Related papers: On Weyl sums over primes in short intervals
This note offers an elementary proof of the Siegel-Walfisz theorem for primes in arithmetic progressions.
We give some results and conjectures about recurrence relations for certain sequences of binomial sums.
The main result is a generalization of Keller's recursion equation for finding a prime number given the previous primes. We also examine the convergence of the limit in Keller's equation and the convergence of the limit in the general…
We prove an analogue of the Prime Number Theorem for short intervals on a smooth projective geometrically irreducible curve of arbitrary genus over a finite field. A short interval "of size E" in this setting is any additive translate of…
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…
We consider the variance of sums of arithmetic functions over random short intervals in the function field setting. Based on the analogy between factorizations of random elements of $\mathbb{F}_q[T]$ into primes and the factorizations of…
Under the assumption of the Riemann Hypothesis (RH), we prove explicit quantitative relations between hypothetical error terms in the asymptotic formulae for truncated mean-square average of exponential sums over primes and in the…
The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.
The basis of this work is a simple, extended corollary of Wilson's theorem. This corollary generates many more quotients than those already generated by Wilson's theorem, and it was of interest to derive how they relate to each other and…
We consider polynomial equations, or systems of polynomial equations, with integer coefficients, modulo prime numbers $p$. We offer an elementary approach based on a counting method. The outcome is a weak form of the Lang-Weil lower bound…
I give some claims on primorial prime numbers for interested readers in number theory.
In this paper, we study the Jacobi sums over Galois rings of arbitrary characteristics and completely determine their absolute values, which extends the work in \cite{feng1}, where the Jacobi sums over Galois rings with characteristics a…
We estimate the sum of products or quotients of $L$-functions, where the sum is taken over all quadratic extensions of given genus over a fixed global function field. Our estimate for the sum of the quotient of two $L$-functions is…
In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.
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…
Based on new explicit estimates for the prime counting function, we improve the currently known estimates for the particular sequence $C_n = np_n - \sum_{k \leq n}p_k$, $n \geq 1$, involving the prime numbers.
An asymptotic formula for the sum of the first n primes is derived. This result improves the previous results of P. Dusart. Using this asymptotic expansion, we prove the conjectures of R. Mandl and G. Robin on the upper and the lower bound…
We prove $q$-variation estimates, $q>2$, on $\ell^{p}$ spaces for averages along primes (with $1<p<\infty$) and polynomials (with $\big| \frac1p - \frac12 \big| < \frac{1}{2(d+1)}$, where $d$ is the degree of the polynomial). This improves…
In recent years some near-optimal estimates have been established for certain sum-product type estimates. This paper gives some first extremal results which provide information about when these bounds may or may not be tight. The main tool…
An overview of the results of new exhaustive computations of gaps between primes in arithmetic progressions is presented. We also give new numerical results for exceptionally large least primes in arithmetic progressions.