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

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4
votes
0answers
50 views

Equidistribution theorem of Weyl

Have you examples of applications of Equidistribution theorem of Weyl in proofs of irrationality of numbers? I don't know if "if and only if" is true for this theorem.
2
votes
0answers
56 views

What is the series expansion of reciprocal of theta function $\frac{1}{\theta(z;q)}$?

"The" theta function is an ambiguous concept, but one definition I have found is: $$ \theta(z;q) = (z;q)_\infty(q/z;q)_\infty = \frac{1}{(q;q)_\infty}\sum_{k \in \mathbb{Z}}z^k q^{\binom{k}{2}} ...
1
vote
0answers
24 views

Sato-Tate measure moments

Let $\mu=\frac{1}{\pi}\sqrt{1-\frac{x^2}{4}}$ be the Sato-Tate measure on the interval $[-2,2]$. Show that $$\displaystyle\int_{-2}^2 x^{2p+1}d\mu=0$$ and $$\displaystyle\int_{-2}^2 ...
5
votes
1answer
105 views

Prime-twins and infinite products

For $n\geq 1$ let the nth twin prime pair $$(p_n,p_n+2).$$ This sequence start as $(3,5),(5,7)$, the next $(11,13)\ldots$. I have two short questions about twin primes and infinite product defined ...
4
votes
1answer
85 views

What is known about the 'Double log Eulers constant', $\lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$?

The Euler constant is defined as $$\gamma = \lim_{n \to \infty}{\sum_{k=1}^n\frac{1}{k}-\ln{n}}$$ Let $$q = \lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$$ I managed to prove that ...
3
votes
1answer
62 views

There are infinitely many powers of $2$ that are at least $10^6$ away from any square and cube

A natural number $x$ is far from squares and cubes if the inequalities $\left|x-k^{2}\right| > 10^{6}$ and $\left|x-k^{3}\right| > 10^{6}$ hold for every natural number $k$. Prove that there ...
5
votes
1answer
43 views

$L$-function absolutely convergent for $\text{Re}(s) > 1$, condition for $L(s, \chi)$ converging for $\text{Re}(s) > 0$?

I have two questions related to here. Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ...
6
votes
1answer
58 views

Product of two absolutely convergent Dirichlet series

We have$$(f * g)(n) = \sum_{d \mid n} f(d)g(n/d).$$How do I see that if the two Dirichlet series$$F(s) = \sum_{n =1}^\infty f(n)n^{-s},\text{ }G(s) = \sum_{n=1}^\infty g(n)n^{-s}$$converge absolutely ...
7
votes
1answer
118 views

Is there a special value for $\frac{\zeta'(2)}{\zeta(2)} $?

The answer to an integral involved $\frac{\zeta'(2)}{\zeta(2)}$, but I am stuck trying to find this number - either to a couple decimal places or exact value. In general the logarithmic deriative of ...
3
votes
0answers
44 views

Compute sum over bounded numbers prime with given number

When I was doing some task of analytic number theory I was stuck on computing this sum $$S:=\frac{1}{L} \sum_{q \in \mathcal{Q}} \phi(q) \overline{a}^{\frac{1}{2}},$$ where $\overline{a}$ is the ...
3
votes
1answer
79 views

What about $\lim_{n\to\infty}\frac{\sum_{k=1}^n s_k\mu(k)}{n}$, for the zeros of Dirichlet eta function $s_k=1+\frac{2\pi k}{\log 2}i$ with $k\geq 1$?

