right)=1}\frac{1}{m^2n^2}.$$ The first thought that came to mind to to try something like this: ...
5
votes
1answer
73 views

Why does Titchmarsh say that we can move the derivative under $\frac{2}{\pi}\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cosh(\alpha t) \, dt$

If we define the Riemann-Xi function as $$ \Xi(t) = \xi(\frac{1}{2} + it)$$ where $$\xi(s) = \frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s),$$ then according to Titchmarsh in his ...
0
votes
0answers
45 views

Comparing a primorial $p\#$ to Dusart's upper bound for the $n$th prime

The number of elements of a reduced residue system modulo a primorial $p$ is $\varphi(p\#)$ I thought that it would be interesting to compare each primorial $p_i\#$ to the Dusart's estimate for the ...
0
votes
1answer
48 views

Chebyshev's theorem on the distribution of primes

I a lecture V. Arnold says that Chebyshev had proved that the limit $$\lim_{n\to \infty}\frac{\pi(n)}{n/\mathrm{log}(n)}$$ if exists is equal to one. Where I can find the proof? Thanks!
1
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0answers
43 views

Reasoning about Pierre Dusart's estimate of the $n$th prime

Cited here, Pierre Dusart established the following lower bound for $p_n$: $$p_n > n(\ln n + \ln \ln n - 1)$$ Using a spreadsheet and plugging in different values of $n$, I noticed that for an ...
3
votes
1answer
38 views

What is the best estimate known for the upper bound for the difference between consecutive primes?

Bertrand's Postulate gives us that: $$p_n < p_{n+1} < 2p_n$$ So that: $$p_{n+1} - p_n < p_n$$ In this answer, this paper is cited which says in Prop 6.8 that: For $x \ge 396738$ ...
0
votes
1answer
49 views

Writing a Gauss sum as a sum over divisors

Let $\chi$ be a Dirichlet character modulo $q$ induced by a primitive character $\chi^*$ modulo $d$ for some divisor $d$ of $q$. Let $n$ be a positive integer, and consider the generalised Gauss sum ...
1
vote
0answers
75 views

Ramanujan conjecture and Langlands program

In the article http://www.thehindu.com/sci-tech/science/the-legacy-of-srinivasa-ramanujan/article2746988.ece, it was mentioned that "This conjecture, later called Ramanujan's conjecture, came to ...
1
vote
1answer
76 views

proof of Perron's formula?

I was reading a journal entry on the proof of Perron's formula, and I got stuck on one of the computations. The following is the journal entry itself: The part I have a problem with is where they ...
4
votes
1answer
51 views

Question about direuler command in Pari/GP

From the Pari/GP users guide: 3.4.16 direuler(p=a,b,expr,{c}). Computes the Dirichlet series associated to the Eulerproduct of expression expr as p ranges through the primes from a to b. expr must ...
2
votes
1answer
54 views

Asymptotic for $\sum a_nb_n$ if asymptotic for $\sum a_n, \sum b_n$ are known

Let us assume that $a_n>0$ and $b_n>0$ for each n. Also let $$ \sum_{n\leq x} a_n \sim f(x) $$ and $$ \sum_{n\leq x} b_n \sim g(x) $$. What can we say about the asymptotic on $\sum_{n \leq x} ...
3
votes
1answer
83 views

$\sum_{n=1}^N\lambda(n)[N/n]=[\sqrt{N}]$ Identity involving Liouville Lambda function

I have to prove $$\sum_{n=1}^N\lambda(n)[N/n]=[\sqrt{N}]$$ I tried using the approach in this question but I don't know how I'll get $\sqrt{N}$. Please help.
1
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0answers
85 views

For Riemann Hypothesis, many people seek physics intuition, why not for Goldbach Conjecture ?

All: As we all know, for Riemann Hypothesis research, many people seek physics intuition, to understand more fundamental reasons why Riemann Hypothesis shall hold. In this direction, we have ...
2
votes
1answer
42 views

Interchanging summands among infinitely many infinite series

I am reading the following lecture notes concerning analytic number theory: http://www.math.uiuc.edu/~hildebr/ant/main4.pdf On the pages 111/112 the partial product $P_N(s)$ is defined. Then some ...
1
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0answers
36 views

Divergence Dedekind zeta function

Let $K$ be a number field, $\mathcal O_K$ be its ring of integers, T a positive integer and $N$ the norm function. Give an upper bound (in T) for $$\sum_{I\leq \mathcal O_K: N(I)\leq T} ...
1
vote
1answer
48 views

Dirichlet Convolution Associativity

I am unsure of the proof of associativity. So far I have: \begin{align} [f\ast (g\ast h)](n)&=\sum\limits_{ab=n}f(a)[g\ast h](b)\notag\\ ...
12
votes
1answer
976 views

What exactly *is* the Riemann zeta function? [duplicate]

I'm doing a little project on the $\zeta$ function, and I am at a complete loss of what it is actually doing. I understand it is way over my head, but when I am plugging say $\zeta(1 + i)$ into ...
0
votes
0answers
13 views

Bounds on function product

I have a question on bounds. If $f(x)$ and $f(y)$ are two functions, and we know that $f(x)$ is bounded by $0$ to $1$ and the value $x$ ranges from some $n_0+1$ to $n$. The function $f(y)$ has $y$ ...
2
votes
0answers
53 views

Convergence of recurrence relation involving divisors

I$\let\leq\leqslant\let\geq\geqslant$ thought up a family of sequences, recursively defined by $$a_{n+1}=\frac{d_n^ra_n+a_{d_n}}{d_n^r+1}\quad(n\geq2)$$ where $r,a_1,a_2\in\mathbb R$ are parameters ...
4
votes
1answer
88 views

Show that if $n$ is composite, then $\phi(n) \leq n-\sqrt{n}$

Please help me showing this: If $n$ is composite, then $\phi(n) \leq n-\sqrt{n}$. I failed to proceed from the definition of Euler function $\phi(n)$. First of all if $n$ is composite, then it ...
5
votes
1answer
70 views

Clarification of Proof involving $\sum_{p \le x} \frac{1}{p}$

For fun I've been doing problems from M. Ram Murty's text "Problems in Analytic Number Theory". I recently encountered the following problem: If $$\lim_{x \rightarrow \infty} \frac{\pi(x)}{x/\log x } ...
1
vote
3answers
76 views

Find an asymptotic formula for $\sum\limits_{n\leq x} d(n)\log n$

Please help me to find the asymptotics for the sum described above: $$\sum\limits_{n\leq x} d(n)\log n.$$ This is a problem in analytic number theory.
0
votes
4answers
45 views

Question about Euler's summation formula as used in Apostol ANT

Given Euler's summation formula in Apostol ANT Theorem 3.1 $$\sum_{y \lt n \leq x} f (n) = \int_y^x f (t) dt + \int_y^x(t- [t])f'(t)dt +f(x)([x]-x) - f(y)([y]-y)$$ Apostol calculates $\sum_{n \leq x} ...
0
votes
2answers
76 views

quadratic Gauss sum over a power of 2

Is there a general formula for the generalized quadratic Gauss sum defined by $$ G(a,b,c)=\frac{1}{c}\sum_{n=0}^{c-1}e\left(\frac{an^2+bn}{c}\right) $$ where $e(x)=\exp(2\pi ix)$ and $c$ is a power of ...
3
votes
0answers
85 views

Is there an equivalent statement of Riemann Hypothesis in term of Random Matrix or physics theory?

We all know that Riemann Hypothesis has many equivalent statements. After Montgomery’s works on pair-relationship, we now know that ZEROs of Riemann Zeta function has similar properties as ...
3
votes
1answer
65 views

error when replacing sum by an integral

I have seen that quite often in analytic number theory, one wants to replace a sum by an integral and then estimate the error. I saw the following estimate but I can't understand how to prove it. ...
1
vote
2answers
133 views

A proof of $\sum{\mu(n)/n}=0$

I am looking for a proof (or references) of the following statement $$\sum_{n=1}^{\infty}{\frac{\mu(n)}{n}}=0$$ where $\mu$ is the Möbius function. Many thanks !
0
votes
1answer
30 views

What's a “Basis of Measurable Sets?”

As defined here http://modular.math.washington.edu/129/ant/html/node82.html Using the notation in the link, one takes sets of the form $\prod\limits_{\lambda} M_{\lambda}$, where each $M_{\lambda}$ ...
3
votes
1answer
53 views

Relation between elliptic curves and Dirichlet L-series

I have read that an elliptic curve $E$ is modular if $a(n) = c(n)$ for all $n$, where $a(n)$ is the $n$-th coefficient in the Dirichlet series of $E$, $L(E,s)$, and $c(n)$ is the $n$-th coefficient in ...
0
votes
0answers
38 views

definition of the L-function $L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is $f$?

