All Questions
Tagged with big-list analytic-number-theory
11 questions
2
votes
0
answers
158
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What rational zeta series with non-integer arguments appear in mathematics?
Background
Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \...
- 4,829
2
votes
0
answers
136
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Elementary functions such that $\sum_{n=2}^{\infty} f(n) \left( \zeta(n)-1 \right)$ can be evaluated, but $\sum_{n=2}^{\infty} f(n)$ can't
Background
The general context for this question is the topic of rational zeta series. What I've found so far, is that it usually the case that sums of the form $$\zeta_{f} := \sum_{n=2}^{\infty} f(n) ...
- 4,829
6
votes
4
answers
2k
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Consequences of Goldbach's conjecture
In a letter of 1742 to Euler, Goldbach expressed the belief that ‘Every integer $N>5$ is the sum of three primes’. Euler replied that this is easily seen to be equivalent to the following statement:...
2
votes
0
answers
206
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What problems are easier assuming zeros of a zeta function don’t behave as we expect?
What are some examples of problems which are easier to solve assuming zeros of zeta functions lie off the critical line or do not have expected vertical distribution.
There are some very well known ...
user156885
0
votes
0
answers
116
views
Reference request for bounds of $n$-th composite
Motivation
I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions.
Recently during trying to understand and prove the ...
user57432
54
votes
4
answers
3k
views
When has the Borel-Cantelli heuristic been wrong?
The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...
- 11.4k
7
votes
0
answers
667
views
What will be the consequences if second Hardy-Littlewood conjecture turns out to be true?
It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is,
What would be the consequences if Second Hardy-Littlewood ...
user57432
9
votes
0
answers
3k
views
"Must read "papers on analytic number theory
Question: What would be some must-read
papers for an aspiring analytic number
theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: Someone ...
43
votes
8
answers
21k
views
Approaches to Riemann hypothesis using methods outside number theory [closed]
Background: Once an analytic number theorist remarked to me that all attempts to prove the Riemann hypothesis using number theoretic methods have failed. Since then that remark stuck in my mind.
The ...
67
votes
17
answers
12k
views
Shortest/Most elegant proof for $L(1,\chi)\neq 0$
Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing ...
- 7,127
157
votes
7
answers
74k
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Consequences of the Riemann hypothesis
I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice ...