Daksh Aggarwal, Unique Subedi, William Verreault, Asif Zaman, Chenghui Zheng
We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers.
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We consider a family of character sums as multiplicative analogues of Kloosterman sums. Using Gauss sums, Jacobi sums and Deligne's bound for hyper-Kloosterman sums, we establish asymptotic formulae for any real (positive) moments of the above character sum as the character runs over all non-trivial multiplicative characters mod $p.$ Moreover, an arcsine law is also established as a consequence of the method of moments. The evaluations of these moments also allow us to obtain asymptotic formulae for moments of such character sums weighted by special $L$-values (at $1/2$ and $1$).
We show that there is significant cancellation in certain exponential sums over small multiplicative subgroups of finite fields, giving an exposition of the arguments by Bourgain and Chang.
We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is "pretentious". This leads to applications in the circle method, and a natural interpretation of the local-global principle.
Large quadratic character sums with multiplicative coefficients
本文在广义黎曼假设下，研究了具有乘法系数的二次狄利克雷特征和的条件大值，并证明了若干Omega结果。
Weyl sums with multiplicative coefficients and joint equidistribution
该研究将Montgomery与Vaughan的带乘性系数指数和结果推广至Weyl和，得到多项式同余根与值的联合等分布结论及混合特征和新结果。
We give a new bound on colinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.
We study sums with multiplicative functions that take values over a non-homogenous Beatty sequence. We then apply our result in a few special cases to obtain asymptotic formulas such as the number of integers in a Beatty sequence representable as a sum of two squares up to a given magnitude.
We investigate the solubility of the congruence xy=1 (mod p), where p is a prime and x,y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.
Multiplicative dependence in the sumset of multiplicative groups
该研究针对有限生成且交集为有限群的代数数乘法群Γ、Δ，探讨其和集元素的乘法相关性，证明二元和的相关结论，并在abc猜想下对三元相关和分类。
In this paper, we study the average size of the product of a number and its multiplicative inverse modulo a prime p. This turns out to be related to moments of L-functions and leads to a curious asymptotic formula for a certain triple exponential sum.
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.
Character sums of Division polynomials twisted by multiplicative functions
研究除法多项式与乘法函数结合的特征和估计，提出非平凡界。
Large values of character sums with multiplicative coefficients
本文研究了带乘性系数的Dirichlet字符和的大值，并在特定区域证明了一个Omega结果。
Let $p$ be an odd prime. Using I. M. Vinogradov's bilinear estimate, we present an elementary approach to estimate nontrivially the character sum $$ \sum_{x\in H}χ(x+a),\qquad a\in\Bbb F_p^*, $$ where $H<\Bbb F_p^*$ is a multiplicative subgroup in finite prime field $\Bbb F_p$. Some interesting mean-value estimates are also provided.
Quadratic character sums with multiplicative coefficients
本文针对带乘法系数的二次特征和$\sum_{n \le N}f(n)χ_d(n)$的极值问题，通过共振方法对适当范围的正数$N$建立了条件$Ω$-结果。
The series of some new estimates for the sums of the type \[ S_{q}(x;f)\,=\,\mathop{{\sum}'}\limits_{n\leqslant x}f(n)e_{q}(an^{*}+bn) \] is obtained. Here $q$ is a sufficiently large integer, $\sqrt{q}(\log{q})\!\ll\!x\leqslant q$, $a,b$ are integers, $(a,q)=1$, $e_{q}(v) = e^{2πiv/q}$, $f(n)$ is a multiplicative function, $nn^{*}\equiv 1 \pmod{q}$ and the prime sign means that $(n,q)=1$.
We present a class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by $1$. The key feature is that they pretend to be the constant function $1$ and that for some prime $q$, $\sum_{k=0}^\infty \frac{f(q^k)}{q^k}=0$. These combined with other conditions guarantee that these functions are periodic and have sum equal to zero inside each period. Further, we study the class of multiplicative functions $f=f_1\ast f_2$, where each $f_j$ is multiplicative and periodic with bounded partial sums.…
Let $k$ be an integer greater than or equal $4$. We show that if a multiplicative function $f$ satisfies \[ f(x_1^2 + x_2^2 + \dots + x_k^2) = f(x_1)^2 + f(x_2)^2 + \dots + f(x_k)^2 \] for all positive integers $x_i$'s, then $f$ is the identity function.
