# 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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### 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 ...
2answers
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### 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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### 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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### 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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### 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 ...
1answer
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### 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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### 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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### 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 ...
1answer
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### 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 ...
1answer
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### 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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### 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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### 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 ...
1answer
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### 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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### 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', ...
2answers
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### 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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### 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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### 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 ...
4answers
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### 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 $n$, ...
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 $n \le 6 \times 10^5$...
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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### 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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### 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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1answer
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### 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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### 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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### 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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### 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 ...
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 ...
0answers
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### 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, $S$ ...
1answer
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### 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 ...