Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [analytic-number-theory]

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

1
vote
1answer
36 views

Hurwitz zeta function for $s=0$ $\zeta(0,1/2)$

I'm studying the Casimir Effect with perfect spherical boundary which involves the use of the Hurwitz zeta function. I've been staring for a while at this equation: \begin{align} \sum_{l=1}^{\...
4
votes
2answers
73 views

Estimating $\sum\limits_{d\mid n}{d+a\choose b}$

Is there any way of estimating a sum like $$\sum_{d\mid n}{d+a\choose b},$$ for positive integers $a$ and $b$? For example, in the OEIS we find that $$\begin{align*} \sum_{d\mid n}{d+1\choose 2} &...
4
votes
1answer
53 views

Density of primes of the form $x^2+my^2$

I was playing with numbers and have the nice conjectures: Let $m$ be a fixed positive integer, and $\pi(N)$ denote the numbers of primes not exceeding $N$ and $\pi_m(N)$ denote the number of prime ...
3
votes
1answer
98 views

Sums of the form $\sum\limits_{a_1=1}^n\sum\limits_{a_2=1}^{a_1}\cdots \sum\limits_{a_r=1}^{a_{r-1}}{[\gcd(a_1,a_2,\dots ,a_r)=1]}$

Background: Consider sums of the form $$\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\cdots \sum_{a_r=1}^{a_{r-1}}{[\gcd(a_1,a_2,\dots ,a_r)=1]},$$ with $[\gcd(a_1,a_2,\dots ,a_r)=1]$ being equal to $1$ if the ...
0
votes
1answer
26 views

Zeta function of the hypersurface of some homogeneous polynomial

Let $f(y)\in \Bbb Z_p[y_0,y_1,....,y_n]$ be a homogeneous polynomial. Let $N_s$ be the number of zeros of $f$ in $\Bbb P^n(F_{p^s})$. Here, $\Bbb P^n(F_{p^s})$ denotes the $n$-th projective space ...
4
votes
1answer
80 views

How to estimate $\sum_{p\leqslant x}\sum_{q\leqslant x}\frac{1}{p+q}$?

How to estimate $$\sum_{p\leqslant x}\sum_{q\leqslant x}\frac{1}{p+q}, \qquad\qquad(1)$$ where $p$, $q$ are prime numbers. We have the Mertens' formula $$ \sum_{p\leqslant x} \frac{1}{p} = \log\log ...
1
vote
1answer
32 views

Prove $F^* = \mu * F$

Let $f: \mathbb{Q} \cap [0,1] \to K$ and set $F(n) = \sum_{k = 1}^n f(\frac k n)$, $F^*(n) = \sum_{k = 1, (k,n) = 1}^n f(\frac k n)$. Show that $F^* = \mu * F$ where $*$ is the Dirichlet product....
0
votes
2answers
61 views

Using Dirichlet's theorem to show existence of number coprime to $n$

I have the following question: Let $n$ be a positive integer and $d$ be divisor of $n$. Use Dirichlet's theorem to show that there exists an integer $k$, where $1\le k\le d-1$ such that the number $m:=...
0
votes
0answers
24 views

Convergence region of local and global zeta functions

Let $\chi = \prod_{v} \chi_{v}: \mathbb{A}^{\times}/F^{\times} \to \mathbb{C}^{\times}$ be a finite order Hecke character and let $\Phi = \prod_{v} \Phi_{v}$ be a Schwartz function on $\mathbb{A}$. ...
1
vote
1answer
34 views

Counting number of ideals in quadratic number field

Let $K$ be a quadratic number field and $R$ be its number ring, and if $a(n)$ denotes number of ideals of norm $n$, if $n$ is a prime number, then number of ideals of norm $n$ is $1+(d|n)$, where $d$ ...
1
vote
0answers
35 views

Do the properties defining the Selberg class imply the distribution of real parts of non trivial zeros of an L-function is strongly unimodal?

Selberg defined what is now known as the Selberg class as a class of L-functions fulfilling for essential properties, which are analyticity, Euler product, functional equation and Ramanujan-Patersson ...
1
vote
0answers
23 views

Bound for sum of squares $r_{2k}(m)$ for $m \geq 1$

I happened to read in (Iwaniec-Kowalkski) Analytic Number Theory book that the Sum of Squares function satisfies the bound $r_{2k}(m) << m^{k-1+\epsilon}$ for $m \geq 1$. But $\epsilon$ is not ...
1
vote
1answer
71 views

The equation $\zeta(q)=0$ for $q$ a quaternion

I know there have been several attempts to define a theory of functions of a quaternionic variable. I would like to know if a coherent and satisfying definition of the "Riemann" zeta function exists ...
1
vote
2answers
36 views

Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$.

Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$. I cannot give a proof to this, although I have try on ...
3
votes
1answer
58 views

Number System with Torsion

Introduction: A number system, long known but seldom seen, is (re)introduced for which some elements are torsion and some are torsion-free. A topology question and an analytic number theory question ...
3
votes
1answer
52 views

the exponent of convergence of $\frac{p_{n+1}}{p_n}$ to $1$

Let $p_k$ be the $k$-th prime. Then $\lim_{n\to\infty}\frac{p_{n+1}}{p_n}=1$ -- this is well known. I was looking for more specific information: What is the exponent of convergence of $\frac{p_{n+...
1
vote
0answers
12 views

application of distribution of primes in arithmetic progressions

I try to understand an application of distribution of primes in arithmetic progressions Let $$f(x) = \sum_{p \leq x p\equiv 3 \bmod 10} 1$$ So computing $f(40) = 3$ i.e. the primes: 3, 13, and 23 ...
3
votes
0answers
63 views

On the proof that $\phi(n)/n$ has a limit law

In this question, $\mathbb{N}$ denotes the set of positive integers. Also, $\overline{\mathrm{d}}$, and $\mathrm{d}$ means upper natural density, and natural densitiy respectively. (They are the ...
2
votes
1answer
37 views

$\sum_{p \le x, p \equiv 3 \bmod 10} \frac{1}{p} = \frac{1}{4} \log\log(x)+A+O(\frac{1}{\log x})$

I wish to prove the following equality $$\sum_{p \le x, p \equiv 3 \bmod 10} \frac{1}{p} = \frac{1}{4} \log\log(x)+A+O(\frac{1}{\log x})$$ For some constant A Own work: Let $$A(x) = \sum_ {p \le x} ...
2
votes
2answers
42 views

$F(x) = L(1, \chi ) \log x + O(1)$

I wish to prove $$F(x) = L(1, \chi ) \log x + O(1)$$ when $A(n) = \sum_{d|n} \chi (d)$ and $F(x) = \sum_{n \leq x} \frac{A(n)}{n}$ I started of course by substituting $A(n)$ in $F(x)$, which becomes ...
0
votes
0answers
17 views

Effectiveness of Landau's Prime Ideal Theorem

Is Landau's Prime Ideal Theorem effective? It seems clear that bounds of the strongest kind known on the error term cannot be made fully effective, due to the possibility of Siegel zeroes. At the same ...
2
votes
1answer
35 views

How to generalize Newman's simplification of O-Tauberian theorem?

Can you prove the next theorem: Let $f$ be Dirichlet series with real, positive coefficients $(a_n>0)$. If $f$ is holomorphic on $\Re(z)\ge1$, but has one singularity at $z=1$, then $\lim_\limits{...
0
votes
0answers
53 views

“on average” in the Bombieri-Vinogradov theorem

TLDR: I don't understand the bit in bold, i.e. where in the formula is the average q? Thanks. The Bombieri-Vinogradov theorem states the following: For any $A > 0$ there exists a $B = B(A)$ such ...
0
votes
0answers
33 views

Bound on log integral

I am looking for an explanation of the bound $$\frac{1}{2\pi}\left(-\frac{T \log T}{1+(t-T)^2} - 2 \int_T^\infty \frac{x \log x (t-x)}{(1+(t-x)^2)^2} dx \right)\ll \left( \frac{1}{t+1} + \frac{1}{T-t+...
0
votes
0answers
44 views

Why aren't holomorphic modular forms bounded?

Let $f$ be any non-zero integral weight (holomorphic) modular form with respect to $SL_2(\mathbb{Z})$ and of weight $k, k\geq 4$. Since it is holomorphic at infinity, for given $\epsilon > 0$, it ...
2
votes
0answers
32 views

Prove the series $\sum n^{-1-it}$ is diverge for all real $t$.