Let for integers $k\geq 1$ the corresponding zeros of Dirichlet eta function of the form $$s_k=1+\frac{2\pi k}{\log 2}i,$$ then we can consider the following puzzle, when we multiply previous ...
4
votes
1answer
41 views

A conditional asymptotic for $\sum_{\text{$p,p+2$ twin primes}}p^{\alpha}$, when $\alpha>-1$

When I've followed a notes that show how obtain a similar asymptotic using Abel summation formula, my case with $a_n=\chi(n)$, the characteristic function taking the value 1 if $p$ is prime (in a twin ...
1
vote
1answer
28 views

Bounding sums of residue classes

Is there a sharper bound for the following sum $$S:=\sum_{d \in (Z/qZ)^{*}} \overline{d},$$ where $\overline{d}$ is the inverse of $d$ modulo $q>0$? Thanks in advance.
2
votes
0answers
38 views

Linear convex combinations of $Li(x)=\int_2^x\frac{1}{\log(t)}dt$ and $\frac{x}{\log(x)}$, and prime counting function

Can provide us a linear convex combination of $Li(x)=\int_2^x\frac{1}{\log(t)}dt$ and $\frac{x}{\log(x)}$ a better approximation for $\pi(x)$, the prime counting function? Or not, is better $Li(x)$ ...
1
vote
1answer
48 views

Can be justified this formula for $\zeta(2n+1)$

Can be justified for integers $n\geq 1$ that $$\zeta(2n+1)=\prod_{\text{p, prime}}\frac{1-\sigma(p^{2n})^{-1}}{1-p^{-2n}}?$$ Truly I don't know if I am wrong another time, when I use for an integer ...
12
votes
1answer
170 views

Putnam 2015 B6, sum involving number of odd divisors on an interval.

For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k})$. What is$$\sum_{k=1}^\infty (-1)^{k-1} {{A(k)}\over{k}}?$$
1
vote
0answers
29 views

A Fact Stated in Davenport's Multiplicative Number Theory

In his text Multiplicative Number Theory on page 9, Davenport mentions that another means of expanding the L-function is known and then mentions the fact that, $$ \mathcal{F} \sum_{n=1,n \; odd} ...
1
vote
1answer
25 views

Laurent expansion at infinity for a weakly modular function with respect to a congruence subgroups

Let $\Gamma\subset \mathrm{SL}_2(\mathbb Z)$ be a congruence subgroup and $h$ the fan width of $\Gamma$ (i.e; the minimum $h>0$ such that $\left(% \begin{array}{cc} 1 & h \\ 0 & 1 \\ ...
2
votes
0answers
50 views

Why do you need to prove the error term goes to zero for the complete derivation of the Euler Product Formula?

I am doing a project on the Riemann-Zeta Function which begins by examining the Euler Product Formula. I understand the proof up until the point where it is made 'rigorous'. In other words, I ...
0
votes
0answers
32 views

Almost primes in short intervals

Define an integer $n$ to be a $k$-almost prime if it has at most $k$ distinct primes factors. A detecting function for the set of such numbers is the generalized von Mangoldt function given by ...
8
votes
0answers
183 views

A map from zeros of $\zeta(s)$ to zeros of $C(s)?$

Let $P(s),C(s),\zeta(s)$ be the prime zeta function, the analogous composite zeta function, and the classical zeta function. I do not know whether it is known that there are infinitely many zeros of ...
0
votes
1answer
46 views

Current Research Using Sieve Methods

I've been learning various basics of Sieve Methods in Analytic Number Theory, and I'm wondering what are some uses of these methods in current research? Not famous, unsolved problems, but areas of ...
3
votes
1answer
50 views

Hints to compute if exists $\lim_{n\to\infty}\sum_{k=1}^n\sigma(k^2)/\sum_{k=1}^n\sigma(k)$, which $\sigma(n)=\sum_{d\mid n}d$, and other question

I would like receive hints at least for one of the following problems, these are going from experiments. Can you provide to me hints for at least one of the following problems? I will try put the ...
1
vote
0answers
38 views

Sketch of a possible equivalence with Riemann hypothesis

From Robin's equivalence (see [1]) and the following trigonometrics identitites, I ask to me if it is feasible write vagues equivalences using this strong result and if these equivalences will be ...
3
votes
1answer
34 views

On a generalization for $\sum_{d|n}rad(d)\phi(\frac{n}{d})$ and related questions

Let $\phi(m)$ Euler's totient function and $rad(m)$ the multiplicative function defined by $rad(1)=1$ and, for integers $m>1$ by $rad(m)=\prod_{p\mid m}p$ the product of distinct primes dividing ...
1
vote
1answer
79 views

Some doubts about easy computations involving nontrivial zeros of Riemann's zeta function

On assumption of Riemann hypothesis when I write a complex zero (nontrivial zero) of zeta function as $\rho=\frac{1}{2}+it_\rho$, and I write $x^\rho$ as $\sqrt{x}e^{it_\rho \log x}$, then multiplying ...
7
votes
0answers
164 views

Fourier transform of the critical line of zeta?

Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along the critical line? I'd love to say that it's a weighted sum of delta distributions, ...
1
vote
2answers
34 views

With $s(n)=\sum_{k=1}^n n \bmod k$, can be justified that $\forall\epsilon>0$ let us $\lim_{n\to\infty}\frac{s(n-1)}{\epsilon+s(n)}=1?$

Denoting as $$s(n)=\sum_{k=1}^n n \bmod k$$ the sum of remainders function (each remainder is defined as in the euclidean division of integers $n\geq 1$ and $k$). See [1] for example. For examples ...
2
votes
1answer
63 views

Sum of arithmetico-geometric series

Could use help trying to find the following sum of series $$ \sum_{n=1}^N r^n\sqrt{a + nd} $$ I have no clue where to begin on this one. Ideally would like solution for all $ r $ but if it helps ...
3
votes
1answer
59 views

Evaluation or asymptotic for $\int_1^x y\sin\left(\frac{2\pi (y-1) x}{y}\right)dy$

Truly, my genuine problem (see Appendix for context) is compute in a closed form or an asymptotic, for real $x\geq 1$, for $$\int_1^x\left(\int_0^{y-1}\cos\left(\frac{2\pi t ...
0
votes
0answers
60 views

Why $\pi(x) \approx \frac{x}{\ln(x)}$? [duplicate]

Is there any proof that $\pi(x) \approx \frac{x}{\ln(x)}$? If possible, I would like the proof involving the Riemann Zeta Function $\zeta(s)$
0
votes
0answers
28 views

Number of solutions to $x^2+yz=1$ (mod $p$). Where is my mistake?

Number of solutions $N$ to $x^2+yz=1$ (mod $p$) where $p>2$ and $p$ is a prime. Where is my mistake? From my lecture notes I know that: ($e(z)=e^{2\pi i z}$) $$ N = ...
1
vote
0answers
39 views

Importance and Applications of Cuban Primes

Are there any applications of cuban primes, or are they only considered to be within the realm of pure mathematics? Is there anything significant about them specifically? Would more research into ...
5
votes
0answers
88 views

On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...
2
votes
1answer
59 views

What is the effective lower bound on gaps between zeta zeros?

In this question here: Upper bound on differences of consecutive zeta zeros by Charles it is said that: "There are many papers giving lower bounds to: $$\limsup_n\ \delta_n\frac{\log\gamma_n}{2\pi}$$ ...
2
votes
0answers
94 views

Zeros of the prime zeta function

A basic confusion about zeros of the prime zeta function $P(s).$ Let $s= \sigma+i~t$ with $\sigma>0.$ Letting $C(s)$ be the corresponding composite zeta function we have ...
1
vote
1answer
29 views

Estimate for sum of negative powers of primes [duplicate]

Specifically, for $a \in (0,1)$, I am interested in the sum $$\sum_{p\leq n} \frac{1}{p^a} $$ as $n$ grows.
0
votes
0answers
41 views

Average of additive character sum over all polynomials of degree at most $d$ over $GF(q)$

