The function $S_n (t) = π\left( \frac{3}{2} - {frac} \left( \frac{\vartheta(t)}π \right) + \left( \lfloor \frac{t \ln \left( \frac{t}{2 πe}\right)}{2 π} + \frac{7}{8} \rfloor - n \right) \right)$ is conjectured to be equal to $S (t_n)_{} = \arg ζ\left( \frac{1}{2} + i t_n \right)$ when $t=t_n$ is the imaginary part of the n-th zero of $ζ$ on the critical line. If $S(t_n)=S_n(t_n)$ then the exact transcendental equation for the Riemann zeros has a solution for each positive integer $n$ which proves that Riemann's hypothesis is true since the counting function for zeros on the critical line is equal to the counting function for zeros on the critical strip if the transcendental equation has a solution for each $n$.
A new definition for the Riemann zeta function for all positive integer number s > 1 is presented. We discover a most elegant expression and easy method for calculating the Riemann zeta function for small even integer values. Through this new reformulation we provide a one-line proof of the value of zeta(2) and demonstrate that zeta(2s) may be calculated by hand using only the cotangent function when the magnitude of the integer s is small.
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.
We extend the results of our previous computer experiment performed on the first 2600 nontrivial zeros $γ_l$ of the Riemann zeta function calculated with 1000 digits accuracy to the set of 40000 first zeros given with 40000 decimal digits accuracy. We calculated the geometrical means of the denominators of continued fractions expansions of these zeros and for all cases we get values very close to the Khinchin's constant, which suggests that $γ_l$ are irrational. Next we have calculated the $n$-th square roots of the denominators $Q_n$ of the convergents of the continued fractions obtaining val…
For Hurwitz zeta function, we obtain power series expression in second variable for its higher order derivatives (with respect to first variable) at non-positive integer arguments and consequently obtain rapidly decreasing series expression for Riemann zeta fuction at positive odd integer arguments. Further, we obtain corresponding results for Dirichlet L-series. We also a unified proof of various classical identities involving Riemann zeta values.
In this article, we consider the expressions for content-parametrized Schur multiple zeta-functions in terms of multiple zeta-functions of Euler-Zagier type and their star-variants, or in terms of modified zeta-functions of root systems. First of all, we focus on the Schur multiple zeta-function of hook type. And then, applying the Giambelli formula and induction argument, we obtain the expressions for general content-parametrized Schur multiple zeta-functions.
Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a derivative function. It is proved that these are equivalent, and that, if the Riemann hypothesis holds, then all zeros of the zeta function on the critical line are simple. The Riemann hypothesis is then shown to imply the third proposition holds, this being a new necessary condition for the Riemann hypothesis. The third proposition is shown to be equivalent to the…
We examine the behaviour of the zeros of the real and imaginary parts of $ξ(s)$ on the vertical line $\Re s = 1/2+λ$, for $λ\neq 0$. This can be rephrased in terms of studying the zeros of families of entire functions $A(s) = {1/2} (ξ(s+λ) + ξ(s - λ))$ and $B(s) = \frac{1}{2i} (ξ(s+λ) - ξ(s - λ))$. We will prove some unconditional analogues of results appearing in \cite{Lag}, specifically that the normalized spacings of the zeros of these functions converges to a limiting distribution consisting of equal spacings of length 1, in contrast to the expected GUE distribution for the same zeros at $…
In the present series of papers, we study the behavior of the r-fold zeta-function of Euler-Zagier type with identical arguments on the real line. In this first part, we consider the behavior on the interval [0,1]. Our basic tool is an "infinite" version of Newton's classical identities. We carry out numerical computations, and draw graphs for real s in [0,1], for several small values of r. Those graphs suggest various properties of the r-fold zeta-function, some of which we prove rigorously. For example, we show that the r-fold zeta-function has r asymptotes, and determine the asymptotic beha…
Explicit extreme values of the argument of the Riemann zeta-function
College of Industrial Technology, Nihon University（日本大学工业技术学部）；National Fisheries University（国立水产大学）；Global Education Center, Waseda University（早稻田大学全球教育中心）
A new zerofree region of the Riemann Zeta-function $ζ$ is identified by using Turán's localization criterion linking zeros of $ζ$ with uniform local suprema of sets of Dirichlet polynomials expanded over the primes. The proof is based on a randomization argument. An estimate for local extrema for some finite families of shifted Dirichlet polynomials, % associated to linearly independent sequences of reals is established by preliminary considering their local increment properties, by means of Montgomery-Vaughan's variant of Hilbert's inequality. A covering argument combined with Turán's localiz…
We make explicit an argument of Heath-Brown concerning large and small gaps between nontrivial zeroes of the Riemann zeta-function, $ζ(s)$. In particular, we provide the first unconditional results on gaps (large and small) which hold for a positive proportion of zeroes. To do this we prove explicit bounds on the second and fourth power moments of $S(t+h)-S(t)$, where $S(t)$ denotes the argument of $ζ(s)$ on the critical line and $h \ll 1 / \log T$. We also use these moments to prove explicit results on the density of the nontrivial zeroes of $ζ(s)$ of a given multiplicity.