Prove that the series $\sum_{n=1}^\infty n^{-1-it}$ diverges for all real $t$. I have shown in the previous exercise that this series is bounded for nonzero $t$, and when $t=0$, it is famous that the ...
1
vote
1answer
54 views

Step by step derivation of Robin's inequality $\sigma(n) < e^\gamma n \log \log n$

Guy Robin proved that $$\begin{equation} \sigma(n) < e^\gamma n \log \log n \end{equation}$$ is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984). The paper where ...
5
votes
1answer
50 views

Lower Bound on the Sum of Reciprocal of LCM

While reading online, I encountered this post which the author claims that \begin{align} S(N, 1):=\sum_{1\le i, j \le N} \frac{1}{\text{lcm}(i, j)} \geq 3H_N-2 \end{align} and $S(N, 1) \geq H_N^2$ ...
1
vote
1answer
64 views

Partial Euler product

The Riemann Zeta function defined as $$\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$$ For $\Re(s)>1$ is convergent and admits the Euler product representation $$\zeta(s) = \prod_p (1-p^{-s})^{-1}$$ For ...
2
votes
1answer
52 views

Inferences about sign of a function from abscissa point

Suppose we have a function $g(x)$ and an integral $$F(s)=\int_1^\infty \frac{g(x)}{x^{s+1}}dx$$ and $F(s)$ converges for $s>\beta$ and diverges for $s = s<\beta.$ Assume also that $\beta$ is ...
2
votes
0answers
23 views

Summatory function of Moebius and Euler's totient function over $y$-smooth numbers

Let $y \geq 1$. We say that a positive integer $n$ is $y$-smooth if $n$ has no prime factors larger than $y$. Let $x \geq y$. Let $\mu$ and $\varphi$ be the Moebius and Euler's totient function ...
0
votes
1answer
20 views

To show a function is bounded by a function when x is large

I want to show $\sum_{p^a\leq x}\log p = O(\sqrt{x}\log^2 x)$,where sum runs over $a\geq2$. I only know that $\sum_{\sqrt{x}<p \leq x}\log p \leq 2x\log x$. I tried using above property but I am ...
0
votes
0answers
18 views

Problem trying to show the following:$\sum_{n\leq x} (\omega(n)-\ln\ln x)^2=O(x\ln\ln x)$

So I have to show the following: $$\sum_{n\leq x} (\omega(n)-\ln\ln x)^2=O(x\ln\ln x)$$ But the problem is in finding a suitable bound for: $$\sum_{n\leq x} (\omega(n))^2$$ I have tried the ...
0
votes
1answer
13 views

If $(a,m)=(b,m)=1$ and if $(\exp_m(a),\exp_m(b))=1$, prove that $\exp_m(ab)=\exp_m(a)\exp_m(b).$

If $(a,m)=(b,m)=1$ and if $(\exp_m(a),\exp_m(b))=1$, prove that $$\exp_m(ab)=\exp_m(a)\exp_m(b).$$ The notation $\exp_m(a)$ is denote the smallest positive integer $n$ such that $a^n\equiv 1\pmod m$. ...
2
votes
1answer
65 views

Limit of a function, given the recurrence relation

Let $f(n)$ be a function defined for $n\ge 2$ and $n\in N$ which follows the recurrence(for $n\ge 3$) $$\displaystyle f(n)=f(n-1) +\frac {4\cdot (-1)^{(n-1)} \cdot \left(\sum_{d \vert (n-1)} (\chi (d))...
2
votes
0answers
45 views

Asymptotic for the gamma function on vertical lines

On page 135 of Joerg Bruedern's "Einfuehrung in die analytische Zahlentheorie" he claims that Stirling's formula implies for fixed $\sigma <0$, any $t\geq 1$, and some constant $C$ (I assume ...
0
votes
0answers
37 views

What information do the moments of the Riemann-Zeta function give us

I have seen an explicit formula for what a moment of the Riemann-Zeta function is but I am unsure what information this give us? If we are looking at the zero's of the function then this can be ...
0
votes
1answer
29 views

Bounding sum (log factor)

I want to prove that $$ \sum_{\substack{1\leq n\leq T \\ n\neq m}}n^{-\frac{1}{2}}\left|\log \frac{m}{n}\right|^{-1}\ll T^{\frac{1}{2}}\log T $$ for any $1\leq m \leq T$. Do you have any hint how I ...
0
votes
0answers
17 views

Equidistributed sequences satisfying the central limit theorem

For a sequence (x_n) of reals that are equidistributed modulo one (such as (prime-) multiples of irrationals, or most geometric sequences) does the (properly scaled) sum of the remainders converge to ...
1
vote
0answers
31 views

The sum $\sum_{r=1}^{p-1} r(r|p)$ when $p$ is an odd prime of the form $4k+3$, $k\geq 1$.

