All Questions
Tagged with analytic-number-theory reference-request
162
questions
1
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1
answer
31
views
Gauss sum over imaginary quadratic field
Define the Gauss sum
$$ G\left( \frac{a}{m}\right) = \sum_{n(\text{mod}\ m)} e\left( \frac{a n^2}{m}\right) ,$$
then I know the following result:
For any $a,m$ with $(2a,m) = 1$,
$$ G\left( \frac{a}{...
- 63
0
votes
0
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110
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Number theoretic partial differential equation
Consider the equation
$$t\frac{\partial^2}{\partial t^2} \sum_{n=1}^\infty \Phi_n(x,t)=-x\frac{\partial}{\partial x}\sum_{n=1}^\infty \Phi_n(x,t)+\sum_{n=2}^{\infty}\Lambda(n)\Phi_n(x,t)$$
where $\...
- 754
1
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0
answers
54
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Large Sieve Inequality
I am currently delving into the Large Sieve Inequality, consulting Chapter 27 of Davenport's Multiplicative Number Theory. Having completed the chapter, I seek a deeper understanding of the practical ...
- 304
1
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1
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97
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Proof of Dirichlet's Theorem on Primes using $\sum_{\substack{p=1\\ p\equiv h\bmod k}}^\infty\frac{\ln p}{p^s}\sim\frac{1}{\varphi(k)(s-1)}$
In my analytic number theory course, my we just finished proving Dirichlet's Theorem using primes in progression using the method's found in Apostol's Introduction to Analytic Number Theory. My ...
- 3,187
4
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64
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Selberg's Trace Formula for Hecke Eigenvalues
I am looking for a reference (if one exists) for an application of Selberg's trace formula after the Hecke operators have been applied. Perhaps to give some context and notation to this, so my ...
- 2,295
3
votes
1
answer
169
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"p-adic completion" of rational function field
Let $q$ be a prime power, $\mathbb{F}_q$ the finite field of order $q$ and let $\mathbb{F}_q(T)$ be the field of rational functions over $\mathbb{F}_q$.
Similarly to $\mathbb{Q}$, the absolute values ...
- 465
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61
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Are there any results in the literature involving square-free factorizations of the integers?
For positive integers $n\in\mathbb{N}^*$, the radical of $n$ is defined as $\text{rad}(n):=\prod_{p\vert n}p$ where the product extends over the prime divisors of $n$. It can also be thought of as the ...
- 1,810
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72
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Exponential sums reference
Looking for reference on exponential sums, in particular Jacobi, Gauss, Kloosterman and Ramanujan sums. The books mentioned in https://mathoverflow.net/questions/65429/exponential-sums-for-beginner ...
- 107
4
votes
2
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264
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Unit square integral $\int_{0}^{1} \int_{0}^{1} \left( - \frac{\ln(xy)}{1-xy} \right)^{m} dx dy $
In an article by Guillera and Sondow, one of the unit square integral identities that is proved (on p. 9) is: $$\int_{0}^{1} \int_{0}^{1} \frac{\left(-\ln(xy) \right)^{s}}{1-xy} dx dy = \Gamma(s+2) \...
- 7,118
6
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229
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Is there a general relationship between definite integrals over functions involving the complete elliptic integral of the first kind and zeta values?
Background
Let $\textbf{K}(k)$ be the complete elliptic integral of the first kind, where $k$ is its elliptic modulus [1]. Moreover, define $k' := \sqrt{1-k^{2}} $ as its complementary modulus. I've ...
- 7,118
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0
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36
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Where can I learn to develop Effective Versions of Asymptotic Theorems?
There are many 'effective' versions of asymptotic theorems. This paper describes an effective version of a theorem as 'an asymptotic with an explicit error term, which holds for x larger than an ...
- 1,950
1
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0
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38
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Generalization of Distribution of Smooth Integers to Algebraic Number Fields
A $n$-smooth (rational) integer is a positive integer with prime factors all less than or equal to $n$. We can define $\Psi (a,n)$ as the number of $n$-smooth integers less than or equal to $a$. It is ...
- 1,950
1
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0
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78
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On the derivation of the fundamental lemma of the combinatorial sieve
Let $b(d)$ be multiplicative function defined on squarefree numbers such that
$$
r_\mathcal A(d)=|\mathcal A_d|-{b(d)\over d}X
$$
relatively small, and $P(z)$ is the product of primes in $\mathcal P\...
- 7,193
2
votes
1
answer
331
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Sum of Dirichlet's Character Over Divisors of Natural Number
It is said in Explanation for a theorem pertaining on Dirichlet character sums it is well-known that $A\left(n\right)=\sum_{d\mid n}\chi\left(d\right)$ is non-negative for $\chi$ is character modulo $...
- 915
4
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1
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90
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A reference for $\displaystyle \sum_{n\leq X}\mu^2(n)= \frac{X}{\zeta(2)}+O(\sqrt{X}\exp(-c\sqrt{\log X}))$
There is this result of counting the number of square-free numbers until $X$, which goes like this:
$$\sum_{n\leq X}\mu^2(n)= \frac{X}{\zeta(2)}+O(\sqrt{X}\exp(-c\sqrt{\log X})).$$
Could someone ...
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