Large character sums with multiplicative coefficients
本文研究带乘法系数的狄利克雷特征和的大值，在特定区间内证明了新的 Omega 结果。
Let $f$ be a multiplicative function which satisfies \[ f(a^2+b^2+c^2+d^2) = f(a^2+b^2)+f(c^2+d^2) \] for positive integers $a$, $b$, $c$, and $d$. We show that $f$ is the identity function provided that $f(3)\,f(11) \ne 0$. Otherwise, $f(n)=0$ for all $n \ge 2$ except for $n=3,9,11$.
We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent $1/k$: There are examples such that the partial sums up to $x$ are $o(x^{1/(k+\frac{1}{2})+ε})$, for all $ε>0$. This generalizes to the $k$-free integers the results of Aymone, `` A note on multiplicative functions resembling the {M}öbius function'', J. Number Theory, 212 (2020), pp. 113--121.
The Landau-Selberg-Delange (LSD) method gives an asymptotic formula for the partial sums of a multiplicative function $f$ whose prime values are $α$ on average. In the literature, the average is usually taken to be $α$ with a very strong error term, leading to an asymptotic formula for the partial sums with a very strong error term. In practice, the average at the prime values may only be known with a fairly weak error term, and so we explore here how good an estimate this will imply for the partial sums of $f$, developing new techniques to do so.
We obtain several asymptotic formulas for the sum of the divisor function $τ(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced residues modulo $q$ and on average over arbitrary sets. The main goal is to obtain nontrivial results for $q \ge x^{2/3}$ with the small amount of averaging over $a$. We recall that for individual values of $a$ the limit of our current methods is $q \le x^{2/3-\varepsilon}$ for an arbitrary fixed $\varepsilon> 0$. Our method builds on an approach due to Blomer (2008) …
We deduce asymptotic formulas for the alternating sums $\sum_{n\le x} (-1)^{n-1} f(n)$ and $\sum_{n\le x} (-1)^{n-1} \frac1{f(n)}$, where $f$ is one of the following classical multiplicative arithmetic functions: Euler's totient function, the Dedekind function, the sum-of-divisors function, the divisor function, the gcd-sum function. We also consider analogs of these functions, which are associated to unitary and exponential divisors, and other special functions. Some of our results improve the error terms obtained by Bordellès and Cloitre. We formulate certain open problems.
On the multiplicative pair correlations of sums of two squares
本文研究了两个平方和的乘积对称性，揭示了在不同缩放下从点测度到泊松行为的过渡过程。
A method of estimating sums of multiplicative functions braided with Dirichlet characters is demonstrated, leading to a taxonomy of the characters for which such sums are large.
Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) = \exp(2 πi z/p)$, which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product $nx$ with $x \in {\mathcal H}$ and $n \in {\mathcal N}$ for a short interval ${\mathcal N}$ of consecutive integers.
There are two main interrelated goals of this paper. Firstly we investigate the sums \[ S_N(α,γ):=\sum_{n=1}^N\frac{1}{n\|nα-γ\|}~~~\text{and}~~~ R_N(α,γ):=\sum_{n=1}^N\frac{1}{\|nα-γ\|}\,, \] where $α$ and $γ$ are real parameters and $\|\cdot\|$ is the distance to the nearest integer. Our theorems improve upon previous results of W. M. Schmidt and others, and are (up to constants) best possible. Related to the above sums, we also obtain upper and lower bounds for the cardinality of \[ \{1\le n\le N:\|nα-γ\|<\varepsilon\} \, , \] valid for all sufficiently large $N$ and all sufficiently small …
Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a positive integer and $a$ be an integer with $(a,\,q)=1$. In this paper, we shall prove that $$\sum_{\substack{n\leq N\\ (n,\,q)=1}}f(n)e({a\bar{n}\over q})\ll\sqrt{τ(q)\over q}N\log\log(6N)+q^{{1\over 4}+{ε\over 2}}N^{1\over 2}(\log(6N))^{1\over 2}+{N\over \sqrt{\log\log(6N)}},$$ where $\bar{n}$ is the multiplicative inverse of $n$ such that $\bar{n}n\equiv 1\,({\rm mod}\,q),\,e(x)=\exp(2πix),\,τ(q)$ is the divisor function.
Short Exponential Sums and Ternary Correlations of Multiplicative Functions
研究一般k-除数有界的乘法函数的三元相关性平均行为，结合圆法与加权短指数和界，利用L-函数积分矩界获得关键输入。
We obtain a new upper bound for binary sums with multiplicative characters over variables belong to some sets, having small additive doubling.
