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Locally harmonic Maass forms and rational period functions
hyperbolic Poincare series harmonic weak Maass forms cusp forms lifting maps Shimura lift Shintani lift
2012/6/21
In this paper we define a new type of modular object and construct explicit examples of such functions. Our functions are closely related to cusp forms constructed by Zagier which played an important ...
The normality of digits in almost constant additive functions
Normal numbers additive function Selberg-Delange method
2012/6/21
We consider numbers formed by concatenating some of the base b digits from additive functions f(n) that closely resemble the prime counting function \Omega(n). If we concatenate the last
Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers
Mersenne numbers cyclotomic cosets of 2 modulo n Poulet pseudoprime super-Poulet pseudoprime overpseudoprime
2012/6/19
We introduce a new class of pseudoprimes-so called "overpseudoprimes to base $b$", which is a subclass of strong pseudoprimes to base $b$. Denoting via $|b|_n$ the multiplicative order of $b$ modulo $...
The autoconjugacy of a generalized Collatz map
The autoconjugacy of a generalized Collatz map Number Theory
2012/6/19
Many of the 2-adic properties of the 3x+1 map generalize to the analogous mx+r map, where m and r are odd integers. We introduce the corresponding autoconjugacy map, prove some simple properties of it...
On Character Sums and Exponential Sums over Generalized Arithmetic Progressions
Character Sums Exponential Sums Generalized Arithmetic Progressions Number Theory
2012/6/19
We study upper bounds for sums of Dirichlet characters. We prove a uniform upper bound of the character sum over all proper generalized arithmetic progressions, which generalizes the classical Polya a...
Complete Residue Systems: A Primer and an Application
Complete Residue Systems A Primer and an Application Number Theory
2012/6/19
Complete residue systems play an integral role in abstract algebra and number theory, and a description is typically found in any number theory textbook. This note provides a concise overview of compl...
Let p, c be distinct odd primes, and l \ge 2 an integer. We find sufficient conditions for the Diophantine equation cy^l=(x^p-1)/(x-1) not to have integer solutions.
New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series
Congruence finite central binomial sum multiple harmonic sum Bernoulli number
2012/6/15
In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences $(\{1\}^a,c,\{1\}^b),$ $(\{2\}^a,c,\{2\}^b)$ and prove a number of congru...
On an incomplete argument of Erdos on the irrationality of Lambert series
Lambert series divisor function q-logarithm Number Theory
2012/6/15
We show that the Lambert series $f(x)=\sum d(n) x^n$ is irrational at $x=1/b$ for negative integers $b < -1$ using an elementary proof that finishes an incomplete proof of Erdos.
The Asymptotic Behavior of Compositions of the Euler and Carmichael Functions
The Asymptotic Behavior Compositions the Euler Carmichael Functions Number Theory
2012/6/14
We compare the asymptotic behavior of Carmichael's lambda function composed with Euler's totient function to the asymptotic behavior of Carmichael's lambda function composed with itself. We establish ...
A Cesaro Average of Hardy-Littlewood numbers
Goldbach-type theorems Hardy-Littlewood numbers Laplace transforms Cesaro averages
2012/6/14
Let $\Lambda$ be the von Mangoldt function and (r_{\textit{HL}}(n) = \sum_{m_1 + m_2^2 = n} \Lambda(m_1),) be the counting function for the Hardy-Littlewood numbers. Let $N$ be a sufficiently large in...
A Diophantine problem with prime variables
Diophantine problems with prime variables Number Theory
2012/6/14
We study the distribution of the values of the form $\lambda_1 p_1 + \lambda_2 p_2 + \lambda_3 p_3^k$, where $\lambda_1$, $\lambda_2$ and $\lambda_3$ are non-zero real number not all of the same sign,...
A Cesaro Average of Goldbach numbers
Goldbach-type theorems Laplace transforms Cesaro averages
2012/6/14
Let $\Lambda$ be the von Mangoldt function and (r_G(n) = \sum_{m_1 + m_2 = n} \Lambda(m_1) \Lambda(m_2)) be the counting function for the Goldbach numbers. Let $N \geq 2$ be an integer. We prove that ...
L^2-norms of exponential sums over prime powers
Primes in short intervals Diophantine problems with primes Number Theory
2012/6/14
We study a suitable mean-square average of primes in short intervals, generalizing Saffari-Vaughan's result. We then apply it to a ternary Diophantine problem with prime variables.
A Diophantine problem with a prime and three squares of primes
Goldbach-type theorems Hardy-Littlewood method diophantine inequalities
2012/6/14
We prove that if $\lambda_1$, $\lambda_2$, $\lambda_3$ and $\lambda_4$ are non-zero real numbers, not all of the same sign, $\lambda_1 / \lambda_2$ is irrational, and $\varpi$ is any real number then,...