In the book Apostol Analytic Number Theory, $(r|p)$ denotes the Legendre Symbol. The exercise tell us to prove when $p\equiv 1\pmod 4$, $$\sum_{r=1}^{p-1}r(r|p)=0.$$ I can solve this quickly, but I ...
3
votes
0answers
40 views

Solving $\int_0^x\int_u^{u+1}\frac1{\ln t}\,dt\,du=0$ (an extension to the Ramanujan-Soldner constant)

For $u,x>0$, let $P$ be the function given by $$P(x)=\int_0^x\int_u^{u+1}\frac1{\ln t}\,dt\,du\tag1.$$ Is there a closed form for the positive root of $P(x)$, denoted by $\nu$? Can it be ...
1
vote
1answer
22 views

Let $P=\{1,2,\cdots,p-1\}$, $P=S\cup T$, prove that $S$ is quadratic residues and $T$ is quadratic nonresidues.

Let $p$ be an odd prime. Assume that the set $\{1,2,\cdots,p-1\}$ can be expressed as the union of two nonempty subsets $S$ and $T$. $S\neq T$, such that the product (mod $p$) of any two elements in ...
0
votes
0answers
11 views

Problem understanding summation of Big-O notation i.e $F(x)=\sum_{n\leq x}O(\frac{x}{n})$

I know that $O(f(x))+O(g(x))=O(g(x))$ if $f(x)=O(g(x))$. But I cant seem to find a bound for F(x) since n changes when x increases. Any ideas. Im assuming that $F(x)=O(\sum_{n\leq x}\frac{x}{n})$ ...
0
votes
1answer
61 views

Show that $\sum_{pq\leq x}\frac{1}{pq}$ = $(\ln \ln x)^2+O(\ln \ln x)$

I know that $\sum_{pq\leq x}\frac{1}{pq}$=$\sum_{p\leq x}\frac{1}{p}\sum_{q\leq\frac{x}{p}} \frac{1}{q}$=$\sum_{p\leq x}\frac{1}{p}(\ln\ln(\frac{x}{p})+A+O(\frac{1}{\ln (\frac{x}{p})}))$. However, I'm ...
6
votes
0answers
143 views

Almost a prime number recurrence relation

For the number of partitions of n into prime parts $a(n)$ it holds $$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$ where $q(n)$ the sum of all different prime factors of $n$. Due to https://oeis....
0
votes
0answers
57 views

Obscure approximate functional equation for the Riemann zeta function

The following result is supposed to follow from an approximate functional equation for the Riemann zeta function, but I've never seen anything close enough to it : For $T \leq t \leq 2T$ where $T$ is ...
2
votes
0answers
38 views

Understanding the Hurwitz-Kronecker Class Number Formula

The Hurwitz-Kronecker Class Number is given by the formula $H(d)=\sum_{Q\in Q_d/(\Gamma=PSL_2(\mathbb{Z}))}\frac{1}{w_Q}$ where $w_Q=card(stab(\alpha_Q))$ with $\alpha_Q$ being the unique zero ...
1
vote
0answers
30 views

Don't know what to do with little-oh!

Let $f:\mathbb{N} \to \mathbb{C}$ be a function for which there exists a positive constant $A$ such that \begin{equation} \lim_{x\to \infty} \frac{1}{x}\sum_{n\leq x} f(n) = A. \end{equation} Prove ...
2
votes
2answers
70 views

Prove that $\left|\sum\limits_{m=N+1}^M\frac{\chi(m)}{m}\right|< \frac{2}{N+1}\sqrt{k}\log k$.

This problem is from Apostol's Analytic Number Theory, Chapter $9$. The problem: Let $\chi$ be a primitive Dirichlet character mod $k$. Prove that if $N< M$ we have $$ \bigg|\sum_{m=N+1}^M\...
1
vote
1answer
25 views

Proof Verification: Let $f(\chi)$ be the conductor of $\chi$, proof that $f(\chi)=f(\chi_1)\cdots f(\chi_r)$.

This is a detailed problem, let me write down the problem and the process I have done: Assume that $k=k_1k_2\cdots k_r$ where $k_i$ and $k_j$ are relatively prime for $i\neq j$. Let $\chi$ be a ...