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

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0
votes
0answers
15 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 ...
1
vote
0answers
11 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
10 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
15 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
4answers
68 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 ...
0
votes
1answer
45 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
37 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
55 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} ...
12
votes
1answer
162 views

Why is $\zeta(1+it) \neq 0$ equivalent to the prime number theorem?

Reading through Titchmarsh's book on the Riemann zeta function, chapter 3 discusses the Prime Number Theorem. One way to prove this result is to check the zeta function has no zeros on the line $z = ...
36
votes
5answers
3k views

What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$ The ...
1
vote
0answers
14 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 ...
2
votes
1answer
85 views

A question from Titchmarsh's The Theory of the Riemann Zeta-Function.

On pages 35-36 here, we have that the integral $$\frac{1}{2i\sqrt{y}}\int_{1/2-i\infty}^{1/2+i\infty}\phi(s-1/2)\phi(1/2-s)(s-1)\Gamma(1+s/2)\pi^{-s/2}\zeta(s)y^sds$$ equals for $\phi(s)=1$ to: ...
7
votes
2answers
125 views

Arithmetic Derivatives: Arithmetic Logarithmic Derivative Problem

In Calculus, whenever we see a constant and want to take the derivative of it, it always is 0. However in Number Theory, we have something called the arithmetic derivative in which we can ...
1
vote
0answers
90 views

Behaviour of the sum of divisors function via logarithmic means versus an elementary problem equivalent to the Riemann Hypothesis due to Lagarias

It is known the following (see [1], here is an open access in his homepage www.math.lsa.umich.edu/~lagarias/doc/elementaryrh.pdf): Theorem (Lagarias, 2002). Let $\sigma(n)$ denote the sum of the ...
4
votes
0answers
34 views

How do you prove that $M(N)=O(N^{1/2+\epsilon})$ from the Riemann Hypothesis?

I understand that if $M(N)=O(N^\sigma)$, then $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}$ and therefore $$ \frac{1}{s\zeta(s)} = \int_0^\infty M(x) x^{-(s+1)} dx $$ for $s>\sigma$, ...
37
votes
3answers
1k views

Small primes attract large primes

$$ \begin{align} 1100 & = 2\times2\times5\times5\times11 \\ 1101 & =3\times 367 \\ 1102 & =2\times19\times29 \\ 1103 & =1103 \\ 1104 & = 2\times2\times2\times2\times ...
2
votes
2answers
54 views

Abel's Summation formula help.

I want to be able to show that, \begin{equation} \sum_{p} \log(p) p^{-s} = s \int_{1}^{\infty} \frac{\theta(t)}{t^{s+1}} dt \end{equation} where $\theta(x) =\sum_{p \le x} \log p$. and $\theta(x) = ...
8
votes
3answers
73 views

Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?

Let $P$ be the set of primes $p$ greater than $3$ such that $p\equiv1 \pmod{4}$. Does the following sum converge or diverge? $$ \sum_{p\in P}\frac{1}{p} $$
0
votes
1answer
35 views

Analytic continuation of Gamma function

I have tried proving the analytic continuation of the gamma function. I am using the notation, \begin{equation} \Gamma(n) = (n-1)! \end{equation} and \begin{equation} (1) \ \ \Gamma(s) = ...
1
vote
3answers
397 views

How to prove $\sum_{d|n} {\tau}^3(d)=\left(\sum_{d|n}{\tau}(d)\right)^2$

$\sum_{d|n} {\tau}^3(d)=\left(\sum_{d|n}{\tau}(d)\right)^2$, where $\tau(d)$ designates the number of positive divisors of d. Now I only know that both sides are multiplicative functions, could ...
1
vote
0answers
23 views

Finding the value of a sum of two Hecke eigenvalues

I did some computations but I am stuck in finding the exression of the sum $$\lambda_f(n^2)+\lambda_f(n)^2 $$ in terms of $\lambda_f(n),$ where $f$ is a modular form for the full modular group. Any ...
2
votes
1answer
59 views

connection between odd primes and a certain q-series

I posed a conjecture about odd primes and a certain q-serieshere.I thought it would be more appropriate ,if I could ask the converse of the aforementioned problem . Is $p$ an odd prime iff ...
11
votes
1answer
274 views

Maximum integer not in $\{ ax+by : \gcd(a,b) = 1 \land a,b \ge 0 \}$

Ryan asked about a variation of the coin problem, which was whether for any coprime natural numbers $x,y$ every sufficiently large natural number is $ax+by$ for some coprime natural numbers $a,b$. ...
8
votes
1answer
219 views

Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function

Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...
4
votes
1answer
72 views

Average order of $\mathrm{rad}(n)$

Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing n. Or equivalently, $$\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ ...
0
votes
0answers
17 views

Question about the sign of a certain sum

Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$ I want to ...
6
votes
2answers
209 views

Prime power Gauss sums are zero

Fix an odd prime $p$. Then for a positive integer $a$, I can look at the quadratic Legendre symbol Gauss sum $$ G_p(a) = \sum_{n \,\bmod\, p} \left( \frac{n}{p} \right) e^{2 \pi i a n / p}$$ where ...
10
votes
2answers
550 views

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth are playing the following game. ...
1
vote
0answers
23 views

From 4 squares to 2 squares

Let $r_k(n)$ denote the number of representations of $n$ as a sum of $k$ squares of integers. Suppose I know that $r_4(n) = 8\sum_{4 \nmid d \mid n} d$ (Jacobi's Theorem). Is there a way to deduce ...
1
vote
0answers
46 views

Cross-correlation of Gaussian and Jacobian sums

I recently came upon the following kind of sum and I'm wondering if anyone has seen it before, or could point out something interesting about them. Let $F$ be a finite field with $q > 2$ elements ...
1
vote
0answers
29 views

