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

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7
votes
1answer
132 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
45 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
80 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
42 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
29 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
45 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
50 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
176 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
30 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
28 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
56 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
38 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 ...
9
votes
0answers
196 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
49 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
52 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
42 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
42 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
82 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
170 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
39 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
64 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
61 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
30 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
42 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
94 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
60 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
106 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
34 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
44 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
28 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
35 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
108 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
48 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
117 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
39 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 ...
5
votes
0answers
79 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
20 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
39 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
90 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 ...
3
votes
0answers
48 views

Average Order of $\frac{1}{\mathrm{rad}(n)}$

Again a question about $\mathrm{rad}(n).$ Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing $n$. Or equivalently, ...
0
votes
0answers
108 views

How to use an inverse Mellin transform to get to the $\mathrm{core}(n)$?

From this question here: Moreover, if multiplicative function $\mathrm{core}(n)$ is defined to map positive integers "$n$" to square-free numbers by reducing the exponents in the prime power ...
2
votes
0answers
17 views

Construct an example of a Dirichlet series with specific abscissa of convergence and absolute convergence

I want, for each $\alpha \in [0,1]$ to construct an example of a Dirichlet series for which $\sigma _ 1 = \alpha$ and $\sigma _0 = 1$ where $\sigma _0$ is the abscissa of convergence and $\sigma _1$ ...
0
votes
0answers
55 views

Integrating a series expansion of $\{x\}\lfloor x\rfloor$ coming from Fourier series of sawtooth function.

Originally, I ran into this problem on some basic math puzzle site (where it was asked to find the definite integral for some specific integer values), but I decided to try and find the indefinite ...
1
vote
0answers
37 views

Locating a pdf of certain paper

One of the references in the book Opera de Cribro by Friedlander and Iwaniec is: J. Friedlander and H. Iwaniec, A polynomial divisor problem, J. Reine Angew. Math. 601 (2006), 109-137 Does anybody ...
3
votes
1answer
95 views

Twin prime conjecture implies $\limsup_{n\to\infty}\frac{\sigma(n)\pi(n)}{n^2}\left(\pi(\log n)-\frac{\pi_2(\log n)}{2C_2}\right)=e^{\gamma}$?

Let $\sigma(n)$ the sum of positive divisor function, $\pi(x)$ is the prime counting function, $\pi_2(x)$ is the twin prime counting function (we will assume that Twin prime conjecture holds), $C_2$ ...
3
votes
1answer
47 views

Estimates for a product involving primes

Is $$\prod_{p\leq z}\bigg(1 + \frac{p}{(p-1)}\frac{\log z}{\log p}\bigg) = O(z)?$$ where $p$ ranges over the primes less than $z$. I believe I can show that it is $O(z\log \log z)$.
10
votes
2answers
227 views

What is your idea about this conjecture?

I conjecture that in a consecutive sequence of $n$ natural numbers all greater than $n$, there exists at least one number which is not divisible by any prime number less than or equal to $n/2$. Can ...