Related papers: Cotangent power sums and character coordinates
We study the conjecture that $\sum_{n\leq x} \chi(n)=o(x)$ for any primitive Dirichlet character $\chi \pmod q$ with $x\geq q^\epsilon$, which is known to be true if the Riemann Hypothesis holds for $L(s,\chi)$. We show that it holds under…
Hybrid character sums are an important class of exponential sums which have nice applications in coding theory and sequence design. Let $\gf_{p^m}$ be the finite field with $p^m$ elements for a prime $p$ and a positive integer $m$. Let…
We construct arithmetic terms representing the partial sums of binomial coefficients, and we extend these results to obtain arithmetic terms representing the multisections of binomial coefficient sums. We also introduce an arithmetic term…
Using elementary methods, we establish old and new relations between binomial coefficients, Fibonacci numbers, Lucas numbers, and more.
We obtain transformation formulas for quadratic character sums with quartic and cubic polynomial arguments.
We investigate the properties of the moments of the cot function using the central factorial numbers. Using a new integral representation of the central factorial numbers, we find a new way to express these moments in terms of recursive…
We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence $\{[\alpha n + \beta] : n =1,2,... \}$, where $\alpha,\beta\in\R$, and $\alpha$ is irrational. Our bounds are nontrivial over the same short…
Jack characters are a one-parameter deformation of the characters of the symmetric groups; a deformation given by the coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions. We study Jack…
In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on…
If two G-manifolds are G-cobordant then characteristic numbers corresponding to the fixed point sets (submanifolds) of subgroups of G and to normal bundles to these sets coincide. We construct two analogues of these characteristic numbers…
Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be…
Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated…
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…
This paper gives new explicit formulas for sums of powers of integers and their reciprocals.
We consider summations over digamma and polygamma functions, often with summands of the form (\pm 1)^n\psi(n+p/q)/n^r and (\pm 1)^n\psi^{(m)} (n+p/q)/n^r, where m, p, q, and r are positive integers. We develop novel general integral…
Permutations can be represented as linear combinations of natural numbers with different powers. In this paper, its coefficient matrix and inverse matrix is derived, and the results show the coefficient matrix is a lower triangular matrix…
We provide estimates for sums of the form \[\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\chi(a+b+c)\right|\] and \[\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\sum_{d\in D}\chi(a+b+cd)\right|\] when $A,B,C,D\subset \mathbb F_p$, the field…
Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular…
Recent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of…
The theory of supercharacters, recently developed by Diaconis-Isaacs and Andre, can be used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and…