Related papers: Euler's Function on Products of Primes in Progress…

The Landau-Selberg-Delange method gives precise asymptotic formulas for the partial sums $\sum_{n \le x} \, a_n$ of a Dirichlet series $\sum_n \, a_n/n^s$ that behaves like a complex power of the Riemann zeta function. However, situations…

We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (N\"{o}rlund polynomials), that give the analytic continuation of these functions to the entire complex plane. Special…

We generalize Ramanujan's expansions of the fractional-power Euler functions (q^{1/5})_{\infty} = [ J_1 - q^{1/5} + q^{2/5} J_2 ](q^5)_{\infty} and (q^{1/7})_{\infty} = [ J_1 + q^{1/7} J_2 - q^{2/7} + q^{5/7} J_3 ] (q^7)_{\infty} to…

Let $\tau(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ \Delta(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}\tau(n) q^n \,,q=e^{2\pi i z} \] (this is the unique holomorphic normalized…

We investigate the first moment of primes in progressions $$ \sum_{\substack{q\leq x/N \\ (q,a)=1}} \Big(\psi(x; q, a) - \frac x{\varphi(q)}\Big) $$ as $x, N \to \infty$. We show unconditionally that, when $a=1$, there is a significant bias…

Theorem 1 Let F:N-->R stand for any function which a) $F$ monotonically weakly increases; b) $F$ tends to infinity; and c) such that $q/F(q)$ tends to infinity. Let Z_F(q) equal the number of divisors of q less than sqrt{F(q)} minus the…

We consider the problem whether the ordinates of the non-trivial zeros of $\zeta(s)$ are uniformly distributed modulo the Gram points, or equivalently, if the normalized zeros $(x_n)$ are uniformly distributed modulo 1. Odlyzko conjectured…

For $i\in \{1,2,3\}$, let $E_i(x)$ denote the error term in each of the three theorems of Mertens on the asymptotic distribution of prime numbers. We show that for $i\in \{1,2\}$ the Riemann hypothesis is equivalent to the condition…

We introduce a Bernoulli operator,let $\mathbf{B}$ denote the operator symbol,for n=0,1,2,3,... let ${\mathbf{B}^n}: = {B_n}$ (where ${B_n}$ are Bernoulli numbers,${B_0} = 1,B{}_1 = 1/2,{B_2} = 1/6,{B_3} = 0$...).We obtain some formulas for…

A conjecture of Erd\H{o}s states that, for any large prime $q$, every reduced residue class $\pmod q$ can be represented as a product $p_1p_2$ of two primes $p_1,p_2\leq q$. We establish a ternary version of this conjecture, showing that,…

The Riemann Hypothesis is a conjecture that all non-trivial zeros of Riemann Zeta function are located on the critical line in the complex plane. Hundreds of propositions in function theory and analytic number theory rely on this…

Recently, we have established the generalized Li criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,a=Sum_(/rho)(1-(1-((/rho-a)/(/rho+a-1))^n) for any real a not…

The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product…

We give a simple Tauberian proof of the Prime Number Theorem using only elementary real analysis. Hence, the analytic continuation of Riemann's zeta function $\zeta$ and its non-vanishing value on the whole line $\{z\in {\mathbb…

In a recent paper Kachi and Tzermias give elementary proofs of four product formulas involving zeta(3), pi, and Catalan's constant. They indicate that they were not able to deduce these products directly from the values of a function…

Hilbert's Satz 90 tells us that for a given cyclic extension $K/k$, a unit of norm $1$ in $K$ can be written as a quotient of conjugate elements in $K$. For the extensions $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ with $p$ prime $> 3$, Newman proved…

We give a conjecture for the moments of the Dedekind zeta function of a Galois extension via the hybrid product method. The moments of the product of primes are evaluated using the Montgomery-Vaughan mean value theorem whilst for the…

The Euler-Mascheroni constant $\gamma=0.5772\dots\!$ is the $K=\mathbb{Q}$ example of an Euler-Kronecker constant $\gamma_K$ of a number field $K.$ In this note we consider the size of the $\gamma_q=\gamma_{K_q}$ for cyclotomic fields…

We prove that a good average order on the Goldbach generating function implies that the real parts of the non-trivial zeros of the Riemann zeta function are strictly less than 1. This together with existing results establishes an…

For any congruence subgroup of the modular group, we extend the region of convergence of the Euler products of the Selberg zeta functions beyond the boundary Re s = 1, if they are attached with a nontrivial irreducible unitary…