Related papers: Two-point correlation function for Dirichlet L-fun…
We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus…
We present a new way to factor the dirichlet convolution for completely multiplicative functions whitch led us to constructing a ring that arise from the operations involved in the factorisation. We will conclude by some identities that was…
We present many novel results in number theory, including a double series formula for the natural logarithm and a proof concerning the H\"older mean based on the functional equation for the Riemann zeta function. We find a harmonic mean…
A new reciprocity formula for Dirichlet $L$-functions associated to an arbitrary primitive Dirichlet character of prime modulus $q$ is established. We find an identity relating the fourth moment of individual Dirichlet $L$-functions in the…
The suggested approach is based on a known representation of Dirichlet $L$-functions via the incomplete gamma functions. Some properties of the Taylor coefficients of the lower incomplete gamma function at infinity seem to be new.…
Extending a result of Heath-Brown, we establish an asymptotic formula for the fourth moment of central values of Dirichlet $L$-functions attached to primitive characters $\pmod q$.
We apply our previous work on Green's functions for the four-dimensional quaternionic Taub-NUT manifold to obtain a scalar two-point function on the homogeneously squashed three-sphere (otherwise known as the Berger sphere), which lies at…
On the critical line the conditional distribution of the zeta function's magnitude around zeta zeros exists and predicts the well-known pair correlation between nontrivial zeta zeros. However, this conditional distribution does not exist at…
We consider the statistical distribution of zeros of random meromorphic functions whose poles are independent random variables. It is demonstrated that correlation functions of these zeros can be computed analytically and explicit…
We consider a sum of the derivatives of Dirichlet $L$-functions over the zeros of Dirichlet $L$-functions. We give an asymptotic formula for the sum.
We fully describe the horofunction boundary $\partial_h L_2$ with the word metric associated with the generating set $\{t,at\}$ (i.e the metric arising in the Diestel-Leader graph $\text{DL}(2,2)$). The visual boundary $\partial_\infty L_2$…
The computation of the correlation numbers in Minimal Liouville Gravity involves an integration over moduli spaces of complex curves. There are two independent approaches to the calculation: the direct one, based on the CFT methods and…
We explicitly evaluate a special type of multiple Dirichlet $L$-values at positive integers in two different ways: One approach involves using symmetric functions, while the other involves using a generating function of the values. Equating…
Let $\mathscr{H}^2$ denote the Hardy space of Dirichlet series $f(s) = \sum_{n\geq1} a_n n^{-s}$ with square summable coefficients and suppose that $\varphi$ is a symbol generating a composition operator on $\mathscr{H}^2$ by…
We give an explicit upper bound for non-principal Dirichlet $L$-functions on the line $s=1+it$. This result can be applied to improve the error in the zero-counting formulae for these functions.
We devise heuristics using multiple Dirichlet series to predict asymptotic formulas for shifted moments of (1) the family of Dirichlet $L$-functions of all even primitive characters of conductor $\leq Q$, with $Q$ a parameter tending to…
We compute the two-point correlation function for spin configurations which are obtained by solving the Euclidean matching problem, for one family of points on a grid, and the second family chosen uniformly at random, when the cost depends…
In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins the general study of…
The aim of this paper is to improve the upper bound for the exceptional zeroes $\beta_0$ of Dirichlet $L$-functions. We do this by improving on explicit estimate for $L'(\sigma, \chi)$ for $\sigma$ close to unity.
We study two dimensional conformal field theories in the semiclassical limit. In this limit, the four-point function is dominated by intermediate primaries of particular weights along with their descendants, and the crossing equations…