Multiplicative Functions and a Taxonomy of Dirichlet Characters

In this paper, we develop a large sieve type inequality for some special characters whose moduli are squares of primes. Our result gives non-trivial estimate in certain ranges.

We present a simple proof and a generalization of a Menon-type identity by Li, Hu and Kim, involving Dirichlet characters and additive characters.

We improve a recent result by Shparlinski and Banks related to sums with convolution of Dirichlet characters.

In our paper, we introduce a new method for estimating incidences via representation theory. We obtain several applications to various sums with multiplicative characters and to Zaremba's conjecture from number theory.

Let $M(\chi)$ denote the maximum of $|\sum_{n\le N}\chi(n)|$ for a given non-principal Dirichlet character $\chi \pmod q$, and let $N_\chi$ denote a point at which the maximum is attained. In this article we study the distribution of $M(\chi)/\sqrt{q}$ as one varies over characters $\pmod q$, where $q$ is prime, and investigate the location of $N_\chi$. We show that the distribution of $M(\chi)/\sqrt{q}$ converges weakly to a universal distribution $\Phi$, uniformly throughou...

We describe a plan how to prove an effective Siegel theorem (about the exceptional Dirichlet character). We give a brief outline in Section 0. We give a more detailed plan in Sections 1-5. The missing details (mostly routine elementary estimations) are in Part Two, which is very long. I am happy to send the pdf-file of Part Two to anybody who requests it by email.

In this paper, we develop a large sieve type inequality with characters to square moduli. One expects that the result should be weaker than the classical inequality, but, conjecturally at least, not by much. The method is generalizable to higher power moduli.

The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading factors of the infinite product over zeta-functions. If rooted at the Dirichlet series for powers, for sums-of-divisors and for Euler's totient, the inheritance of multiplicativity through Dirichlet convolution or ordinary multiplication of pairs...

In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Polya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Polya-Vinogradov inequality may be sharpened assuming the GRH. We give a simple proof of their estimate, and provide an improvement for characters of od...

Consider a multiplicative function f(n) taking values on the unit circle. Is it possible that the partial sums of this function are bounded? We show that if we weaken the notion of multiplicativity so that f(pn)=f(p)f(n) for all primes p in some finite set P, then the answer is yes. We also discuss a result of Bronstein that shows that functions modified from characters at a finite number of places.