Questions tagged [analytic-number-theory]
On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.
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Rate of convergence of the Riemann zeta function and the Euler product formula
We consider the Euler product formula $$\sum_{n=1}^\infty \frac{1}{n^s}=\prod_p \frac{1}{1-p^{-s}}$$
I have two questions about this equality:
1)Does the rate of convergence of each side ...
- 472
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Possible implications of the bound $\sum_{n\leq x}\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)=O\left(x\right)$
Let $\lambda(n)$ be the Liouville function and consider the Goldbach-type problem $\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)$. Assume the Riemann hypothesis, the ...
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Dirichlet polynomial approximation of $\zeta(\sigma+it)$ for $\sigma = \sigma_t \sim 0$
It is known by Theorem 4.11 of Titchmarsh that if $\sigma \geq \sigma_0 >0$ and $t \leq 2\pi x/C$ where $C>1$ is a constant, then
$$\zeta(\sigma+it)=\sum_{n \leq x} n^{-\sigma-it} - \frac{x^{1-\...
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General argument strategy for counting functions in AP's
For concreteness I'll refer to the divisor function but just pick your favourite multiplicative $f:\mathbb N\rightarrow \mathbb C$.
If I want to estimate $$\sum _{n\leq x}d(n)$$ then I can write it as ...
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Can exist a positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ are not prime for all $n \ge 1$?
Using my computer, I found that the most of positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ is prime number after a few iterations. But exist some positive integer numbers, my ...
- 1,454
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Distribution of traces and max entries of words of fixed length in $\operatorname{SL}_2(\mathbb{N})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\tr{\mathsf{tr}}$$\SL_2(\mathbb{N})$ is a free monoid on the generators
$$
L=\begin{pmatrix}1&0\\1&1\end{pmatrix},\quad R=\begin{pmatrix}1&1\...
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Formula for $\pi$ involving exponents of Mersenne primes
Can someone provide a proof for the following claim?
$$\pi=\dfrac{4S_2}{M_3M_5} \cdot\left(\displaystyle\prod_{p \equiv 1 \pmod{4} } \frac{p}{p-1}\right) \cdot \left(\displaystyle\prod_{p \equiv 3 \...
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Systematic way to compute $\sum_{n=1}^\infty P(n) / Q(n)$ for polynomials $P$ and $Q$
This may be well known so feel free to downvote.
When $P = 1$ and $Q(x) = x^k$, this is of course the Riemann zeta function. But what about other cases?
For instance is it always possible to express $\...
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Division based recurrence with negative coefficients, e.g. $F(n)= -F(\lfloor n/2\rfloor) - F(\lfloor n/3\rfloor)$
A famous problem of Erdos dealt with the division-based recurrence $a_n = a_{\lfloor n/2\rfloor}+a_{\lfloor n/3\rfloor}+a_{\lfloor n/6\rfloor}$ with $a_0=1$ (and was about the limit $\lim_{n\to\infty} ...
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Can we rule out the possibility that $\sqrt[3]{2}$ is small modulo every prime?
Consider a prime $p$ such that the polynomial $X^3-2$ splits into linear factors over $\mathbb{F}_p$: $X^3-2 = (X-\alpha_p)(X-\beta_p)(X-\gamma_p)$. It seems reasonable to expect that (identifying $\...
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Subconvexity bounds for Dedekind zeta functions of cyclotomic fields
I am looking for hybrid subconvexity bounds on $|\zeta_K(1/2+it)|$ where $K=\mathbb{Q}(\mu_n)$ is the cyclotomic field with $n$th roots of unity. Basically, I am looking for
$$|\zeta_K(1/2+it)| \ll |\...
- 885
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Other kind of supercongruences for rational Ramanujan-like series
We can write rational Ramanujan-like series with rational parameters in the following form:
$$
\sum_{n=0}^{\infty} \left( \prod_{i=0}^{2m} \frac{(s_i)_{n}}{(1)_{n}} \right) z_0^{n} \sum_{k=0}^m a_k n^...
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Record for determining complete list of imaginary quadratic fields with small class number
In 2003, Mark Watkins (Class numbers of imaginary quadratic fields) determined all imaginary quadratic fields having class number at most 100.
Has this list been improved? That is, what is the largest ...
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The exponential sum over primes on average
In https://academic.oup.com/blms/article-abstract/20/2/121/266256?redirectedFrom=fulltext Vaughan shows the following bounds for the $L^1$-mean of the exponential sum over primes $$\sqrt x\ll \int _0^...
- 1,256
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The location of the zeros of the "new" function $\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$
Defining the Psi function as $$\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$$ and by studying, from a numerical point of view, the location of the zeros in the complex plane up to $0<\Im(s)<...
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