As early as the 1930s, Pál Erdős conjectured that: {\em for any multiplicative function $f:\mathbb{N}\to\{-1,1\}$, the partial sums $\sum_{n\leq x}f(n)$ are unbounded.} Considering this conjecture, in this paper we consider multiplicative functions $f$ satisfying $$\sum_{p\leq x}f(p)=c\cdot\frac{x}{\log x}(1+o(1)).$$ We prove that if $c>0$ then the partial sums of $f$ are unbounded, and if $c<0$ then the partial sums of $μf$ are unbounded. Extensions of this result are also discussed.
We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet character. As an application we improve current bounds on odd order character sums. Furthermore, conditionally on the Generalized Riemann Hypothesis we obtain a bound for odd order character sums which is best possible.
Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, χ$ be a non-principal Dirichlet character modulo $q$. Let $\varepsilon$ be a sufficiently small positive constant, $A$ be a large constant, $q^{\frac12+\varepsilon}\ll N\ll q^A$. In this paper, we shall prove that $$ \sum_{n\leq N}f(n)χ(n+a)\ll N\frac{\log\log q}{\log q} $$ and that $$ \sum_{n\leq N}f(n)χ(n+a_1)\cdotsχ(n+a_t)\ll N\frac{\log\log q}{\log q}, $$ where $t\geq 2, a_1, \ldots, a_t$ are distinct integers modulo $q$.
We provide a simple proof that the partial sums $\sum_{n\leq x}f(n)$ of a Rademacher random multiplicative function $f$ change sign infinitely often as $x\to\infty$, almost surely.
In the paper we obtain new estimates for binary and ternary sums of multiplicative characters with additive convolutions of characteristic functions of sets, having small additive doubling. In particular, we improve a result of M.-C. Chang. The proof uses Croot-Sisask almost periodicity lemma.
Logarithmic exponential sums with multiplicative coefficients
该论文针对1-有界乘法函数的对数指数和，证明了含最优常数的上界估计，解决了Granville与Lamzouri的两个猜想，提出双素数Ramaré分解恒等式作为核心新方法。
On multiplicative functions with small partial sums
该研究聚焦解析数论中积性函数素数取值与平均值的反向关联，证明了除数有界且部分和小的积性函数，其素数平均取值可由对应Dirichlet级数在ℜ(s)=1上的零点虚部表示，推广了已有相关结果。
We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which previously known methods do not apply. In particular, in the multiplicative characters modulo a prime $p$ we break the barrier of $p^{1/4}$ for ranges of individual variables.
The Landau-Selberg-Delange method provides an asymptotic formula for the partial sums of a multiplicative function whose average value on primes is a fixed complex number $v$. The shape of this asymptotic implies that $f$ can get very small on average only if $v=0,-1,-2,\dots$. Moreover, if $v<0$, then the Dirichlet series associated to $f$ must have a zero of multiplicity $-v$ at $s=1$. In this paper, we prove a converse result that shows that if $f$ is a multiplicative function that is bounded by a suitable divisor function, and $f$ has very small partial sums, then there must be finitely ma…
A folklore conjecture on multiplicative exponential sums
本文证明了带乘法系数的指数和的一个民间猜想，还证明了两个相关猜想界，并指出特定条件下该界可能不成立。
We prove a stratification result for certain families of $n$-dimensional (complete algebraic) multiplicative character sums. The character sums we consider are sums of products of $r$ multiplicative characters evaluated at rational functions, and the families (with $nr$ parameters) are obtained by allowing each of the $r$ rational functions to be replaced by an "offset", i.e. a translate, of itself. For very general such families, we show that the stratum of the parameter space on which the character sum has maximum weight $n+j$ has codimension at least $j\lfloor(r-1)/2(n-1)\rfloor$ for $1\le …
Bilinear Kloosterman sums in function fields and the distribution of irreducible polynomials
本文通过改进双线性Kloosterman和的界，拓展了不可约多项式分布的水平，并给出了超越1/2的Bombieri-Vinogradov定理的类比。
Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a prime number and $a$ be an integer with $(a,\,q)=1$, $χ$ be a non-principal Dirichlet character modulo $q$. In this paper, we shall prove that $$ \sum_{n\leq N}f(n)χ(n+a)\ll {N\over q^{1\over 4}}\log\log(6N)+q^{1\over 4}N^{1\over 2}\log(6N)+{N\over \sqrt{\log\log(6N)}}. $$ We shall also prove that \begin{align*} &\sum_{n\leq N}f(n)χ(n+a_1)\cdotsχ(n+a_t)\ll {N\over q^{1\over 4}}\log\log(6N)\\ &\quad+q^{1\over 4}N^{1\over 2}\log(6N)+{N\over \sqrt{\log\log(6N)}}, \end{align*} where $t\geq 2$, $a_1,\,\cdots,\,…
This article is an expanded version of the talk given by the first author at the conference "Exponential sums over finite fields and applications" (ETH, Zürich, November, 2010). We state some conjectures on archimedian and $p$-adic estimates for multiplicative character sums over smooth projective varieties. We also review some of the results of J. Dollarhide, which formed the basis for these conjectures. Applying his results, we prove one of the conjectures when the smooth projective variety is ${\mathbb P}^n$ itself.
