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

learn more… | top users | synonyms

1
vote
0answers
34 views

modern proof of the conditional three prime theorem by Hardy and Littlewood

Hardy and Littlewood proved the three prime theorem under the GRH(generalized Riemann hypothesis) in an old paper: Some problems of `Partitio numerorum'; III: On the expression of a number as a sum of ...
0
votes
0answers
30 views

A combinatorics question arising from taking powers of a sum

Let $N \in \mathbb{N}$ and let $f$ be an arithmetic function. For a positive integer $k \geq N$ consider the sum $$ S_k(N) = \left( \sum_{n=1}^N \mathbf{1}_{f(n) = 0} \right)^k = \sum_{1 \leq n_1, ...
11
votes
1answer
145 views

Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges

I am going through A. J. Hildebrand's lecture notes on Introduction to Analytic Number Theory. I'm currently stuck at the exercises at the end of Chapter 3 (Distribution of Primes I - Elementary ...
3
votes
1answer
64 views

Looking an asymptotic for $\sum_{k\leq x}\Lambda(k)e^k$, where $\Lambda (n)$ is the von Mangoldt function

Using Abel's identity (see Theorem 4.2 in page 77 of [1]) and Prime Number Theorem (Theorem 4.4 in page 75) I compute $$\frac{1}{x}\sum_{k\leq x}\Lambda(k)e^k\sim 1\cdot e^x-\frac{1}{x}\int_1^x ...
3
votes
1answer
49 views

Numbers as sum of two relatively prime composite numbers

It is not hard to prove by analytical method the existence of a positive integer $n$ such that for all integers $m > n$ the following assertion is true: There exist two positive integers $a$ and ...
3
votes
0answers
61 views

What are the (possible) fundamental reasons for all zeros of zeta function to be on the critical line?

What are the (possible) fundamental reasons for all zeros of zeta function to be on the critical line ? Are there particular mechanism to make this happen ? Because of Levinson and Corney's work, we ...
9
votes
0answers
71 views

Sums of the form $\sum_{d|n} x^d$

Let $$S(x,n) = \sum_{d|n} x^d, n \in \Bbb N$$ Do these sums appear in the literature? What are they called if they do and what is known about them?
1
vote
0answers
37 views

How to derive a formula related to the Gauss sum

Let $\chi$ be a Dirichlet character modulo $m$ induced by $\chi'$ modulo $m'$. We define $$ \tau(\chi):= \sum_{a(mod \ m)} \chi(a) e(a/m). $$ Could someone please show me how to derive the formula: ...
1
vote
1answer
38 views

Showing that $\int_0^1 \vert 1 - e^{2\pi i z} \vert^{-1} dx \ll \log(2 + y^{-1})$

I am trying to modify some work of Henryk Iwaniec involving bounds on the Fourier coefficients of modular forms. There is a bound which he gets that I'm having trouble understanding. So I want to ...
6
votes
2answers
113 views

Riemann Hypothesis: Is $1/2$ of critical line same as the $1/2$ of square-root accurately of error term of prime number theorem?

Here is a question about Riemann Hypothesis: Is $1/2$ of critical line same as the $1/2$ of square-root accurately of error term of prime number theorem $?$ In other words, (just for some brain ...
1
vote
0answers
44 views

How to produce Riemann zeta zero spectrum with the Fourier transform in Mathematica?

All: I post a question generating Riemann Zeta zero spectrum using Mathematica on board of mathematica.stackexchange.com: ...
3
votes
1answer
46 views

Are any zeros of Riemann zeta function and the zeros of the derivatives of Riemann zeta function same?

All: Are any zeros of Riemann zeta function and the zeros of the derivatives of Riemann zeta function same ? They shall be all different, right ? Is there a proof of this statement ? Thank you.
1
vote
0answers
28 views

Can the Von-Mangoldt function and the Chebyshev function be defined for entire complex plane?

Can the von-Mangoldt function and the Chebyshev function be defined for the entire complex plane ? I assume so, but I had not seen the definition. Can anyone provide some links for this? Thank you.
1
vote
0answers
37 views

Where to find Brun's original combinatoric treatment of Brun Sieve?

I tried to understand Brun's original combinatoric treatment of Brun Sieve. (Unfortunately, I do not understand German), so I could not read Brun's original paper as in following: Viggo Brun (1915). ...
1
vote
1answer
44 views

Number of primitive characters modulo $m$.

Let $N(m)$ be the number of primitive Dirichlet characters modulo $m$. Could someone please explain me why it satisfies the following relation?: $$ \phi(m) = \sum_{d|m} N(m). $$ Thank you very ...
2
votes
1answer
61 views

What are some fundamental symmetries of Riemann Zeta (Xi) function are important to Riemann Hypothesis?

What are some fundamental symmetries of Riemann Zeta (Xi) function are important to Riemann Hypothesis ? (Beside the obvious symmetry of Riemann Xi function, s <--> 1-s reflection) IMHO, at the ...
0
votes
0answers
23 views

What are some recursive properties of Merten function or Summatory Liouville function?