Let $F$ be a finite field with $q$ elements, let $E$ be its degree $n \geq 2$ extension and let $\psi$ be the canonical additive character of $E$. For $d \leq n $ let $$ \mathcal{P}_{\leq d} = \{f(x) ...
1
vote
2answers
25 views

The Number $p(n)$ of triplets $(x,y,z)$ : $x+2y+3z=n$

I need some ideas for studying Diophantine equation (linear or exponential ) with elementary probability. For example this one : Find the Number $p(n)$ of triplets $(x,y,z)\in\mathbb{N}^3$ such ...
3
votes
0answers
34 views

Binary Strings with Primitive Suffixes

Let $wp$ be a binary word of length $n$ such that $p$ is primitive (that is, $p \neq r^m$ for any word $r$ and $m > 1$). I want to calculate the expected length of $p$. 1) The number of primitive ...
6
votes
0answers
106 views

For a given integer $n$, how many primes $p_1,p_2 \leq n$ such that $\tau(p_1-1)=\tau(p_2-1)$

This is a curiosity question. Let $N$ be positive integer, I just want to know how many (an approximation) pair of primes $(p_1,p_2)$ that are less than $n$ and verify the following identity: ...
2
votes
1answer
47 views

Equation $x^2=y^p+1$

can you help me please for solving this dophantine equation $$x^2=y^p+1$$ and if you can give me a general method to studying such equation $$x²=y^p+t$$ Thanks
3
votes
1answer
95 views

Prove a complex number is real

Let $z$ be a complex such that $|z-1| =1$, and consider the complex numbers $v$ and $w$ such that: $w = z^2 -z$ and $3\arg(v) = 2\arg(w)$, where arg is the argument of a complex number. Show that $$ ...
1
vote
0answers
36 views

A shortcut for analytic continuation?

Let $P(x)$ be a nonconstant integer polynomial with nonnegative coëfficiënts such that the equation $y= P(y)$ has only one real solution $q$. Let $x_1=P(0)$ and $x_n = P(x_{n-1})$. $$f(z) = ...
1
vote
0answers
13 views

Estimates related to a sum over primes from a fixed, sparse set

Let $E$ be a fixed infinite sequence of primes such that $\sum_{p \in E} \frac{1}{p} = \infty$. Assume that $\sigma > 1$ depends on some parameter $x \rightarrow \infty$ in such a way that $\sigma ...
0
votes
0answers
15 views

Relation between Bombieri theorem and p-adic squares

Koblitz states in his book on p-adic numbers on page 84: Suppose that $\alpha \in \mathbb Q$ is such that $1 + \alpha$ is the square of a nonzero rational number $a/b$. Let $S$ be the set of all ...
0
votes
0answers
17 views

Additive character sum over additive subgroups of finite fields, with special monomial arguments

Let $F$ be a finite field with $q = p^n$ elements, let $\psi$ be a non-trivial additive character of $F$, let $m$ be an integer coprime to $q-1$, and let $K$ be a large subspace of $F$, say $K$ is the ...
3
votes
0answers
35 views

Additive character sum over intersection of additive and multiplicative subgroups of finite fields

Let $H$ be a multiplicative subgroup of the finite field $\mathbb{F}_q$ with $q$ elements, say $H$ is the subgroup of $d$-th powers, $d \mid q-1$. Let $L$ be a subspace of $\mathbb{F}_q$ over some ...
2
votes
1answer
82 views

Are there any intuitive reasons for Goldbach conjecture to be true?

One thing puzzled me is that, despite its simple form, I have not seen any intuitive reasons for Goldbach conjecture to be true. Typical heuristic reason is based on probability arguments. Such ...
1
vote
0answers
19 views

Does Linnik's approximation to Goldbach's problem also work for the power of 3, 5, 7, etc ?

Linnik proved in 1951 the existence of a constant K such that every sufficiently large even number is the sum of two primes and at most K powers of 2. Roger Heath-Brown and Jan-Christoph ...