Arithmetic function

In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n. There is a larger class of number-theoretic functions that do not fit the above definition, for example, the prime-counting functions. This article provides links to functions of both classes. Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum. Multiplicative and additive functions An arithmetic function a is

completely additive if a(mn) = a(m) + a(n) for all natural numbers m and n;
completely multiplicative if a(mn) = a

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The aim of this course is to present the basic techniques of analytic number theory.

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

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Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions

Philippe Michel

We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums.

European Mathematical Soc2014

TFHE: Fast Fully Homomorphic Encryption Over the Torus

This work describes a fast fully homomorphic encryption scheme over the torus (TFHE) that revisits, generalizes and improves the fully homomorphic encryption (FHE) based on GSW and its ring variants. The simplest FHE schemes consist in bootstrapped binary gates. In this gate bootstrapping mode, we show that the scheme FHEW of Ducas and Micciancio (Eurocrypt, 2015) can be expressed only in terms of external product between a GSW and an LWE ciphertext. As a consequence of this result and of other optimizations, we decrease the running time of their bootstrapping from 690 to 13 ms single core, using 16 MB bootstrapping key instead of 1 GB, and preserving the security parameter. In leveled homomorphic mode, we propose two methods to manipulate packed data, in order to decrease the ciphertext expansion and to optimize the evaluation of lookup tables and arbitrary functions in RingGSW-based homomorphic schemes. We also extend the automata logic, introduced in Gama et al. (Eurocrypt, 2016), to the efficient leveled evaluation of weighted automata, and present a new homomorphic counter called TBSR, that supports all the elementary operations that occur in a multiplication. These improvements speed up the evaluation of most arithmetic functions in a packed leveled mode, with a noise overhead that remains additive. We finally present a new circuit bootstrapping that converts LWE ciphertexts into low-noise RingGSW ciphertexts in just 137 ms, which makes the leveled mode of TFHE composable and which is fast enough to speed up arithmetic functions, compared to the gate bootstrapping approach. Finally, we provide an alternative practical analysis of LWE based schemes, which directly relates the security parameter to the error rate of LWE and the entropy of the LWE secret key, and we propose concrete parameter sets and timing comparison for all our constructions.

SPRINGER2020

Computation over Gaussian networks with orthogonal components

Michael Christoph Gastpar, Chien-Yi Wang

Function computation of arbitrarily correlated discrete sources over Gaussian networks with multiple access components but no broadcast is studied. Two classes of functions are considered: the arithmetic sum function and the frequency histogram function. The arithmetic sum function in this paper is defined as a set of multiple weighted arithmetic sums, which includes averaging of sources and estimating each of the sources as special cases. The frequency histogram function counts the number of occurrences of each argument, which yields many important statistics such as mean, variance, maximum, minimum, median, and so on. For a class of networks, an approximate computation capacity is characterized. The proposed approach first abstracts Gaussian networks into the corresponding modulosum multiple-access channels via lattice codes and linear network coding and then computes the desired function by using linear Slepian–Wolf source coding.

2013

Dirichlet series

In mathematics, a Dirichlet series is any series of the form \sum_{n=1}^\infty \frac{a_n}{n^s}, where s is complex, and a_n is a complex sequence. It is a sp

Riemann hypothesis

In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Man

Möbius function

The Möbius function μ(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. It is ubiquitous in elemen

The aim of this course is to present the basic techniques of analytic number theory.

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.