Let P(x,d,a) denote the number of primes p<=x with p=a(mod d). Chebyshev's bias is the phenomenon that `more often' P(x;d,n)>P(x;d,r) than the other way around, where n is a quadratic non-residue mod d and r is a quadratic residue mod d. If P(x;d,n)>=P(x;d,r) for every x up to some large number, then one expects that N(x;d,n)>=N(x;d,r) for every x. Here N(x;d,a) denotes the number of integers n<=x such that every prime divisor p of n satisfies p=a(mod d).
In this paper we develop some tools to deal with this type of problem and apply them to show that, for example, N(x;4,3)>=N(x;4,1) for every x. In the process we express the so called second order Landau-Ramanujan constant as an infinite series and show that the same type of formula holds true for a much larger class of constants.
In a sequel to this paper the methods developed here will be used and somewhat refined to resolve a conjecture from P. Schmutz Schaller to the extent that the hexagonal lattice is `better' than the square lattice (see p. 201 of Bull. Amer. Math. Soc. 35 (1998), 193-214).
We investigate a relationship between MacMahon's generalized sum-of-divisors functions and Chebyshev polynomials of the first kind. This determines a recurrence relation to compute these functions, as well as proving a conjecture of MacMahon about their general form by relating them to quasi-modular forms. These functions arise as solutions to a curve-counting problem on Abelian surfaces.
We investigate the analogues, in $\mathbb{F}_q[t]$, of highly composite numbers and the maximum order of the divisor function, as studied by Ramanujan. In particular, we determine a family of highly composite polynomials which is not too sparse, and we use it to compute the logarithm of the maximum of the divisor function at every degree up to an error of a constant, which is significantly smaller than in the case of the integers, even assuming the Riemann Hypothesis.
On the Mean Value of a Weighted Composite Arithmetic Function
FIMA Laboratory, Department of Mathematics, Faculty of Matter Sciences and Computer Sciences, University of Khemis Miliana (UDBKM), Algeria（费马实验室，数学系，物质科学与计算机科学学院，赫米斯米利亚纳大学）
What is an interesting number theoretic or a combinatorial characterization of the divisors of 24 amongst all positive integers? In this paper I will provide one characterization in terms of modular multiplication tables. This idea evolved interestingly from a question raised by a student in my elementary number theory class. I will give the characterization and then provide 5 proofs using various techniques: Chinese remainder theorem, structure theory of units, Dirichlet's theorem on primes in an arithmetic progression, Bertrand-Chebyshev theorem, and results of Erdos and Ramanujan on the pi(…
We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $ϕ(n)$ and the sum-of-proper-divisors function $s(n):=σ(n)-n$. We prove that the values $ϕ(n)$, for $n\le x$, that are coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5 \le p \le (\log{x})^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $p\le (\log{x})^A$. These appear to be the first results of their kind where th…
Boğaziçi University（博阿齐希大学）；Institut Élie Cartan de Lorraine & Institut Universitaire de France, Université de Lorraine（洛林伊利·卡当研究所及法国大学学院，洛林大学）；Universität Bielefeld（比勒费尔德大学）；Department of Mathematics, University of Georgia（佐治亚大学数学系）；Mathematics Institute, Utrecht University（乌得勒支大学数学研究所）
Lehmer's totient problem asks if there exist composite integers n satisfying the condition phi(n)|(n-1), (where phi is the Euler-phi function) while Carmichael numbers satisfy the weaker condition lambda(n)|(n-1) (where lambda is the Carmichael universal exponent function). We weaken the condition further, looking at those composite n where each prime divisor of phi(n) also divides n-1. (So rad(phi(n))|(n-1).) While these numbers appear to be far more numerous than the Carmichael numbers, we show that their distribution has the same rough upper bound as that of the Carmichael numbers, a bound …
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 $n$, we show that a composite $n$ is overpseudoprime if and only if $|b|_d$ is invariant for all divisors $d>1$ of $n$. In particular, we prove that all composite Mersenne numbers $2^{p}-1$, where $p$ is prime, are overpseudoprime to base 2 and squares of Wieferich primes are overpseudoprimes to base 2. Finally, we show that some kinds of well known numbers are overpseudoprime to a base $b$.
