Related papers: Mertens' theorem and prime number theorem for Selb…
Two elementary formulae for Mertens function $M(n)$ are obtained. With these formulae, $M(n)$ can be calculated directly and simply, which can be easily implemented by computer. $M (1) \sim M (2 \times 10^7) $ are calculated one by one.…
We prove a general zero density theorem on the Selberg class of functions. The result verifies the Density Hypothesis in the strip when the real part of the variable is at least 0.9 under the assumption that the degree of the function does…
We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their powers. The formula is comprised of an infinite series of oscillatory terms, one for each zero of the zeta function on the…
We study the moments of the function that counts the number of representations of an integer as sums of two prime squares. We refine some of the previous arguments and apply the Selberg sieve to get an unconditional upper bound for all…
In this paper, we present a simple analytic proof of Siegel's theorem that concerns the lower bound of $L(1,\chi)$ for primitive quadratic $\chi$. Our new method compares an elementary lower bound with an analytic upper bound obtained by…
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…
Euler presents a third proof of the Fermat theorem, the one that lets us call it the Euler-Fermat theorem. This seems to be the proof that Euler likes best. He also proves that the smallest power x^n that, when divided by a numer N, prime…
This is an expanded version of the author's lecture at the XX Congresso U.M.I., held in Siena in September 2015. After a brief review of L-functions, we turn to the classical converse theorems of H.Hamburger, E.Hecke and A.Weil, and to some…
A new method for continuing the usual Dirichlet series that defines the Riemann zeta function ${\zeta}(s)$ is presented. Numerical experiments demonstrating the computational efficacy of the resulting continuation are discussed.
We represent the Euler alternating series (sometimes called the "Dirichlet eta function"), and generally $(b^s-b)\zeta(s)/b^s$ for $b>1$ an integer, in the half-plane $\Re s>0$, via series dominated by geometric series, with arbitrarily…
We provide explicit bounds in the theory of the Riemann zeta-function at the line $\Re{s}=1$, assuming that the Riemann hypothesis holds until the height $T$. In particular, we improve some bounds, in finite regions, for the logarithmic…
Many Dirichlet series of number theoretic interest can be written as a product of generating series $\zeta_{\,d,a}(s)=\prod\limits_{p\equiv a\pmod{d}}(1-p^{-s})^{-1}$, with $p$ ranging over all the primes in the primitive residue class…
We introduce a generic framework to provide bounds related to the pair correlation of sequences belonging to a wide class. We consider analogues of Montgomery's form factor for zeros of the Riemann zeta function in the case of arbitrary…
The Riemann hypothesis states that all nontrivial zeros of the zeta function lie on the critical line $\Re(s)=1/2$. Hilbert and P\'olya suggested a possible approach to prove it, based on spectral theory. Within this context, some authors…
In the present paper we prove lower bounds for L-functions from the Selberg class, by this means improving earlier results obtained by the second author together with J\"orn Steuding. We formulate two theorems which use slightly different…
Arzel\`a's bounded convergence theorem (1885) states that if a sequence of Riemann integrable functions on a closed interval is uniformly bounded and has an integrable pointwise limit, then the sequence of their integrals tends to the…
Beginning from the formal resolution of Riemann Zeta function, by using the formula of inner product between two infinite-dimensional vectors in the complex space, the author proved the world's baffling problem -- Riemann hypothesis raised…
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…
This analysis which uses new mathematical methods aims at proving the Riemann hypothesis and figuring out an approximate base for imaginary non-trivial zeros of zeta function at very large numbers, in order to determine the path that those…
We represent the Riemann zeta function in the half-plane $\Re s >1$ via series whose terms admit geometrically decreasing bounds. Due to an underlying recurrence relation, which is used to compute coefficients entering into the terms, the…