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

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5
votes
1answer
65 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
62 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
42 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
57 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
47 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
57 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
103 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
23 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
40 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
25 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
50 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
54 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
92 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
0answers
26 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
26 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
48 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
37 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
1answer
87 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
31 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
90 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
44 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
45 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
74 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
27 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 ...
0
votes
1answer
30 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
78 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
18 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
77 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
98 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
28 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
50 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
55 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
50 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
13 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} ...
1
vote
1answer
27 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 ...
0
votes
0answers
18 views

Recommendable books to study the Selberg zeta function.

I've study on the Riemann zeta function and some zeta functions which have analytic properties directly. And now I want to know about the Selberg's zeta function which has some geometric properties. ...
0
votes
1answer
65 views

Elementary proof that $\omega(n)$ is bounded $\frac{\log n}{\log( \log n)}$ in the limit?

I'm trying to show that $\omega(n)$ is less than $\frac{\log n}{\log(\log n)}$ as it's stated without proof in an analytic number theory text. It's a corollary of the PNT, but I want to not use that ...
1
vote
1answer
33 views

$\zeta_m(s)=\prod\limits_{p\nmid m} \frac{1}{\left(1-\frac{1}{p^{f(p)s}}\right)^{g(p)}}$ is a Dirichlet series with non-negative coefficients

Let $p$ be a prime number, $m$ be any integer, $f(p)$ be the order of $p$ in $(Z/mZ)^*$, $i.e.$ $p^{f(p)} \equiv 1 \pmod m$ with $f(p)$ smallest. Let $g(p)=\frac{\phi(m)}{f(p)}$ is a integer where ...
4
votes
0answers
28 views

Derivatives of a Dirichlet polynomial

I am new here, so I don't know how this works exactly. If I do something wrong, please let me know. I'd like help to solve a problem I am studying: Let $A$ be finite set of positive integers and ...
2
votes
0answers
26 views

Relationship between asymptotic distribution and logarithmic sums of elements of subset of the natural numbers

Consider a subset $A$ of the natural numbers analogous to the primes (but rarer). Let $a_n$ denote the $n$th element of $A$, and $a(n)$ denote the number of elements of $A$ less than or equal to $n$ ...
1
vote
1answer
35 views

Understanding a series representation of the logarithm of the zeta function

I am reading through M. Ram Murty's Problems in Analytic Number Theory and have the following question regarding the first step in his proof of Dirichlet's Theorem. Given this definition for the zeta ...
2
votes
1answer
36 views

Asymptotic formula for sums related to primes

Suppose $0 < \alpha < 1$. What is the asymptotic formula for the sum $$\displaystyle \sum_{p \leq x} \frac{\log p}{p^\alpha}?$$ Thanks for any insights.
-1
votes
1answer
57 views

“Multivariable” version of this lemma about showing analytically that a number is irrational.

Lemma: let $\alpha \in \mathbb{R}^+$ and $p_1,p_2,\dots, q_1, q_2, \ldots \in \mathbb{N}$ such that $\left|\alpha q_n - p_n \right| \neq 0$ for all $n \in \mathbb{N}$ and $$ \lim_{n ...
3
votes
1answer
62 views

Asymptotic expression for $3$ term arithmetic progression in the primes

I have found an asymptotic for the following sum using the circle method: \begin{align} R(n)=\sum_{\substack{p_1,p_2,p_3 \le n \\p_1+p_2=2p_3 }} \log (p_1) \log (p_2) \log ...
2
votes
1answer
60 views

Asymptotics of $\sum_{\mathfrak{a}}\frac{n^{k-\epsilon}}{\mathfrak{N}\left(\mathfrak{a}\right)^{r\left(k-\epsilon\right)}}$

In this paper by Brian D. Sittinger, the following claim is made: For an algebraic number field $K$ with norm $\mathfrak{N}$, let $\epsilon=\left[K:\mathbb{Q}\right]^{-1}$. Then, taking the sum over ...
1
vote
0answers
39 views

Do you know any answer for equation y^2 = x^3 + k? [duplicate]

As you know, the equation y^2 = x^3 + k for k like (4n-1)^3 - 4m^2 that m , n are integers & no prime number that p is congruent to 1 modulo 4 count m, don't have any answer & it's proof is by ...
2
votes
1answer
55 views

Asymptotics of $\sum_{n\leq x}\tau_{k}\left(n\right)$

We define $\tau_{k}\left(n\right)$ to be the number of ordered $k$-tuples of positive integers with product equal to $n$. It is easily shown that this satisfies the recurrence relation ...
2
votes
0answers
81 views

Any formula for the exact number of primes below a given bound?

Reading The music of the primes, the author relates that Riemann had figured out a formula giving exact number of primes up to a certain bound with no errors. Does such formula really exist? If ...
1
vote
0answers
53 views

Explicit formula for floor(x)?

In number theory we have so-called explicit formula's in terms of the Riemann zeta zero's. For instance to count the sum of the logarithms of the primes below some given integer. ( second Chebyshev ...
3
votes
1answer
66 views

Moving the integral $Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds$ past Re(s) = 1.

Given the integral $$Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds,$$ I know that the integrand is holomorphic except for simple poles at ...