- Riemann zeta function
- Riemann zeta function - Wikipedia
- Techniques for Adding the Numbers 1 to 100
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The Riemann hypothesis would imply that this proportion is 1. The fact that. For sums involving the zeta-function at integer and half-integer values, see rational zeta series. There are a number of similar relations involving various well-known multiplicative functions ; these are given in the article on the Dirichlet series. The critical strip of the Riemann zeta function has the remarkable property of universality.

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This zeta-function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable. The first proof of universality was provided by Sergei Mikhailovitch Voronin in There are some theorems on properties of the function S t.

Among those results [19] [20] are the mean value theorems for S t and its first integral. Earlier similar results were obtained by Atle Selberg for the case. An extension of the area of convergence can be obtained by rearranging the original series. The Mellin transform of a function f x is defined as. There are various expressions for the zeta-function as Mellin transform-like integrals.

If the real part of s is greater than one, we have. By modifying the contour, Riemann showed that. We can also find expressions which relate to prime numbers and the prime number theorem. These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. The Riemann zeta function can be given by a Mellin transform [23].

However, this integral only converges if the real part of s is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all s except 0 and Abel—Plana formula. Another series development using the rising factorial valid for the entire complex plane is [ citation needed ]. On the basis of Weierstrass's factorization theorem , Hadamard gave the infinite product expansion. A simpler infinite product expansion is. To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.

Euler summation :. The series only appeared in an appendix to Hasse's paper, and did not become generally known until it was discussed by Jonathan Sondow in Peter Borwein has developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations. Here p n is the primorial sequence and J k is Jordan's totient function.

Hence this algorithm is essentially as fast as the Riemann-Siegel formula. Similar algorithms are possible for Dirichlet L-functions. The zeta function occurs in applied statistics see Zipf's law and Zipf—Mandelbrot law.

## Riemann zeta function

Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory. In one notable example, the Riemann zeta-function shows up explicitly in one method of calculating the Casimir effect. The zeta function is also useful for the analysis of dynamical systems. The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.

Other sums include. There are yet more formulas in the article Harmonic number. There are a number of related zeta functions that can be considered to be generalizations of the Riemann zeta function. These include the Hurwitz zeta function.

## Riemann zeta function - Wikipedia

For other related functions see the articles zeta function and L -function. The Lerch transcendent is given by. The multiple zeta functions are defined by. One can analytically continue these functions to the n -dimensional complex space. The rightmost digit is the "ones" column, and the one next to it is the tens column. Each digit we place to the left gets a value 10 times as great as the one to the right. In another base, the value with be n times as great, where n is the base. In other words, what we're looking at is a sequence of powers.

## Techniques for Adding the Numbers 1 to 100

With base 10, it's powers of With base 26, it'll be powers of Then thousands, and so on. Now, to find the value of a number, we multiply the digit in that spot by the power of ten which it corresponds to, and add up them up A nice way to think of it is to look inside a counter, like the odometer in a car or the counter on some tape recorders, with a wheel for each digit. Each wheel has ten digits, 0 through 9; when it turns past 9 back to 0, it turns the wheel to its left one place, meaning "we've just counted ten more; I've started over at zero, so please keep track of the number of tens for me.

Another way to think of it is that we count things by grouping them into stacks of ten, then stacks of ten into stacks of ten tens, and so on. When we write "" it means we have one stack of , two stacks of 10, and three single items. You can do the same with other bases besides ten Here it is, but I wouldn't call it a formula.

It is more like an algorithm. That means it is a method that always works, kind of like long division To convert a binary number to a decimal number you must first understand what each digit in the binary number means. To explain this let's look at the decimal number The '2' in represents two hundred because it is a two in the hundreds position two times a hundred is two hundred.

In similar fashion, the '4' in represents forty because it is a four in the tens position four times ten is forty. Finally, the '7' represents seven because it is a seven in the units position seven times one is seven. In a decimal number, the actual value represented by a digit in that number is determined by the numeral and the position of the numeral within the number.

