Questions tagged [analytic-number-theory]

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

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20 views

Where to find a proof of the Pólya-Jensen criterion in English?

The Pólya-Jensen criterion for the Riemann Hypothesis asserts that RH is equivalent to the hyperbolicity of certain Jensen polynomials for all degrees d ≥ 1 and all shifts n. This criterion was used ...
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2answers
94 views

Construction of an exact function for counting primes in intervals.

I have constructed an exact function for counting primes in intervals and am curious to know if it 1) has any importance? 2) Has been derived already? I have no formal education in number theory, and ...
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31 views

Rules of Analytic Continuation

Context This summer, I have been reading Tom Apostol's "Introduction to Analytic Number Theory." I have yet to take a formal analysis class, but have been able to follow along until the very end. ...
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26 views

Growth of a series involving the floor function to a certain exponent.

Let $\theta \geq 0$ and consider the sum $$\sum_{n \leq x} \lfloor \frac{x}{n} \rfloor^{-\theta}.$$. How do I find a constant $c(\theta)$ so that this sum equals $$c(\theta)x+O(1),$$ where the ...
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29 views

A curious identity summing over primes in an interval.

Let $K \geq1$ be an arbitrary positive constant. Why do we have the equality $$\sum_{x/K \leq p \leq x} \dfrac{log(p)}{p} = log(K)+O(1),$$ where the error term is uniform in $K?$ Here the summation is ...
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1answer
41 views

Prove that the asymptotic density of $w(n)|n$ is 0

Note $w(n)$ is number of primes dividing $n$. I know the definition of asymptotic density, but I'm not sure how to start with this problem. I can prove that the sets $w(n)|n$ and $w(n)\nmid n$ are ...
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104 views

Chebyshev function: variational formulation

The Wikipedia article on the Chebyshev function $\psi(x)$ states that, for $x=e^t$, it minimizes the functional $$J[f] = \int_0^\infty \dfrac{f(s)\zeta'(s+c)}{\zeta(s+c)(s+c)}ds - \int_0^\infty \...
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1answer
34 views

Simplifying an alternating sum of a product of factorials

For integers $a$ and $b$, I am curious how to simplify an expression of the form $$\sum_{k=1}^n (-1)^k (a+k)! (b+k)!$$ I assume there is some simplification using properties of gamma and beta ...
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93 views

How was this formula made by Srinavasa Ramanujan derived?

I'm wondering how this formula was derived. I'm looking for a proof. My guess is that there was a substitution made, and through contour integration or something, it was shown that the limits don't ...
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46 views

Heuristics on the Circle Method

I hope I'm not missing anything too obvious here, but I have a question on the overall setup of the circle method itself. Just recently started glossing over Vaughan's book and other sources online, ...
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3answers
695 views

Does there exist positive rational $s$ for which $\zeta(s)$ is a positive integer?

Does there exist positive rational $s$ for which the Riemann Zeta function $\zeta(s) \in N$ or equivalently, does there exist finite positive integers $\ell,m$ and $n$ such that $$\zeta\left(1+\dfrac{\...
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57 views

Curious closed forms of the $q$-Gamma function.

I found on Wikipedia the $q$-Gamma function, defined as $$\Gamma_q(x)=(1-q)^{1-x}\prod_{n\ge0}\frac{1-q^{n+1}}{1-q^{n+x}}$$ for $|q|<1$. There is a definition for $|q|>1$, but we won't need ...
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111 views

Wolstenholme Number

A Wolstenholme number is the (reduced) numerator of the fraction $1+{1\over4}+\cdots+{1\over n^2}$. The first few are $1, 5, 49, 205, 5269, 5369, 266681, 1077749$. Are there Wolstenholme numbers that ...
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1answer
35 views

Bounding coefficients of Dirichlet Series

Consider the exponentiated Riemann-Zeta function $\zeta(s)^p$. If it is represented as $$\zeta(s)^p = \sum_{n=1}^\infty\frac{a_n}{n^s}$$ Is there any upper bound we can put on $|a_n|$ in terms of ...
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1answer
96 views

sign unchanged for Dirichlet polynomials?