To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...
0
votes
1answer
76 views

Obtaining the expression for nth prime from PNT with remainder

Given the Prime number theorem with the error term : $π(x) = li(x) + O(x.e^{−c\sqrt{log(x)}}))$ , how do you go about obtaining an expression for the nth prime? Any help would be much appreciated
0
votes
1answer
55 views

Is $\sum_{n \geq 2} \frac{1}{\pi (n^2)}$ convergent or divergent?

I wonder if $$\sum_{n \geq 2} \frac{1}{\pi (n^2)},$$ where $\pi(\cdot)$ is the prime-counting function, is convergent or not. Please help me solve and understand this problem. Is related to analytic ...
0
votes
3answers
113 views

show that $\sum_{p\leq x} \frac{1}{p \log p} = O(1)$

Please my knowledge in this field is very low so could you help me solve this question in analytic number theory.
0
votes
1answer
45 views

coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$

I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is greater than 1( The ...
1
vote
0answers
30 views

Averages of $L(1,\chi)$

Let $(\frac{m}{n})$ denote the usual quadratic Jacobi symbol and $\mu(n)$ be the Moebius function. The series $$ \sum_{\substack{m,n \in \mathbb{N} \\ m,n\equiv 1 \mod{4}}} ...
4
votes
1answer
65 views

What are the subjects an analytic number theorist must be well versed with after undergraduate studies?

I am a mathematics major and I aspire to be an analytic number theorist. In general, what are the subjects an analytic number theorist must be well versed with after undergraduate studies (i.e. in ...
0
votes
1answer
54 views

Does analytic continuation apply only to analytic functions?

I'm a high school senior attempting to do a project on the riemann zeta function. I've looked online, tried reading college textbooks but still don't have a completely clear idea of analytic ...
3
votes
2answers
95 views

I find “closed formula” for divisors of an positive integer, it is useful or not? [closed]

I find this formula, but i don't know this is worth or not. $$d(n)=\sum_{i=1}^{n} \lim_{j\to\infty} (cos\left(\frac{\pi n}{i} \right))^{2j}$$ It is possible to improve it to deduce formula for $P_n$ ...
0
votes
1answer
37 views

Proof Janusz Algebraic number fields, convergence of Dirichlet Series.

The book Algebraic number fields, Janusz Please, Could you explain the proof of the part b) a little more? Thank you all.
2
votes
1answer
91 views

Prove that $\pi(n^2)-\pi(\frac{n^2+2n}{2})>0$

I'd like to know if there's a better way to prove that: $$\pi\left(n^2\right)-\pi\left(\frac{n^2+2n}{2}\right)>0$$ than using "There's always a prime in $(m-m^{23/42},m)$" by Iwaniec-Pintz: (I ...
0
votes
2answers
45 views

Properties of $\underset{k\geq1}{\sum}\frac{1}{\left(2k-1\right)^{s}}$

Is this function $$\underset{k\geq1}{\sum}\frac{1}{\left(2k-1\right)^{s}},\,Re(s)>1$$ well known? In particular I'm interessed about analytic continuation and its zeros and poles. Have this ...
2
votes
1answer
48 views

Fourier transform and dual vector space

In Serre's A Course In Arithmetic, it says the following: I don't know what it is talking about, I know the definition of $f'$, but what is This is in the last sentence refered to? $f'$ is a ...
2
votes
2answers
84 views

Non vanishing of an infinite product

I need to prove that the infinite product $$\prod_n \left(1-\frac{1} {(a^n+1)^2} \right)^{\frac{a^n}{n}} $$ with $a$ an integer $\geq 3$, converges to a real number $L$ such that $0<L<1$. It's ...
2
votes
1answer
32 views

Why is $f(z)y^k$ bounded for $f$ a cusp form?

For $f$ is a cusp form of weight $2k, k>0$ ($f(z)=(cz+d)^{-2k}f(\frac{az+b}{cz+d}$)), then why is $f(z)y^k$ bounded? If expanded $f$ in $\sum a_nq^n$, it's domain is a open disc, hence I can't ...
1
vote
1answer
38 views

Writing Dirichlet series in infinite product.

In Serre's $A \, Course\, In \,Arithmetic$, it says the following: $\sum\limits_{n=1}^{\infty}c(n)/n^s= \prod\limits_{p \,\rm prime}\frac{1}{1-c(p)p^{-s}+p^{2k-1-2s}}$ $\Longleftrightarrow$ ...