In this paper, two approximation algorithms are given. Let N be an odd composite number. The algorithms give new directions regarding primality test of given N. The first algorithm is given using a new method called digital coding method. It is conjectured that the algorithm finds a divisor of N in at most O(ln^{4} N), where ln denotes the logarithm with respect to base 2. The algorithm can be applied to find the next largest Mersenne prime number. Some directions are given regarding this. The second algorithm finds a prime divisor of N using the concept of graph pairs and it is proved that th…
We introduce a new class of pseudoprimes-so called "overpseudoprimes" which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime iff value of h(n) is invariant of all divisors d>1 of n. In particular, we prove that all composite Mersenne numbers 2^p-1,where p is prime, and squares of Wieferich primes are overpseudoprimes. We give also a generalization of the results on arbitrary base a>1 and prove that every overpseudoprime is strong pseudoprime of the same base.
A composite positive integer $n$ is said to be a {\it weak Carmichael number} if $$ \sum_{\gcd(k,n)=1\atop 1\le k\le n-1}k^{n-1}\equiv φ(n) \pmod{n}. \leqno(1) $$ It is proved that a composite positive integer $n$ is a weak Carmichael number if and only if $p-1\mid n-1$ for every prime divisor $p$ of $n$. This together with Korselt's criterion yields the fact that every Carmichael number is also a weak Carmichael number. In this paper we mainly investigate arithmetic properties of weak Carmichael numbers. Motivated by the investigations of Carmichael numbers in the last hundred years, here we …
We define a primitive index of an integer in a sequence to be the index of the term with the integer as a primitive divisor. For the sequences $k^u+h^u$ and $k^u-h^u$, we discern a formula to find the primitive indexes of any composite number given the primitive indexes of its prime factors. We show how this formula reduces to a formula relating the multiplicative order of $k$ modulo $N$ to that of its prime factors. We then introduce immediate consequences of the formula: certain sequences which yield the same primitive indexes for numbers with the same unique prime factors, an expansion of t…
For n>1, let G(n)=σ(n)/(n log log n), where σ(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) \ge \max(G(N/p),G(aN)), for all prime factors p of N and all multiples aN of N. The proof uses Robin's and Gronwall's theorems on G(n). An alternate proof of one step depends on two properties of superabundant numbers proved using Alaoglu and Erdős's results.
Let r_j(π,s) denote the number of cells, colored j, in the s-residue diagram of partition π. The GBG-rank of πmod s is defined as r_0+r_1*w_s+r_2*w_s^2+...+r_(s-1)*w_s^(s-1), where w_s=exp(2*Π*I/s). We will prove that for (s,t)=1, v(s,t) <= binomial(s+t,s)/(s+t), where v(s,t) denotes a number of distinct values that GBG-rank mod s of t-core may assume. The above inequality becomes an equality when s is prime or when s is composite and t<=2p_s, where p_s is a smallest prime divisor of s. We will show that the generating functions for 4-cores with the prescribed values of GBG-rank mod 3 are all …
The Prime state of $n$ qubits, $|\mathbb{P}_n\rangle$, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than $2^n$. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of $|\mathbb{P}_n\rangle$ up to $n=30$ qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free i…
Let $p$ be a prime number. A chain $\{p,2p+1,4p+3,\cdots,(p+1)2^{l(p)-1}-1\}$ is called the Cunningham chain generated by $p$ if all elements are prime number and $(p+1)2^{l(p)}-1$ is composite. Then $l(p)$ is called the length of the Cunningham chain. It is conjectured by Bateman and Horn in 1962 that the number of prime $p\leq N$ such that $l(p)\geq k$ is asymptotically equal to $B_k N/(\log N)^k$ with a real $B_k>0$ for all natural number $k$. This suggests that $l(p)=Ω(\log p/\log\log p)$. However, so far no good estimation is known. It has not even been proven whether $\limsup_{p\to\infty…
Gronwall's function $G$ is defined for $n>1$ by $G(n)=\frac{σ(n)}{n \log\log n}$ where $σ(n)$ is the sum of the divisors of $n$. We call an integer $N>1$ a \emph{GA1 number} if $N$ is composite and $G(N) \ge G(N/p)$ for all prime factors $p$ of $N$. We say that $N$ is a \emph{GA2 number} if $G(N) \ge G(aN)$ for all multiples $aN$ of $N$. In arXiv 1110.5078, we used Robin's and Gronwall's theorems on $G$ to prove that the Riemann Hypothesis (RH) is true if and only if 4 is the only number that is both GA1 and GA2. Here, we study GA1 numbers and GA2 numbers separately. We compare them with super…
Prime Certificates for Exact Vertex-Coprime Ramsey Numbers
Tsinghua Shenzhen International Graduate School, Tsinghua University（清华大学深圳国际研究生院）；School of Electrical and Electronic Engineering, The University of Sheffield（谢菲尔德大学电气与电子工程学院）