Both Merten function and Summatory Liouville function show some kinds of "scale invariance" properties. (Those functions also display some kind of "periodic" behavior.( Just wonder if those "scale ...
0
votes
0answers
24 views

What are equivalent statements or nature occurrences of Goldbach Conjecture in other math branches? [duplicate]

What are equivalent statements or nature occurrences of Goldbach Conjecture in other math branches ? We all know that Riemann Hypothesis(RH) has many (maybe over one hundred) equivalent statements. ...
1
vote
1answer
54 views

Proving the Complex Conjugate is Analytic [duplicate]

Let G be a region and define $G^∗ = \{z : z̅ ∈ G\}$. If $f : G \to \mathbb{C}$ is analytic, prove that $f^* : G^∗ → C$, defined by $f^*(z) = \overline{f(\overline{z})}$ is also analytic.
1
vote
0answers
44 views

Elementary theorems in number theory which need analytical methods (analysis)

A well known theorem of Dirichlet says: if $a,b$ are relatively prime positive integers, then there are infinitely many primes of the form $an+b$. The original proof by Dirichlet (possibly) uses ...
1
vote
1answer
29 views

Using partial summation to bound $\sum n^{-s} \log^\ell n$

I am having trouble obtaining the following estimate in a book: $$ \sum_{n > X} n^{-s} (\log n)^l = \int_{X}^{\infty} t^{-s} (\log t)^l \ dt + O(|s|(\log X)^{l+1}/X). $$ Here $s$ is a complex ...
1
vote
0answers
30 views

Upper bound for the $t$-th moment in terms of lower moments

Let $a_1, \ldots, a_n$ be positive integers. For positive integers $t$ and $m$ define the sum $$ M_t(m) = \dfrac{1}{n} \sum_{k=1}^n |a_k - m|^t. $$ I'm interested in upper bounds for $M_t(m)$ in ...
1
vote
0answers
63 views

Techniques in analytic number theory

I'm fairly new to the subject and trying to figure things out. Would be nice to hear some ideas and trickery for what follows. Suppose we wish to show there exists an integer $x$ in some finite set ...
2
votes
1answer
77 views

How do primes determine the locations of zeros of Zeta function?

All: We all know that zeros of Zeta function determine (or predict) the locations of primes through explicit formula. Is there anything similar that shows how primes determine the locations of zeros ...
2
votes
1answer
62 views

Estimating a sum related to a short Euler product

The Question Is $$\sum_{\substack{n>y\\ p\mid n\Rightarrow p\leq y}}\frac{\Lambda(n)}{n^s\log n}=O(1/\log T)$$ where $y=(\log T)^{100}$ and $T$ is large? Background Assume that ...
0
votes
0answers
28 views

data for zeros of the derivative of the Riemann zeta function

People have computed a large number of zeros of the Riemann zeta function. Do we have data for zeros of the first derivative of the Riemann zeta function?
6
votes
3answers
295 views

The order of li(x)

So I've been investigating $\mathrm{li}(x)$, that is the logarithmic integral function. I am unsure if this is true, but it seems as if $\mathrm{li}(x) = O\left(\frac{x}{\log x}\right)$ for $x$ ...
0
votes
1answer
39 views

Simplifying an expression involving log using euler's constant $\gamma$

I was wondering if someone could give me a hint or solution on how to obtain the following estimate: $$ x \log^2 x - 2 x \log x + 2 x + O(\log^2 x) = x \log x (\sum_{k \leq x} 1/k) - \sum_{k \leq x} ...
1
vote
0answers
40 views

Why is $M(x)=\frac{1}{x}\sum_{n\leq x}\mu(n)=o(1)$ ($x\to\infty$) equivalent to the Prime Number Theorem

Where $\mu$ is the mobius function and $o(\phi)$ is a Lindau Symbol, $f=o(\phi)$ if and only if $f/\phi\to 0$. Found it stated as an obvious fact in a math journal. I'd like a hint if it really is ...
2
votes
1answer
34 views

A basic question on real multiplicative character

Suppose I have a real multiplicative character module $q$, say $\chi$. Could some one please explain me why the following is true: given $n$, $$ \prod_{p^{\alpha} || \ n} (1 + \chi(p) + ... + ...
2
votes
3answers
57 views

Computing the tail of the zeta function $\sum_{n>x}n^{-s}$

I want to compute $$ f_s(x)=\sum_{n>x}n^{-s} $$ for some $s>1$ (in my case, $s=3$). Of course $$ f_s(x)=\zeta(s)-\sum_{n\le x}n^{-s} $$ but for $x$ large this is hard to compute. Are there good ...
4
votes
2answers
109 views

Estimate for a specific series

For a positive integer $m$ define $$ a_m=\prod_{p\mid m}(1-p), $$ where the product is taken over all prime divisors of $m$, and $$ S_n=\sum_{m=1}^n a_n. $$ I am interested in an estimate for $|S_n|$. ...
0
votes
1answer
65 views

Ramanujan Theta Function Identity

I have honestly tried a variety of things for this identity. I just straight plugged stuff in and tried to simplify as much as possible (for a while), and I also tried an induction proof, but I cannot ...
0
votes
1answer
44 views

A particular cases of second Hardy-Littlewood conjecture

Can someone solve this problem? (or does someone known a proof of this problem, if it exists?) for every $n\geq 2$, $\pi(2n)\leq 2\pi(n)$. Thanks a lot!
0
votes
1answer
63 views

If f(k) is the least number such that none of [f(k), f(k)+1,…f(k)+k-1] are k power free, how fast does f(k) grow?