Let $\chi_{q}$ be a primitive Dirichlet character with modulus $q$ (see definition at wikipedia ). For example for $q=5$ we have \begin{equation} \begin{aligned} \chi_{5,1}&=(1, 1, 1, 1, 0),\\ ...
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78 views

Unique pattern in addition of digits

Problem: For positive integers $n,k$, let $$S(n,k)=\sum_{i=1}^{n}i^k$$ and for positive integers $m,b$, with $b>1$, let $D(m,b)$ be the sum of the base-$b$ digits of $m$. Q$1$- Show that ...
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28 views

Non-vanishing Taylor coefficients and Poincaré series

I'm studying these algebraic numbers in the table below and have succesfully shown what i wanted with the given values I had. The table is found in the book "The 1-2-3 of Modular Forms" by Jan ...
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1answer
229 views

Dirichlet character modulo p

How can I prove that if $\chi$ is a non-principal character modulo $p$ prime, then $\chi (-1) = \overline{\chi} (-1)= \pm 1$ and $\sum_{x=1}^p \chi (x) e^{2\pi i x}=0$? For the first question, I just ...
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29 views

Polya-Vinogradov inequality over non-consecutive integers

Let $N$ and $a$ be positive integers. Consider the Kronecker symbol $\left( \frac{n}{m} \right)$, which is a character modulo $m$. I have seen it several times that 'by Polya-Vinogradov inequality', ...
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2answers
45 views

For the divisor function is $d(n^2)$ related to $d(n)$ knowing also n?

The divisor function d(n) is defined as 'the number of positive divisors of n (including 1 and n)' according to Underwood Dudley. Is the divisor function $d(n^2)$ related to $d(n)$? for example d(...
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0answers
95 views

Why does this interval contain at least $x/(5\log(x))$ primes?

Let $$A_x = (x/2,x] \cup (x/4,x/3] \cup (x/6,x/5] \cup \cdots.$$ I can prove that $$\sum_{n \in A_x} \Lambda(n) = \log(2)x+O(\log(x)),$$ and that further, the error term cannot be improved. Given ...
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77 views

Generalization on a feature of 21

Let $n,m \in \mathbb{N}$ $$n=\prod_{i=1}^{r}p_{i}^{a_i}$$ where $p_i$ are prime factors and $f$ , $g$ and $h$ are the functions $$f(n,m)=\sum_{j=1}^{n}j^m$$ And $$g(n)=\sum_{i=1}^{r}a_i.p_i$$ If we ...
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4answers
7k views

Euler totient function sum of divisors. Theorem 2.2 Apostol

Prove that : If $ n\ge 1 $, then $ \sum_{d|n}\phi(d)=n $. Let $S$ denote the set $\{1,2,...,n\}$. We distribute the integers of $S$ into disjoint sets as follows. For each divisor $d$ of $n$, ...
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1answer
248 views

Are there infinitely many zeroes of $\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) $?

Let $\mu(n)$ be the Möbius function and $S(x)$ be the number of positive integers $n \le x$ such that $$ \sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) = 0 $$ My experimental data for $n \le 6 \times 10^5 $...
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1answer
572 views

What is an upper bound for number of semiprimes less than n?

A semi prime is a number which is product of two distinct prime number. What is an upper bound for number of numbers in the form pq less than n? $p,q$ are prime numbers smaller than $n$.
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41 views

Why is the following equality involving big $O$-notation true?

Suppose I know that $\Delta(x)$ is a function on $\mathbb{R}$ such that $\Delta(x) = O(\sqrt{x}).$ Suppose that $x$ is a large real number and that $h < x/2.$ Given this, why does the equality $$\...
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1answer
763 views

Change of order of summation.

I feel like an idiot for asking this, so bear my stupidity. I have the sum $\sum_{n\leq N} \sum_{p | n ; \ p \ prime} 1$, and I want to change the order of summation of these two sums I think it ...
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1answer
35 views

Prove that von Mangoldt function satisfies $\sum_{n \le x} \Lambda(n) \lfloor{\frac{x}{n}}\rfloor= x \ln(x)-x+O(\ln x)$

The picture above is what I have, which has an error that is too big.
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50 views

Regarding definitions of modular functions and modular forms

I am self studying Analytic number theory from Apostol Dirichlet series and modular functions in number theory and I have a doubt regarding defintions of modular forms and modular functions. My ...
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1answer
46 views

Why logarithmic density of sets are studied?

Natural density of a subset $S $ of positive integers is defined as $$\lim_{n \to \infty}\frac{1}{n}\sum\limits_{k\in S, k\le n}1 $$ whenever it exists. The logarithmic density of $S$ is defined as $...
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8 views

Regarding Dirichlet approximation theoram in diophantine approximation

I am self studying analytic number theory from apostol Dirichlet series and modular forms in number theory. I am unable to solve this problem in Ch-7. Let $\alpha = ( 1+ \sqrt 5) / 2 $ and $$\beta = ...
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16 views

Diophantine approximation and Farey sequences

I am self studying Apostol Dirichlet Series and Modular Functions in Number Theory and could not solve this question given in Chapter 7. Please help. Question is – Let $ \Theta $ be an irrational ...
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0answers
9 views

Regarding properties of Farey sequences related to lattice points.

I am self studying Apostol's Dirichlet series and Modular Functions in Number Theory and need help in this question. Question is – let $n \ge 1$ and $T_n$ denotes the set of lattice points $(x, y)$ ...
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15 views

Is the following equality concerning an $L$-function really true?