It works the same way with a binary number Binary addition is the simplest of the binary operations, so let's start there. The next easiest operation is multiplication Actually, you subtract in binary pretty much the same way that you do in base You subtract digit by digit starting on the right side. If the subtraction cannot be made for example, you cannot subtract 1 from 0 , you must then "borrow", just as you do in base 10 subtraction.

But when you borrow a "one" from the 4's digit, it turns into two 2's. This borrowing by two's rather than 10's is what makes it quite different from base 10 subtraction Doctor Robert, Doctor Anthony Binary subtraction I need help understanding the rules of subtracting binary numbers when the subtrahend is larger then the minuend.

For example, I see from the answer provided in a textbook the that - is but I can't figure out how the leading -0 was obtained in the answer. Let's start by thinking about the more familiar base You'll find that the same problem exists there, but you probably don't stop to think about it because the solution is more familiar there. We can't subtract a larger number from a smaller one columnwise, because the sign gets mixed up. Instead, you reverse the order of the numbers, subtract, and take the negative.

Now, there are a couple of alternative ways to do this Doctor Peterson Binary Divisibility by 10 How you can tell whether a binary number of arbitrary size is divisible by 10 without looking at the whole number?

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I'll first show you the strictly binary method, since it can be instructive, then I'll show you the better way, which in fact is just like the decimal rule for divisibility by 3. You can use the same algorithm as long division in decimal, but the values will go in either one time or 0 times. Doctor TWE Binary to hexadecimal Is there a simple way to convert from binary base 2 numbers to hexadecimal base 16 numbers?

A big part of the reason that we use hexadecimal is that it is relatively easy to convert between binary and hexadecimal. Hexadecimal numbers are closely related to binary, but they are shorter and easier to read than binary. Group the binary digits into groups of 4 starting from the right Doctor Rick Changing number bases Could I have some information on hexadecimal and binary for my classes? We usually deal with base ten, which is just a way that's convenient for us to write down numbers. If there are numbers after the decimal point, you just continue the pattern: So in base 2 binary we use 2 digits, 0 and 1, and in base 16 hexadecimal we use 16 digits: 0,1,2,3,4,5,6,7,8,9,a,b,c,d,e, and f.

What about the binary complement? To complement a number in base 10, you subtract it from a row of 9's: the complement of is Likewise, in base 2, the complement of a number is obtained by subtraction from a row of 1's. Let's see if we can relate base two to something you can picture easily. You've probably seen an odometer in a car, or a tape counter in a cassette player, or things like that. They have a set of wheels, each of which has the ten digits 0, 1, Peterson Converting bases 1 How do you convert hexadecimal, binary, and decimal numbers? First, be sure you understand exactly what numbers mean in the old familiar decimal system.

Now there's nothing magic about 10 - it was chosen because we happen to have 10 fingers. There is a trick you can use, and after a little practice you won't have to write down the steps any more. Let's go from base 10 decimal to base 16 hex with the number decimal I want to be able to follow a formula Converting base 9 to base 10 is the same kind of process, only with more arithmetic Is there a simple formula? The easiest half is to convert from binary to decimal. Going the other way is slightly harder Please use the following numbers in an example: and We have a common denominator of 2 and I'll write this in a different colour here.

So we are going to have k times k plus 1 plus 2 times k plus 1. Now at this step right over here you can factor out a k plus 1. So let's factor this out. Let me colour code those. So you would know what I'm doing. So this 2 is this 2 right over there and this k is this k right over there. We factored it out. And it's going to be all of this over 2. Now, we can rewrite this. This is the same thing.

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This is equal to. This is the same thing as k plus 1, that's this part right over here. Times k plus 1 plus 1. All of that over 2. Why is this interesting to us? Well we have just proven it. So we showed , we proved our base case. This expression worked for the sum for all of positive integers up to and including 1. And it also works if we assume that it works for everything up to k.

Or if we assume it works for integer k it also works for the integer k plus 1. And we are done. That is our proof by induction.

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That proves to us that it works for all positive integers. Why is that? We have proven it for 1 and we have proven it that if it works for some integer it will work for the next integer. So if you assume it worked for 1 then it can work for 2. Well we have already proven that it works for 1 so we can assume it works for 1.