For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive integral argument in first variable.On using functionalequations this results in instant evaluation of Riemann Zeta function at positive even integral values of its argument and of Dirichlet L series at positive integral values of its argument,when the argument and the corresponding Dirichlet character are both even or both odd.We also obtain finite sum expressio…
We study the behavior of $r$-fold zeta-functions of Euler-Zagier type with identical arguments $ζ_r(s,s,\ldots,s)$ on the real line. Our basic tool is an "infinite'' version of Newton's classical identities. We carry out numerical computations, and draw graphs of $ζ_r(s,s,\ldots,s)$ for real $s$, for several small values of $r$. Those graphs suggest various properties of $ζ_r(s,s,\ldots,s)$, some of which we prove rigorously. When $s \in [0,1]$, we show that $ζ_r(s,s,\ldots,s)$ has $r$ asymptotes at $\Re s=1/k$ ($1\leq k\leq r$), and determine the asymptotic behavior of $ζ_r(s,s,\ldots,s)$ clo…
We have used the first 2600 nontrivial zeros gamma_l of the Riemann zeta function calculated with 1000 digits accuracy and developed them into the continued fractions. We calculated the geometrical means of the denominators of these continued fractions and for all cases we get values close to the Khinchin's constant, what suggests that gamma_l are irrational. Next we have calculated the n-th square roots of the denominators q_n of the convergents of the continued fractions obtaining values close to the Khinchin-Levy constant, again supporting the common believe that gamma_l are irrational.
We investigate relations among Schur multiple zeta functions and zeta-functions of root systems attached to semisimple Lie algebras. Schur multiple zeta functions are defined as sums over semi-standard Young tableaux. Then, assuming the Young tableaux is of anti-hook shape, we show that they can be written in terms of modified zeta-functions of root systems of type $A$. Our proof is quite computational, but we also give a pictorial interpretation of our argument in terms of Young tableaux. It is also possible to understand that one of our theorems gives an expression of Schur multiple zeta fun…
In this article, we give, under the Riemann hypothesis, an upper bound for the exponential moments of the imaginary part of the logarithm of the Riemann zeta function on the critical line. Our result, which gives information on the fluctuations of the distribution of the zeros of $ζ$, has the same accuracy as the result obtained by Soundararajan in his paper entitled "Moments of the Riemann zeta function".
We present an infinite family of Borwein type $+ - - $ conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.
We solve problem x proposed by O. Oloa, AMM xxx 2012 {\bf 119?} (to appear), p. yyy for certain definite logarithmic integrals. A number of generating functions are developed with certain coefficients $p_n$, and some extensions are presented. The explicit relation of $p_n$ to Nörlund numbers $B_n^{(n)}$ is discussed. Certain inequalities are conjectured for the $\{p_n\}$ sequence of coefficients, including its convexity, and an upper bound is demonstrated. It is shown that $p_n$ values may be used to express the Stieltjes constants for the Hurwitz and Riemann zeta functions, as well as values …
We discuss a formal derivation of an integral expression for the Li coefficients associated with the Riemann xi-function which, in particular, indicates that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We conjecture the validity of this and related expressions without the need for the Riemann Hypothesis and discuss a physical interpretation of this result within the Hilbert-Polya approach. In this context we also outline a relation between string theory and the Riemann Hypothesis.
