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

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2
votes
1answer
39 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. ...
0
votes
1answer
19 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
2answers
25 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
18 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
27 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
-1
votes
0answers
33 views

Showing that $ \sum_{\substack{p ~ \text{prime} \\ p \leq x}} \frac{1}{p \ln(p)} = O(1) $. [duplicate]

I have trouble proving that $$ \sum_{\substack{p ~ \text{prime} \\ p \leq x}} \frac{1}{p \ln(p)} = O(1). $$ The hint given was to use Abel’s Summation Formula.
0
votes
2answers
42 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
2answers
59 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.
-1
votes
0answers
24 views

Let $p_n$ be the nth prime. show that $\frac{2}{9}n \log n < p_n$ [on hold]

Please help me to prove this in analytic number theorem. I do not even know how to go about it.
0
votes
0answers
23 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
24 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}}} ...
3
votes
1answer
38 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
30 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
82 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
26 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
85 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
36 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
40 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
68 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
22 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
26 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
67 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
76 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
93 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
23 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
51 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
44 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
11 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
25 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
14 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
62 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
28 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
27 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
25 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
34 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
34 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.
2
votes
1answer
58 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
55 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
38 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
48 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
55 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
47 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
63 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 ...
3
votes
1answer
103 views

What is the inverse of the divisor sum function $\sigma $?

Let $(A, +, *)$ be the commutative ring of arithmetic functions with Dirichlet convolution as the multiplicative operation *. The element $$\sigma(n)=\prod_i \frac{p_i^{k_i+1}-1}{p_i-1}, \text { ...
2
votes
1answer
63 views

On the proof of Fejér-Riesz theorem

I'm having a course about Analytic Number Theory, and I'm having trouble understanding the proof of Fejér-Riesz Theorem: http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf First of all, I didn't ...
2
votes
1answer
183 views

How to calculate this sum like Gauss sum.

I would like to calculate the following sum, which looks like a Gauss sum. Let $n$ be a natural number and let $a,b$ be integers. Denote by $e(x)=e^{2\pi i x/n}$. Consider the sum $$ \sum_{1 \leq j, ...
0
votes
0answers
40 views

What is $B$ in $\varsigma(s)=e^{A+Bs} \prod_\rho \left(1-\dfrac{s}{\rho}\right) e^{s/\rho}$

$$\varsigma(s)=e^{A+Bs} \prod_\rho \left(1-\dfrac{s}{\rho}\right) e^{s/\rho}$$ (I have proved that $\rho$ is nontrivial zero of the Riemann zeta function in first part of question.) Show that ...
9
votes
3answers
77 views

Is there a monotonic $f$ such that $\sum f(n)$ diverges but $\sum f(p)$ converges?

(where the former summation is over natural numbers $n$ and the latter is over prime numbers $p$, and $f: \mathbb{N} \to \mathbb{R}$ is a monotonic function.) For the class of functions $f_s(n) = ...