Let $\psi$ be a Dirichlet character defined mod $q.$ I have seen the claim that for $s = \sigma +it$ fixed and $\sigma >0,$ that $$\sum_{n=1}^y \psi(n)n^{-s} = L(\psi,s) + \underline{O}(y^{-\sigma})...
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1answer
253 views

Relationship between GCD, LCM and the Riemann Zeta function

Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased, $$ \frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\text{lcm}(k,i)}\bigg)^s \approx \...
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447 views

A Special Observation on Prime Numbers and $\pi (n)$

$\eth(n)$ is a little algorithm I made, which may appear to be quite complex, so I will start with an example middle of the post. Questions are at the end of the post. Definition Let $...
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36 views

Can a upper bound of $\sum_{b=1}^{p-1}\left(\frac{b^2-a^2}{p}\right)\left(\frac{b^2-1}{p}\right)$, $a\in{Z}$, be strictly less than $\sqrt{p}$?

By weil estimate I can only say, one bound is $\sqrt{p}$. Can it be strictly less than $\sqrt{p}$? I want to see whether one better bound be given or not.
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1answer
61 views

Minimum and maximum of a partial Euler product?

Question: If if $n\in\mathbb{N}$ and $s\in \mathbb{C},$ say $s=\sigma+t\sqrt{-1},$ then Dirichlet Beta function is defined to be $$ \beta(s)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}; $$ which for ...
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43 views

Why is the series representation of the logarithm of the zeta function analytic?

I try to prove, that the logarithm of the $\zeta$-function has the following representation for $z \in \mathbb{C}$ with $\text{Re}(z) > 1$ $$\log\zeta(z)=-\sum_p\log\left(1-\frac{1}{p^z}\right).$$ ...
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24 views

Probability for an L-function to be RS-primitive

Assuming an L-function is any element of the intersection $\mathcal{L}$ of the Selberg class $\mathcal{S}$ and the class of automorphic L-functions $\mathcal{A}$, define the notion of Galois class of ...
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1answer
939 views

primitive roots of composite numbers

I am looking for a way to find primitive roots of composite numbers by primitive roots of its prime factors.im looking for a analytic way no algebraic. I meant a way without meanings of abstract ...
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1answer
80 views

Is $\sum_{n=1}^{\infty} \frac{\mu(n)}{n} = 0$? [duplicate]

Does $\sum_{n=1}^{\infty} \frac{\mu(n)}{n}$ converge and does it converge to $0$ ?. I know that $\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$ converges on $Re(s) > 1$ to $\prod_{j=1}^{\infty}{(1-\frac{1}...
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1answer
59 views

Are these equalities wrong $\sum_{n\le x}\mu(n)o(\frac xn)=o(x\sum_{n\le x}\frac 1n)=o (x\log x) $?

I simply found an asymptotic relation for $\sum_{n\le x}\mu(n)o(\frac xn)$ like below: $$\sum_{n\le x}\mu(n)o(\frac xn)=o(x\sum_{n\le x}\frac 1n)=o (x\log x) $$ But Basil Gordon, an American ...
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74 views

Weyl's equidistribution theorem in the case of rational numbers

Let θ be a non-zero real number and let N be a positive integer. Consider the fractional parts of nθ for 0 ≤ n < N. This creates a maximum of N possible intervals. Show that there as only three ...
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19 views

On Property of Ramanujan Tau function

I have been self studying number theory from Apostol Dirichlet series and modular forms and I could not solve this question from chapter 4 Pg92. There is a solution of this question on stack exchange ...
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0answers
22 views

On proving a result on Jacobi theta function

I have been self studying Apostol Dirichlet series and Modular forms and I have could not solve this problem from Chapter 4 - Prove that $\theta(-1/t) = \sqrt{-it}\ \theta(t)$ , t belongs to upper ...
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4answers
511 views

Reference requests: Jitsuro Nagura

I spent some time today looking for any biographical information on Jitsuro Nagura and came up empty-handed. Any suggestions welcome. Also, the Wiki note on the Chebyshev $\psi$ function says that ...
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0answers
59 views

Density of Numbers with Exactly One Prime Factor of Multiplicity 1

Let $S$ be the set of positive integers $n$ with the property that exactly one prime factor of $n$ has multiplicity $1$ and every other prime factor has multiplicity greater than $1$ (to be clear, $S$ ...
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1answer
57 views

Interlacing with arithmetic progressions?

Given integers $0<a_1<\dots<a_t$ and $0<b_1<\dots<b_t$ with $a_t<b_t<M$ can we find integers $m,n,m',n'\in\mathbb Z$ such that $$ma_i+n<m'b_i+n'<ma_{i+1}+n$$ holds at ...