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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Shifted Dirichlet series
If $\sum_{n=1}^\infty \frac{a_n}{n^s} $ converges, does
$\sum_{n=1}^\infty \frac{a_n}{(n+1)^s} $ also converge?
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Asymptotically multiplicative functions and matrices
Hi,
Let $\mathbb{N}_{cop}^2$ denote the set of all pairs of coprime natural numbers. A function $f:\mathbb{C}\rightarrow\mathbb{C}$ is called asymptotically multiplicative, iff $\epsilon_{m,n}:=f(mn)...
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Prime numbers $p$ not of the form $ab + bc + ac$ $(0 < a < b < c )$ (and related questions)
If we ask which natural numbers n are not expressible as $n = ab + bc + ca$ ($0 < a < b < c$) then this is a well known open problem. Numbers not expressible in such form are called Euler'...
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Does there exist a meromorphic function all of whose Taylor coefficients are prime?
More precisely, does there exist an unbounded sequence $a_0, a_1, ... \in \mathbb{N}$ of primes such that the function
$\displaystyle O(z) = \sum_{n \ge 0} a_n z^n$
is meromorphic on $\mathbb{C}$?
...
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Change rates of the 2nd Chebyshev function
Let $\psi(x):=\sum_{n\leq x}\Lambda(n)$ denotes the 2nd Chebyshev function, where $\Lambda$ stands for the von Mangoldt function. Are there any known (and 'nice') estimates for the change rates $\psi(...
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Is the maximum domain to which a Dirichlet series can be continued always a halfplane?
Let $f(s)=\sum_n a_n n^{-s}$ be a Dirichlet series whose coefficients satisfy $\lvert a_n\rvert\leq n^{C}$. Then $f(s)$ converges absolutely in some halfplanes, and is conditionally convergent in (...
- 7,836
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Very strong multiplicity one for Hecke eigenforms
In Invent. math. 116, 645-649 (1994) Dinakar Ramakrishnan proves a theorem which I understand to imply that the following statement (in light of the fact that elliptic curves over $\mathbb{Q}$ are ...
- 7,062
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Is the set of primes "translation-finite"?
The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be ...
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Equivalent forms of the Grand Riemann Hypothesis
I have long been curious about equivalent forms of the Riemann hypothesis for automorphic L-functions.
In the case of the ordinary Riemann hypothesis, one gets a very good error term for the prime ...
- 7,062
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Strong Bertrand postulate
Is it known that for every epsilon there is N_0 such that all intervals of the form [N, (1+\epsilon)*N], where N > N_0, contain ...
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Bertrand's postulate [closed]
I believe there was an old conjecture that there's always a prime number between N and 2N.
What's the history and how is this ...
- 15.1k
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Why do zeta functions contain so much information?
Is there some intuitive explanation why Dedekind zeta functions contain so much information about their number field?
For example the residue at the pole $s=1$ relates several invariants of the ...
- 261
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Primes are pseudorandom?
I've been reading the wonderful slides by Terry Tao and thought about this question.
Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes ...
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The large sieve for primes
Let $\Lambda(n)$ be the von Mangoldt function, i.e., $\Lambda(n) = \log p$ for $n$ a prime power $p^k$ and $\Lambda(n) = 0$ for all $n$ that not prime powers. Let
$$S(\alpha) = \sum_{n \leq N} \...
- 20.2k
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Arithmetic progressions without small primes
The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? :
Is it known that there are infinitely many primes p for which ...
- 156k
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Zeta-function regularization of determinants and traces
The short answer to my question may be a pointer to the right text. I will give all the background I know, and then ask my questions in list form.
Let A be an operator (on an infinite-dimensional ...
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