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

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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
45 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
57 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
48 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
44 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
72 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
vote
0answers
76 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
41 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
vote
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
47 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
918 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
50 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
82 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
75 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
44 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
1answer
66 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
66 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
59 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
114 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
25 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
46 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
30 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
72 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
101 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
41 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
60 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
50 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 ...
2
votes
2answers
90 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
34 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
46 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
79 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
31 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
36 views

Writing Dirichlet series in infinite product.

In Serre's $A \, Course\, In \,Arithmetic$, it says the following: $\sum\limits_{n=1}^{\infty}c(n)/n^s= \prod\limits_{p \,\rm prime}\frac{1}{1-c(p)p^{-s}+p^{2k-1-2s}}$ $\Longleftrightarrow$ ...
4
votes
1answer
98 views

Why doesn't Mertens's second theorem prove the Prime Number Theorem?

Mertens's second theorem states that $$\sum_{p \le x} \frac 1p = \log \log x+O(1).$$ Defining $p_x=p_{\lfloor x \rfloor}$ for all real $x \ge 1$, we can replace the sum by the integral $$\int_1^x ...
1
vote
0answers
19 views

Asymptotic for the height of the derivatives of a rational function

Let $\phi=\frac{P(z)}{Q(z)}$ be a homogeneous rational function of degree $d\ge 2$ over $\overline{\mathbb{Q}}$. If $h$ is the absolute logarithmic height, it seems that for each $z\in ...
3
votes
1answer
78 views

How to evaluate $\sum\limits_{m\in Z}(\sum\limits_{n\in Z} \frac{1}{(m-1+nz)(m+nz)})$ with $Im(z)>0$

How to prove $\phi(z)=\sum\limits_{m\in Z}(\sum\limits_{n\in Z} \frac{1}{(m-1+nz)(m+nz)})=2-\frac{2\pi i}{z}$ with $Im(z)>0$ and $(m,n)\neq(0,0),(1,0)$? For $m$ fixed, $a_m=\sum\limits_{n\in Z} ...
1
vote
1answer
101 views

Curve profile for the logarithm-integral sum term of Riemann explicit formula?

I am considering the following term from the Riemann explicit formula (see here >>>): $$\sum_{\rho(\Im>0)}{\mathrm{li}(x^\rho)}$$ with $\rho$ non-trivial zeros of $\zeta$-function. I have a plot ...
0
votes
1answer
33 views

Residue theorem and Angle of modular function

Let $f$ be a meromorphic function on the region $Im(z)>0$, $v_p(f)$ be the order of $p$. (The number $n$ such that $\frac{f(z)}{(z-p)^n}$ is holomorphic and non-zero at $p$.) Moreover, assume $f$ ...
1
vote
3answers
51 views

Does a positive constant $\nu$ exist so that $\varphi(n)>\nu\cdot n$ for all $n$?

Does a positive constant $\nu$ exist so that $\varphi(n)>\nu\cdot n$ for all $n$? Clearly this problem is exactly the same as asking if $\prod\limits_{i=1}^\infty \frac{p_i-1}{p_i}=0$. This is ...
2
votes
1answer
62 views

Asymptotic bound of the series $\sum_{n\leq x}\log n / \varphi(n)$

Could someone give me a hint on the computation of the asymptotic bound for the following series $$ \sum_{n\leq x}\frac{\log n }{ \varphi(n)}\,, $$ where $\varphi(n)$ is the Euler totient function? ...
0
votes
1answer
51 views

Elliptic curves $\mathbb C/\Gamma , \mathbb C/\Gamma'$ are isomorphic iff $\Gamma=\lambda\Gamma'.$

Let, $\Gamma, \Gamma'$ be $lattices$ of $\mathbb C$, define $elliptic$ $curves$ by $\mathbb C/\Gamma , \mathbb C/\Gamma'$, then $\mathbb C/\Gamma , \mathbb C/\Gamma'$ are isomorphic ...
0
votes
0answers
18 views

Getting rid of weights after using smoothed versions of Perron's formula

In order to get better convergence properties in integrals that arise when estimating sums of arithmetic functions, instead of Perron's formula we can use summation formulas like $$ \sum_{n\leq x} ...
2
votes
1answer
31 views

Second-order asymptotics for $\pi(n), \theta(n)$

Let $\pi, \vartheta$ be respectively the prime counting function and the first chebyshev function. As you know, $ \pi(x) \sim x/\log x$, and $\vartheta(x) \sim x$, so that, at first order, seems ...