We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random walks on a finite number of sites. Relations among multiplicative nested sums, which are generalizations of relations between harmonic series and multiple zeta functions, can be easily derived from identities of the index matrices. Combinatorial identities and their generalizations can also be derived from this computation.
Divisibility of the Multiplicative Order Modulo Monic Irreducible Polynomials Over Finite Fields
本文研究有限域上使给定有理函数模首一不可约多项式的乘法阶被固定整数整除的多项式集合，刻画其在三种密度概念下的密度存在性并给出公式。
We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit new phenomena concerning Chebyshev-type biases of such sums when the number of irreducible factors is very large.
On exponential sums with multiplicative coefficients
该研究针对带乘法系数的指数和，在特定有理近似条件下证明了相关上界估计，完全乘法性可优化常数，证实了民间猜想，还得到对数加权估计并给出反例。
In the paper we obtain some new upper bounds for exponential sums over multiplicative subgroups G of F^*_p having sizes in the range [p^{c_1}, p^{c_2}], where c_1,c_2 are some absolute constants close to 1/2. As an application we prove that in symmetric case G is always an additive basis of order five, provided by |G| > p^{1/2} log^{1/3} p. Also the method allows us to give a new upper bound for Heilbronn's exponential sum.
We study sums of a random multiplicative function; this is an example, of number-theoretic interest, of sums of products of independent random variables (chaoses). Using martingale methods, we establish a normal approximation for the sum over those n \leq x with k distinct prime factors, provided that k = o(log log x) as x \rightarrow \infty. We estimate the fourth moments of these sums, and use a conditioning argument to show that if k is of the order of magnitude of log log x then the analogous normal limit theorem does not hold. The methods extend to treat the sum over those n \leq x with a…
We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from $\mathbb{F}_q$-linear formal power series. Since the logarithmic derivatives connect power sums to elementary symmetric functions via the Newton identities, we establish, as applications, new identities between important quantities of function field arithmetic, such as the Bernoulli-Carlitz fractions and power sums as well as their multi-variable generalizations. Resulting understanding of their factorizations has arithmetic significance, as w…
We prove an estimate for multi-variable multiplicative character sums over affine subspaces of $\mathbb A^n_k$, which generalize the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character sums.
We improve an existing result on exponential quadrilinear sums in the case of sums over multiplicative subgroups of a finite field and use it to give a new bound on exponential sums with quadrinomials.
Low moments of automorphic random multiplicative function sums
本文确定了满足素平方消去条件的自守随机乘法函数部分和低阶矩的阶，推广了Harper的结果，适用于酉型和辛型不可约紧群表示。
Bounded Exponential Sums with Multiplicative Coefficients
研究具有乘法系数的指数和有界条件，证明其仅在函数接近扭曲狄利克雷字符时成立，并对完全乘法函数和无理数α进行了完整分类。
Almost sure upper bound for sums of random multiplicative functions and critical chaos
该研究运用临界混沌理论方法，改进了Steinhaus与Rademacher型随机乘性函数部分和的几乎必然上界，以强形式证明了Harper的相关猜想并确定了精确对数指数。
In this work we establish a Burgess bound for short multiplicative character sums in arbitrary dimensions, in which the character is evaluated at a homogeneous form that belongs to a very general class of "admissible" forms. This $n$-dimensional Burgess bound is nontrivial for sums over boxes of sidelength at least $q^β$, with $β> 1/2 - 1/(2(n+1))$. This is the first Burgess bound that applies in all dimensions to generic forms of arbitrary degree. Our approach capitalizes on a recent stratification result for complete multiplicative character sums evaluated at rational functions, due to the s…
Restricted sumsets in multiplicative subgroups
本文研究了有限域上乘法子群及整数完美幂中的受限和集，扩展了Shkredov的结果，并提出了一个受限和集的van Lint-MacWilliams猜想的类比。
Dynamics of multiplicative groups over fields and Folner-Kloosterman sums
研究无限域上乘法群的动力学及Følner-Kloosterman求和的等分布性，揭示其乘法不对称性及组合结论。
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