By a k-powerfree number, I mean a positive integer that is not divisible by the kth power of any prime number. For example, 32 is 6 powerfree but not 5 powerfree. The interval of integers ...
1
vote
1answer
29 views

A number-theory question on the deficiency function $2x - \sigma(x)$

Let $\sigma(x)$ be the sum of the divisors of a (positive) integer $x$. (For example, $\sigma(2) = 1 + 2 = 3$.) Define the deficiency function $D(x)$ to be the number $$D(x) = 2x - \sigma(x).$$ Let ...
1
vote
0answers
49 views

A conjecture on integers coprime to $2^n-1$ and having a prescribed Hamming weight in their binary representation

I wonder if anyone has seen this before or would have some ideas on how to go about proving it. I have done several experiments with the computer and seems to hold. For an integer $k$, denote by ...
1
vote
2answers
41 views

Simplifying logarithmic integral $Li(x)$

I was wondering if someone could suggest me a hint on how to obtain the following expression $$ Li(x) = \int_{0}^x \frac{1}{\log y} dy = \int_{1}^x (1 - \frac{1}{ y}) \frac{1}{ \log y} dy + \log \log ...
1
vote
1answer
233 views

Derivation of Perron's formula

I tried to derive Perron's formula, but got really screwed up. I know of other ways to derive it, but I'm not quite sure why this way isn't working. I would appreciate some pointers on where I'm going ...
0
votes
0answers
21 views

Convergence of a sum involving the divisor function and characters

Could someone please show me that $\sum_{n = 1}^{\infty} \sigma_{x}(n)n^{-(x + 1)/2} \chi(n)$ converges where $\chi $ is a Dirichlet character of modulus $m$?
0
votes
0answers
26 views

Limit of the ratio of the square root of a Mersenne number to the product of its prime factors

Mersenne numbers with prime exponents are numbers of the form $M_p = 2^p-1$, where $p$ is prime. Suppose that $p$ is such that $M_p$ has exactly two prime factors, $\rho, P$. Given $\epsilon > 0$, ...
1
vote
0answers
34 views

Limit of an euler product

Before I can ask my question, I have to state a couple of definitions. Let $f$ be a multiplicative function and let $$ D_f(s) = \sum_1^{\infty} \frac{f(n)}{n^s}, $$ and define $\Lambda_f(n)$ as ...
3
votes
1answer
30 views

A certain zeta function; or, the determinant of the Laplacian plus a constant on the circle

I am interested in a certain "zeta function," a meromorphic function of $s \in \mathbb{C}$ that depends on a real parameter $\alpha \neq 0$. It's defined for the real part of $s$ large by $$ ...
0
votes
0answers
17 views

Kloosterman Sums and Lattice Hyperbolas

Part of this blog discussing the twin prime conjecture mentions a connection between three objects: $ \sum_{x \leq n \leq 2x} \tau\Big(n(n+2)\Big) $ average over twin primes where $\tau(n) = (1 ...
2
votes
1answer
27 views

Mellin transform of rescaled delta distributions

There's something about the Mellin transform I don't get, so hopefully someone can tell me what it is that I'm doing wrong. Let's define the Mellin transform of $f(t)$ as $\mathcal{M}\{f(t)\}(s) = ...
4
votes
3answers
81 views

What do following asymptotic symbols mean?

What do these symbols mean? I see them in analytic number theory. $$\ll$$ $$\gg$$ $$\ll_\epsilon$$ $$\gg_\epsilon$$ $$\asymp$$ $$\sim$$ All these appear in here ...
1
vote
1answer
67 views

Estimate for partial sums of a series equivalent to the Riemann hypothesis

The sums $$S_N=\sum_{n=1}^N\frac{\mu(n)}{n},$$ where $\mu$ is the Moebius function, are known to tend to 0 as $N\to+\infty$. As far as I remember, there was an estimate on $S_N$ equivalent to the ...
1
vote
2answers
40 views

Can someone help me understand this number sequence?

In the link below there is a number sequence and I do not know how it is put together. It is also arranged in an irregular triangle on the page, I tried looking up how the "Abramowitz and Stegun ...
2
votes
1answer
59 views

How to show $\binom{2n}{n} \ge \prod_{n < p \le 2n} p $?

What is the best way to show \begin{equation} \binom{2n}{n} \ge \prod_{n < p \le 2n} p \end{equation} for prime $p$. I know that $ 2^{2n} = (1+1)^{2n} \ge \binom{2n}{n}$. and \begin{equation} ...
1
vote
0answers
19 views

Symmetry of Hecke L-function zeroes

For the Riemann zeta function, it is known by the functional equation and $\zeta(s)=\overline{\zeta(\bar s)}$ that the zeroes of $\zeta(s)$ are symmetric about the critical line $1/2$ and the real ...