What Google puts in writing
The page to send anybody who raises this is Google’s own explainer on what you need to know about online marketing. It asks the question then answers it in a single word, Nope, before stating that running online ads has no impact on where your website appears in organic search results. The same page notes that ads will never appear within the organic search results themselves, which is worth knowing when somebody shows you a screenshot as evidence.
Google Search Central makes the matching point about crawling, in its guide to how Search works. Google does not accept payment to crawl a site more frequently or rank it higher. The guide adds that if anyone tells you otherwise they are wrong. Two first party pages saying the same thing is about as settled as anything gets in this field.
Why the belief is so persistent
Because the two things do move together, for reasons that have nothing to do with the ranking system. A brand that starts spending on ads is usually a brand that has also tidied its site, written new landing pages, fixed its product data and started doing some PR. Organic improves over the following months. The ads were running throughout. The invoice gets the credit.
There is a reporting version of the same illusion. Run ads on your own brand name and some clicks that would have arrived through the organic listing arrive through the ad instead, which makes organic look weaker in the report while nothing about the ranking has changed. Pause the ads and those clicks come back. Neither movement is a ranking effect, so judge both together or you will keep mistaking one for the other.
The connection that is real
Paid search is the fastest market research in this industry. The search terms report shows the exact wording behind the clicks you paid for, which is the phrasing your content plan, your product titles and your page headings should be using. Google’s guidance on that report is to add anything not relevant enough to what you offer as a negative keyword. Those real phrases are the ones organic work otherwise spends months guessing at.
The traffic runs the other way too. Landing page experience is one of the three components Google names in its definition of Quality Score, described as how relevant and useful the page is to people who click the ad. So the site and content work you do for organic reasons tends to lower what you pay for a click. That is the honest commercial reason to run both at once. It is also a better argument than the one about rankings.