Several Sequence and New Conjectures of Zeros of Riemann Zeta Functions

In this survey, we will focus on some results related to the explicit location of zeros of the Riemann zeta function. We have few techniques to know the exact behavior of zeros of zeta functions. For the author's study 011 zeros of zeta functions, the Riemann-Siegel formula. We have collected several examples whose structures look like the Riemaim-Siegel formula and which satisfy the analogue of the Riemann hypothesis. The Riemann-Siegel formula can be simplified by wheresatisfies certain nice conditions. Due to the symmetry, zeros of this formula tend to lie onThus, our strategy to the Riemann hypothesis (RH) is to find a nice representation of the Riemann zeta function satisfying the above formula: if we are able to prove that complex zeros ofare in or inthen we essentially derive RH. Forwe have whereis the number of zeros of the Riemann zeta function in. Thus, the average gap of consecutive zeros of the Riemann zeta function inis What can we say about gaps of zeros of the Riemann zeta function? This question is one of the most important questions in studying the behavior of zeros of the Riemann zeta function. Together with the Riemann-Siegel formula. Montgomery's pair Riemann zeta function.

Since the series does not converge forit it is difficult to picture, on the face of it, what relationship, if any, would exist between the zeros of the truncated zeta-function andin this half plane. The following plot, made possible through the application of the zero finder, illustrates a few points. What is first noticeable is a string of zeros ofnear the critical line implies thatis roughly approximated bvnear the critical line for, but t also large enough so thatis 'small'. The strip of zeros is in accordance with Spira's observation. Aboveabout 1326, in Figure 1, the zeros scatter more wildly. Fig. 1

Single parameter curves. Figure 1 shows the positions of the first '2000 zeros of in the upper half plane. Although the zeros are not recurring in a completely regular pattern, their positioning does appear to have a semi periodic nature. The algorithm used to locate these zeros used a homotopy. In its more general form, for, starting from the known position of the zeros of the end terms Fig.2

zeros along the pat as t increased from 0 to 1. That this method worked so well in locating all of the zeros ofup to heights tested, suggests that the error estimate in above could be improved to. In any case, in such finite exponential sums, the largest integer in the expansion, N, is an indicator or the number of zeros to be found up to a height T in much the same way as the degree of a polynomial determines its number of zeros in the complex plane.

Let us recall that zeta vanishes at negative odd integers. These zeros are called the trivial zeros of. The functional equation

(1)

entails that other zeros of(they are called the non-trivial zeros) are symmetric with respect to the critical line: for each non trivial zero the valueit is also a zero of

We show that all the non-trivial zeros oflie in the critical strip defined by values of the complex number s such thatBecause of the functional equation, it suffices to show that does not vanish on the closed half plane

thatdoes not vanish for(a convergent infinite product cannot converge to zero because its logarithm is a convergent series). Thus it suffices now to prove that does not vanish on the line This property is in fact the key in the proof of the prime number theorem, and Hadamard and obtained this result independently in 1896 by different mean (this problem is in fact a first step in a determination of a zero-free region, important to obtain good error terms in the prime number theorem). We present here the argument ofPoussin which Is simpler to expose and more elegant.

line The starting point is the relation

(2)

for all values of the real numberThe Euler infinite product writes as thus for the complex numberwe have This relation entails so with (2) we deduce

(3)

Now suppose thatIs a zero ofLettingwe have thatthus as entails thattends to infinity, which is impossible sinceis analytic aroundThus we have proved thathas no zeros on the line Other proofs of this result can be found in [3].

The present thesis deals with the Riemann zeta- functionand it is intended ad an exposure of some of the most relevant results obtained until today. The zeta-function plays a fundamental role in number theory and one of the most important open question in mathematics is the Riemann Hypothesis (RH), which states that all non-trivial zeros of have real part equal toThe zeta-function appeared for the first time in 1859 on a Riemann's paper originally devoted to the explicit formula connecting the prime counting function,with the logarithmic integralnevertheless, the paper contained other outstanding results, like the analytical continuation ofthrough the whole complex plane, and deep conjectures, each of them involving the non-trivial zeros ofAll these conjectures, thanks to von Mangoldt and Hadamard, afterwards became theorem except, as said before, the conjecture about the displacement of non-trivial zeros along the critical

[1] I. Hacking. Representing arul Intervening. Cambridge University Press. 1983. ISBN 0521282462. [2] Yu. Matiyasevich. An artless method for calculating approximate values of zeros of Riemann's zeta function. WWW site of ongoing research, http://logic.pdmi.ras.ru/~yumat/ personaljournal/artlessmethod/artlessmethod.php. [3] WiebBosma. Jolrn Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. J- Symbolic Comput., 24(3- [4] J. B. Conrey. More than two fifths of the zeros of the Riemann zeta function arc on the critical line. J. Heine Angcw. Math., 399:1-26. 1989, [5] .J. B. Conrey, "More than two fifths of the zeros of the Riemaim zeta function are on the critical line". ./. reine angew. Math.. 399 (1989), pp. 1-26. [6] H. M. Edwards, Riemann's Zeta Function, Academic Press, 1974. [7] E. C. Titchmarsh. The theory of the Riemann Zeta-function. Oxford Science publications, second edition, revised by D. R. Heath-Brown (1986).