Gauss sum of a multiplication of two multiplicative characters of a finite field

Let $F$ be a finite field with $q$ elements and characteristic $p$. Let $E$ be a proper extension over $F$ of degree $n$. Let $\psi$ be the canonical additive character of $E$ defined by $\psi(x) = ...
1
vote
0answers
28 views

Increasing sequences and $\zeta$-type functions

The Riemann zeta function is defined as the sum $\zeta(s) = \sum_{n \geqslant 0} n^s$. The question is whether it globally characterizes the sequence of all natural numbers, in the following sense: ...
1
vote
0answers
14 views

Prime Factorisation Probability Decrease: upper bound

Suppose we have a probability distribution $p$ over $\{0, 1, 2\}$, with probabilities $p_0$, $p_1$ and $p_2$, and $p_0 + p_1 + p_2 = 1$. Now suppose we repeatedly choose an element randomly from this ...
5
votes
1answer
137 views

Behaviour of $\zeta(s)$ near $s=1$

I would appreciate if somebody could run this over and see if it works out? any suggestions or pointers would be appreciated. I denote the standard eta function $\eta$ by $\zeta^{*}$. I have not used ...
1
vote
1answer
54 views

Group of Dirichlet Characters Modulo $q$ is Isomorphic to $(\mathbb{Z} / q\mathbb{Z})^*$

I'm currently reading a book on analytic number theory, and shortly after defining Dirichlet characters, the author stated that one can prove that for a given $q\in\mathbb{N}$, the group of Dirichlet ...
2
votes
2answers
53 views

What do the zero's of L-functions entail?

I don't know exactly how, but I've read the Riemann Zeta function's nontrivial zero's imply something about an error term for an approximation function thing for the Prime Counting Function. I found ...
34
votes
2answers
3k views

Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. The following are exluded: Books by mathematical ...
1
vote
2answers
126 views

References for Riemann Hypotheis giving the best bound for Prime Number Theorem

Which books cover the proof that Riemann Hypothesis is equivalent to the best error bound for the Prime Number Theorem? My understanding is that Riemann Hypothesis is equivalent to the best bound of ...
2
votes
0answers
34 views

Which mathematical objects generate the zeroes of $L$-functions?

I've studied analytic and algebraic number theory for years and years, and I encountered a hard question about Riemann zeta function and other kinds of $L$-functions - which might be one of the most ...
2
votes
1answer
46 views

Upper bound for $|\zeta'(s)|$ near the line $\sigma=1$, a detailed proof

In page 285 Apostol leaves as a reader's asigment the proof that $|\zeta'(s)|=O(\log^{2}t)$, this is for every $T>0$ there exists a positive constant $K$ (depending on T) such that ...
0
votes
0answers
16 views

The number of monic irreducible

I have some questions about the number of monic irreducible polynomial. Let $a_n$ be the number of monic irreducible polynomial of degree $n$ over finite field of $q$ elements $F_q[X]$. Then we ...
5
votes
0answers
28 views

Equidistribution estimates after Zhang

I was curious what were the best equidistribution estimates toward the Elliott-Halberstam conjecture $$\sum_{q\le Q}\max\left|\Theta(x;q,a)-\frac{x}{\phi(q)}\right|<<\frac{x}{(log x)^A}$$ with ...
3
votes
0answers
35 views

An inequality for $|\zeta (s,a)|$, a detailed proof

In page 272 of [1], Apostol leaves as a reader's assigment to complete a proof of a related statement with Hurwitz zeta function, defined initially for $\sigma >1$ by the series ...
15
votes
2answers
840 views

A Conjecture of Schinzel and Sierpinski

Melvyn Nathanson, in his book Methods in Number Theory (Chapter 8: Prime Number's) states the following: A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can ...
4
votes
2answers
59 views

Divide a square into different parts

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with geometry, which perhaps yields the shortest, simplest proofs, but other ...
4
votes
1answer
78 views

On the series $\sum_{n=1}^{\infty} (H_{n}+\exp(H_{n})\log(H_{n}))/n^{s}$, where $H_{n}$ is the $n$th harmonic number

It is known the following (see [1], here is an open access PDF on his homepage): Theorem (Lagarias, 2002). Let $\sigma(n)$ denote the sum of the positive divisors of $n$. Riemann Hypothesis ...
2
votes
2answers
94 views

$\zeta(2n)$ proof [duplicate]

Can anybody pass me on a good source to see the steps in proving, \begin{equation} \zeta(2n) = \frac{(-1)^{k-1}B_2k (2 \pi)^{2k}}{2(2k)!} \end{equation} I know how we start by looking at the product ...
2
votes
1answer
46 views

Character sum of a type of “almost linear” surjective mappings over finite fields

Let $F$ be a finite field of characteristic a prime $p$ with $q$ elements and let $E/F$ be a finite extension of degree $n > 1$ over $F$. Let $\chi$ be the additive canonical character on $F$ ...
7
votes
1answer
944 views

Why the Riemann hypothesis doesn't imply Goldbach?

I'm interested in number theory, and everyone seems to be saying that "It's all about the Riemann hypothesis (RH)". I started to agree with this, but my question is: Why then doesn't RH imply the ...
0
votes
1answer
48 views

Why is the Bernoulli Number $B_1$ sometimes $+ \frac{1}{2} $?

By using the recursive formula, \begin{equation} \sum_{i=0}^{n} \binom{n+1}{i} B_i = n+1 \end{equation} we find $B_1$ to be $\frac{1}{2}$ and not $- \frac{1}{2}$. Why is this?