In this article, we derive an expression for the complex magnitude of the Dirichlet beta function $β(s)$ represented as a Euler prime product and compare with similar results for the Riemann zeta function. We also obtain formulas for $β(s)$ valid for an even and odd $k$th positive integer argument and present a set of generated formulas for $β(k)$ up to $11$th order, including Catalan's constant and compute these formulas numerically. Additionally, we derive a second expression for the complex magnitude of $β(s)$ valid in the critical strip from which we obtain a formula for the Euler-Mascheroni constant expressed in terms of zeros of the Dirichlet beta function on the critical line. Finally, we investigate the asymptotic behavior of the Euler prime product on the critical line.
We give closed-form expressions for the Dirichlet beta function at even positive integers and for the Dirichlet lambda function at odd positive integers, based on the function J(s) defined via convergent integral. We also show fundamental relations between Dirichlet lambda and beta functions and the function J(s).
A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values, provides a general method for obtaining all special constants associated with Dirichlet beta function. We also show various new and fundamental relations between the polygamma function, Riemann zeta, the even-indexed euler numbers, the Dirichlet beta functions in a way never seen or imagined before.
small In this paper, we define $q$-analogues of Dirichlet's beta function at positive integers, which can be written as $β_q(s)=\sum_{k\geq1}\sum_{d|k}χ(k/d)d^{s-1}q^k$ for $s\in\N^*$, where $q$ is a complex number such that $|q|<1$ and $χ$ is the non trivial Dirichlet character modulo 4. For odd $s$, these expressions are connected with the automorphic world, in particular with Eisenstein series of level 4. From this, we derive through Nesterenko's work the transcendance of the numbers $β_q(2s+1)$ for $q$ algebraic such that $0<|q|<1$. Our main result concerns the nature of the numbers $β_q(2…
In this paper, we study the alternating Euler $T$-sums and $§$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $§$-sums, from which, the parity theorems of Hoffman's double and triple $t$-values and Kaneko-Tsumura's double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $§$-sums are expressible in…
We prove that more than nine percent of the central values $L(\frac{1}{2},χ_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges over primes and $χ_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain the order of magnitude of the second moment of $L(\frac{1}{2},χ_p)$, and conditionally we obtain the order of magnitude of the third moment. Assuming the Generalized Riemann Hypothesis, we show that our lower bound for the second moment is asymptotically sharp.
We prove a lower bound of the correct order of magnitude in the conductor aspect for small rational moments of Drichlet $L$-functions. Such bounds require new techniques, which is visible from the relationship to non-vanishing results for $L(1/2, χ)$.
In this paper, we establish the expected order of magnitude of the $k$th-moment of quadratic Dirichlet $L$-functions associated to hyperelliptic curves of genus $g$ as well as of prime moduli over a fixed finite field $\mathbb{F}_{q}$ for all real $k \geq 0$.
We prove an asymptotic formula for the fourth power mean of Dirichlet L-functions averaged over primitive characters to modulus q and over t\in [0,T] which is particularly effective when q \ge T. In this range the correct order of magnitude was not previously known.
We study the second moment of Dirichlet $L$-functions to a large prime modulus $q$ twisted by the square of an arbitrary Dirichlet polynomial. We break the $\frac{1}{2}$-barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than $q^{51/101} = q^{1/2 +1/202}$. As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet $L$-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.
Improving earlier work of Balasubramanian, Conrey and Heath-Brown, we obtain an asymptotic formula for the mean-square of the Riemann zeta-function times an arbitrary Dirichlet polynomial of length $T^{1/2 + δ}$, with $δ= 0.01515....$ As an application we obtain an upper bound of the correct order of magnitude for the third moment of the Riemann zeta-function. We also refine previous work of Deshouillers and Iwaniec, obtaining asymptotic estimates in place of bounds. Using the work of Watt, we compute the mean-square of the Riemann zeta-function times a Dirichlet polynomial of length going up …
We continue our examination the effects of certain hypothetical configurations of zeros of Dirichlet $L$-functions lying off the critical line ("barriers") on the relative magnitude of the functions $π_{q,a}(x)$. Here $π_{q,a}(x)$ is the number of primes $\le x$ in the progression $a \mod q$. In particular, we construct barriers so that $π_{q,1}(x)$ is simultaneously greater than, or simultaneously less than, each of $k$ functions $π_{q,a_i}(x)$ ($1\le i\le k$). We also construct barriers so that only a small number of the $r!$ possible orderings of functions $π_{q,a_i}(x)$ ($1\le i\le r$) occ…
Euler discovered a formula for expressing the value of the Riemann zeta function for all even positive integer arguments. A closed-form expression for the Riemann zeta function for all odd integer arguments, based on the values of the Dirichlet beta function, euler numbers and pi, reveals a new evidence about the self-recursive nature of Riemann zeta function at odd integers. We demonstrate for the first time that the Riemann zeta function at odd integers always produces a recurrence relation that is self-recursive.
