Related papers: Lower order terms in the full moment conjecture fo…

We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.

We describe a numerical algorithm for evaluating the numbers of roots minus the number of poles contained in a region based on the argument principle with the function of interest being written as a Mellin transformation of a usually…

We propose a conjecture for the exact expression of the dynamical zeta function for a family of birational transformations of two variables, depending on two parameters. This conjectured function is a simple rational expression with integer…

Additive divisor sums play a prominent role in the theory of the moments of the Riemann zeta function. There is a long history of determining sharp asymptotic formula for the shifted convolution sum of the ordinary divisor function. In…

Conrey, Ghosh and Gonek studied the first moment of the derivative of the Riemann zeta function evaluated at the non-trivial zeros of the zeta function, resolving a problem known as Shanks' conjecture. Conrey and Ghosh studied the second…

We establish sharp upper bounds for the $2k$th moment of the Riemann zeta function on the critical line, for all real $0 \leqslant k \leqslant 2$. This improves on earlier work of Ramachandra, Heath-Brown and Bettin-Chandee-Radziwi\l\l

Given good knowledge on the even moments, we derive asymptotic formulas for $\lambda$-th moments of primes in short intervals and prove "equivalence" result on odd moments. We also provide numerical evidence in support of these results.

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) := E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x)…

Assuming the Riemann hypothesis, we obtain asymptotic formulas for $\sum_{0<\gamma<T}\zeta(\rho+\delta)\zeta(1-\rho+\overline{\delta})$ in the region $-\frac{a}{\log T} \leq \Re \delta \leq \frac{1}{2}+\frac{a}{\log T}$, $|\Im \delta|\ll…

We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.

The individual terms of the series representing the Riemann zeta function are examined geometrically from their accumulated plot in the complex plane. Symmetry is identified and determined mathematically for comparison with more traditional…

We give lower bounds for the small moments of the sum of a random multiplicative function, which improve on some results of Bondarenko and Seip and constitute further progress towards (dis)proving a conjecture of Helson. We also prove…

In this short note, we obtain error estimates for Riemann sums of some singular functions.

Discrete moments of the Riemann zeta function were studied by Gonek and Hejhal in the 1980's. They independently formulated a conjecture concerning the size of these moments. In 1999, Hughes, Keating, and O'Connell, by employing a random…

We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure…

A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.

We have looked at the evaluation of the Riemann Zeta function at odd arguments and have provided a simple formula to approximate the value with exponential convergence. We have compared it with various other formulae present in literature.…

We prove an equivalent of the Riemann hypothesis in terms of the functional equation (in its asymmetrical form) and the $a$-points of the zeta-function, i.e., the roots of the equation $\zeta(s)=a$, where $a$ is an arbitrary fixed complex…

For functions belonging to the classes $C^{2}[0, 1]$ and $C^{3}[0, 1]$, we establish the lower estimate with an explicit constant in approximation by Bernstein polynomials in terms of the second order Ditzian-Totik modulus of smoothness.…

Assuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet $L$-functions. We use this to obtain conditional upper bounds on high moments of theta functions. Both of these results strengthen…